helm/software/helena/examples/automath/grundlagen_0.aut

   1 # Landau's "Grundlagen der Analysis", formal specification in AUTOMATH
   2 # Copyright (C) 1977, L.S. van Benthem Jutting
   3 #               1992, revised by F. Wiedijk (http://www.cs.ru.nl/~freek/aut/)
   4
   5 +l
   6 @[a:'prop'][b:'prop']
   7 imp:=[x:a]b:'prop'
   8 [a1:a][i:imp(a,b)]
   9 mp:=<a1>i:b
  10 a@refimp:=[x:a]x:imp(a,a)
  11 b@[c:'prop'][i:imp(a,b)][j:imp(b,c)]
  12 trimp:=[x:a]<<x>i>j:imp(a,c)
  13 @con:='prim':'prop'
  14 a@not:=imp(con):'prop'
  15 wel:=not(not(a)):'prop'
  16 [a1:a]
  17 weli:=[x:not(a)]<a1>x:wel(a)
  18 a@[w:wel(a)]
  19 et:='prim':a
  20 a@[c1:con]
  21 cone:=et([x:not(a)]c1):a
  22 +imp
  23 b@[i:imp(a,b)][j:imp(not(a),b)]
  24 th1:=et(b,[x:not(b)]<<trimp(con,i,x)>j>x):b
  25 b@[n:not(a)]
  26 th2:=trimp(con,b,n,[x:con]cone(b,x)):imp(a,b)
  27 b@[n:not(b)][i:imp(a,b)]
  28 th3:=trimp(con,i,n):not(a)
  29 b@[a1:a][n:not(b)]
  30 th4:=[x:imp(a,b)]<a1>th3(n,x):not(imp(a,b))
  31 b@[n:not(imp(a,b))]
  32 th5:=et([x:not(a)]<th2(x)>n):a
  33 th6:=[x:b]<[y:a]x>n:not(b)
  34 b@[n:not(b)][i:imp(not(a),b)]
  35 th7:=et(a,th3(not(a),b,n,i)):a
  36 -imp
  37 b@[i:imp(not(b),not(a))]
  38 cp:=[x:a]th7".imp"(b,not(a),weli(x),i):imp(a,b)
  39 @obvious:=imp(con,con):'prop'
  40 obviousi:=refimp(con):obvious
  41 b@ec:=imp(a,not(b)):'prop'
  42 [n:not(a)]
  43 eci1:=th2".imp"(not(b),n):ec(a,b)
  44 b@[n:not(b)]
  45 eci2:=[x:a]n:ec(a,b)
  46 +ec
  47 b@[i:imp(a,not(b))]
  48 th1:=i:ec(a,b)
  49 b@[i:imp(b,not(a))]
  50 th2:=[x:a][y:b]<x><y>i:ec(a,b)
  51 -ec
  52 b@[e:ec(a,b)]
  53 comec:=th2".ec"(b,a,e):ec(b,a)
  54 [a1:a]
  55 ece1:=<a1>e:not(b)
  56 e@[b1:b]
  57 ece2:=th3".imp"(not(b),weli(b,b1),e):not(a)
  58 +*ec
  59 c@[e:ec(a,b)][i:imp(c,a)]
  60 th3:=trimp(c,a,not(b),i,e):ec(c,b)
  61 e@[i:imp(c,b)]
  62 th4:=comec(c,a,th3(b,a,c,comec(e),i)):ec(a,c)
  63 -ec
  64 b@and:=not(ec(a,b)):'prop'
  65 [a1:a][b1:b]
  66 andi:=th4".imp"(not(b),a1,weli(b,b1)):and(a,b)
  67 b@[a1:and(a,b)]
  68 ande1:=th5".imp"(not(b),a1):a
  69 ande2:=et(b,th6".imp"(not(b),a1)):b
  70 comand:=andi(b,a,ande2,ande1):and(b,a)
  71 +and
  72 b@[n:not(a)]
  73 th1:=weli(ec,eci1(n)):not(and)
  74 b@[n:not(b)]
  75 th2:=weli(ec,eci2(n)):not(and)
  76 b@[n:not(and)][a1:a]
  77 th3:=ece1(et(ec,n),a1):not(b)
  78 n@[b1:b]
  79 th4:=ece2(et(ec,n),b1):not(a)
  80 n@th5:=th3"l.imp"(and(b,a),and(a,b),n,[x:and(b,a)]comand(b,a,x)):not(and(b,a))
  81 c@[a1:and(a,b)][i:imp(a,c)]
  82 th6:=andi(c,b,<ande1(a1)>i,ande2(a1)):and(c,b)
  83 a1@[i:imp(b,c)]
  84 th7:=andi(a,c,ande1(a1),<ande2(a1)>i):and(a,c)
  85 -and
  86 b@or:=imp(not(a),b):'prop'
  87 [a1:a]
  88 ori1:=th2".imp"(not(a),b,weli(a1)):or(a,b)
  89 b@[b1:b]
  90 ori2:=[x:not(a)]b1:or(a,b)
  91 +or
  92 b@[i:imp(not(a),b)]
  93 th1:=i:or(a,b)
  94 b@[i:imp(not(b),a)]
  95 th2:=[x:not]et(b,th3"l.imp"(not(b),a,x,i)):or(a,b)
  96 -or
  97 b@[o:or(a,b)][n:not(a)]
  98 ore2:=<n>o:b
  99 o@[n:not(b)]
 100 ore1:=et(th3".imp"(not(a),b,n,o)):a
 101 o@comor:=[x:not(b)]ore1(x):or(b,a)
 102 +*or
 103 b@[n:not(a)][m:not(b)]
 104 th3:=th4"l.imp"(not(a),b,n,m):not(or(a,b))
 105 b@[n:not(or(a,b))]
 106 th4:=th5"l.imp"(not(a),b,n):not(a)
 107 th5:=th6"l.imp"(not(a),b,n):not(b)
 108 a@th6:=refimp(not(a)):or(a,not(a))
 109 -or
 110 c@[o:or(a,b)][i:imp(a,c)][j:imp(b,c)]
 111 orapp:=th1".imp"(c,i,trimp(not,b,c,o,j)):c
 112 c@[d:'prop']
 113 +*or
 114 o@[i:imp(a,c)]
 115 th7:=trimp(not(c),not,b,[x:not(c)]th3"l.imp"(a,c,x,i),o):or(c,b)
 116 o@[i:imp(b,c)]
 117 th8:=trimp(not(a),b,c,o,i):or(a,c)
 118 d@[o:or(a,b)][i:imp(a,c)][j:imp(b,d)]
 119 th9:=th7(a,d,c,th8(a,b,d,o,j),i):or(c,d)
 120 b@[o:or(a,b)]
 121 th10:=o:imp(not(a),b)
 122 th11:=comor(o):imp(not(b),a)
 123 b@[o:or(not(a),b)]
 124 th12:=trimp(a,wel(a),b,[x:a]weli(x),o):imp(a,b)
 125 b@[i:imp(a,b)]
 126 th13:=trimp(wel(a),a,b,[x:wel(a)]et(x),i):or(not(a),b)
 127 b@[o:or(not(a),not(b))]
 128 th14:=weli(ec,th12(not(b),o)):not(and)
 129 b@[n:not(and)]
 130 th15:=th13(not(b),et(ec,n)):or(not(a),not(b))
 131 b@[a1:and(not(a),not(b))]
 132 th16:=th3(ande1(not(a),not(b),a1),ande2(not(a),not(b),a1)):not(or(a,b))
 133 b@[n:not(or(a,b))]
 134 th17:=andi(not(a),not(b),th4(n),th5(n)):and(not(a),not(b))
 135 -or
 136 b@orec:=and(or(a,b),ec(a,b)):'prop'
 137 [o:or(a,b)][e:ec(a,b)]
 138 oreci:=andi(or(a,b),ec(a,b),o,e):orec(a,b)
 139 +orec
 140 b@[a1:a][n:not(b)]
 141 th1:=oreci(ori1(a1),eci2(n)):orec(a,b)
 142 b@[n:not(a)][b1:b]
 143 th2:=oreci(ori2(b1),eci1(n)):orec(a,b)
 144 -orec
 145 b@[o:orec(a,b)]
 146 orece1:=ande1(or(a,b),ec,o):or(a,b)
 147 orece2:=ande2(or(a,b),ec,o):ec(a,b)
 148 comorec:=oreci(b,a,comor(orece1),comec(orece2)):orec(b,a)
 149 +*orec
 150 o@[a1:a]
 151 th3:=ece1(orece2,a1):not(b)
 152 o@[b1:b]
 153 th4:=ece2(orece2,b1):not(a)
 154 o@[n:not(a)]
 155 th5:=ore2(orece1,n):b
 156 o@[n:not(b)]
 157 th6:=ore1(orece1,n):a
 158 -orec
 159 b@iff:=and(imp(a,b),imp(b,a)):'prop'
 160 [i:imp(a,b)][j:imp(b,a)]
 161 iffi:=andi(imp(a,b),imp(b,a),i,j):iff(a,b)
 162 +iff
 163 b@[a1:a][b1:b]
 164 th1:=iffi([x:a]b1,[x:b]a1):iff(a,b)
 165 b@[n:not(a)][m:not(b)]
 166 th2:=iffi(th2"l.imp"(n),th2"l.imp"(b,a,m)):iff(a,b)
 167 -iff
 168 b@[i:iff(a,b)]
 169 iffe1:=ande1(imp(a,b),imp(b,a),i):imp(a,b)
 170 iffe2:=ande2(imp(a,b),imp(b,a),i):imp(b,a)
 171 comiff:=iffi(b,a,iffe2,iffe1):iff(b,a)
 172 +*iff
 173 i@[a1:a]
 174 th3:=<a1>iffe1:b
 175 i@[b1:b]
 176 th4:=<b1>iffe2:a
 177 i@[n:not(a)]
 178 th5:=th3"l.imp"(b,a,n,iffe2):not(b)
 179 i@[n:not(b)]
 180 th6:=th3"l.imp"(n,iffe1):not(a)
 181 b@[a1:a][n:not(b)]
 182 th7:=th1"l.and"(imp(a,b),imp(b,a),th4"l.imp"(a1,n)):not(iff(a,b))
 183 b@[n:not(a)][b1:b]
 184 th8:=th2"l.and"(imp(a,b),imp(b,a),th4"l.imp"(b,a,b1,n)):not(iff(a,b))
 185 -iff
 186 a@refiff:=iffi(a,refimp,refimp):iff(a,a)
 187 b@[i:iff(a,b)]
 188 symiff:=comiff(i):iff(b,a)
 189 c@[i:iff(a,b)][j:iff(b,c)]
 190 triff:=iffi(a,c,trimp(iffe1(i),iffe1(b,c,j)),trimp(c,b,a,iffe2(b,c,j),iffe2(i))):iff(a,c)
 191 +*iff
 192 b@[i:iff(a,b)]
 193 th9:=[x:not(a)]th5(i,x):imp(not(a),not(b))
 194 th10:=[x:not(b)]th6(i,x):imp(not(b),not(a))
 195 th11:=iffi(not(a),not(b),th9,th10):iff(not(a),not(b))
 196 b@[i:imp(not(a),not(b))][j:imp(not(b),not(a))]
 197 th12:=iffi(cp(j),cp(b,a,i)):iff(a,b)
 198 b@[o:orec(a,b)]
 199 th13:=iffi(not(b),orece2(o),comor(orece1(o))):iff(a,not(b))
 200 th14:=th13(b,a,comorec(o)):iff(b,not(a))
 201 b@[i:iff(a,not(b))]
 202 th15:=oreci(comor(b,a,iffe2(not(b),i)),iffe1(not(b),i)):orec(a,b)
 203 b@[i:iff(b,not(a))]
 204 th16:=comorec(b,a,th15(b,a,i)):orec(a,b)
 205 c@[i:iff(a,b)][j:imp(a,c)]
 206 thimp1:=trimp(b,a,c,iffe2(i),j):imp(b,c)
 207 i@[j:imp(c,a)]
 208 thimp2:=trimp(c,a,b,j,iffe1(i)):imp(c,b)
 209 i@[e:ec(a,c)]
 210 thec1:=th3"l.ec"(c,b,e,iffe2(i)):ec(b,c)
 211 i@[e:ec(c,a)]
 212 thec2:=th4"l.ec"(c,a,b,e,iffe2(i)):ec(c,b)
 213 i@[a1:and(a,c)]
 214 thand1:=th6"l.and"(c,b,a1,iffe1(i)):and(b,c)
 215 i@[a1:and(c,a)]
 216 thand2:=th7"l.and"(c,a,b,a1,iffe1(i)):and(c,b)
 217 i@[o:or(a,c)]
 218 thor1:=th7"l.or"(c,b,o,iffe1(i)):or(b,c)
 219 i@[o:or(c,a)]
 220 thor2:=th8"l.or"(c,a,b,o,iffe1(i)):or(c,b)
 221 i@[o:orec(a,c)]
 222 thorec1:=oreci(b,c,thor1(orece1(a,c,o)),thec1(orece2(a,c,o))):orec(b,c)
 223 i@[o:orec(c,a)]
 224 thorec2:=oreci(c,b,thor2(orece1(c,a,o)),thec2(orece2(c,a,o))):orec(c,b)
 225 -iff
 226 @[sigma:'type'][p:[x:sigma]'prop']
 227 all:=p:'prop'
 228 [a1:all(sigma,p)][s:sigma]
 229 alle:=<s>a1:<s>p
 230 +all
 231 p@[s:sigma][n:not(<s>p)]
 232 th1:=[x:all(sigma,p)]<<s>x>n:not(all(sigma,p))
 233 -all
 234 p@non:=[x:sigma]not(<x>p):[x:sigma]'prop'
 235 some:=not(non(p)):'prop'
 236 [s:sigma][sp:<s>p]
 237 somei:=th1".all"(non(p),s,weli(<s>p,sp)):some(sigma,p)
 238 +some
 239 p@[n:not(all(sigma,p))][m:non(non(p))][s:sigma]
 240 t1:=et(<s>p,<s>m):<s>p
 241 %set etared
 242 m@t2:=<[x:sigma]t1(x)>n:con
 243 %reset etared
 244 n@th1:=[x:non(non(p))]t2(x):some(non(p))
 245 p@[s:some(non(p))][a1:all(sigma,p)][t:sigma]
 246 t3:=weli(<t>p,<t>a1):not(not(<t>p))
 247 a1@t4:=<[x:sigma]t3(x)>s:con
 248 s@th2:=[x:all(sigma,p)]t4(x):not(all(sigma,p))
 249 p@[n:not(some(sigma,p))]
 250 th3:=et(non(p),n):non(p)
 251 [s:sigma]
 252 th4:=<s>th3:not(<s>p)
 253 p@[n:non(p)]
 254 th5:=weli(non(p),n):not(some(sigma,p))
 255 -some
 256 p@[s:some(sigma,p)][x:'prop'][i:[y:sigma]imp(<y>p,x)]
 257 +*some
 258 i@[n:not(x)][t:sigma]
 259 t5:=th3"l.imp"(<t>p,x,n,<t>i):not(<t>p)
 260 n@t6:=mp(some(sigma,p),con,s,th5([y:sigma]t5(y))):con
 261 -some
 262 i@someapp:=et(x,[y:not(x)]t6".some"(y)):x
 263 +*some
 264 p@[q:[x:sigma]'prop'][s:some(sigma,p)][i:[x:sigma]imp(<x>p,<x>q)]
 265 th6:=someapp(s,some(q),[x:sigma][y:<x>p]somei(q,x,mp(<x>p,<x>q,y,<x>i))):some(q)
 266 -some
 267 c@or3:=or(a,or(b,c)):'prop'
 268 [o:or3(a,b,c)][n:not(a)]
 269 +or3
 270 th1:=ore2(or(b,c),o,n):or(b,c)
 271 -or3
 272 [m:not(b)]
 273 or3e3:=ore2(b,c,th1".or3",m):c
 274 o@[n:not(b)]
 275 +*or3
 276 n@th2:=th2"l.or"(c,a,[x:not(a)]or3e3(x,n)):or(c,a)
 277 -or3
 278 n@[m:not(c)]
 279 or3e1:=ore2(c,a,th2".or3",m):a
 280 o@[n:not(c)]
 281 +*or3
 282 n@th3:=th2"l.or"([x:not(b)]or3e1(x,n)):or(a,b)
 283 -or3
 284 n@[m:not(a)]
 285 or3e2:=ore2(th3".or3",m):b
 286 +*or3
 287 o@th4:=th1"l.or"(b,or(c,a),[x:not(b)]th2(x)):or3(b,c,a)
 288 th5:=th4(b,c,a,th4):or3(c,a,b)
 289 -or3
 290 c@[a1:a]
 291 or3i1:=ori1(a,or(b,c),a1):or3(a,b,c)
 292 c@[b1:b]
 293 or3i2:=ori2(a,or(b,c),ori1(b,c,b1)):or3(a,b,c)
 294 c@[c1:c]
 295 or3i3:=ori2(a,or(b,c),ori2(b,c,c1)):or3(a,b,c)
 296 +*or3
 297 c@[o:or(a,b)]
 298 th6:=th4"or3"(c,a,b,ori2(c,or(a,b),o)):or3(a,b,c)
 299 c@[o:or(b,c)]
 300 th7:=ori2(or(b,c),o):or3(a,b,c)
 301 c@[o:or(c,a)]
 302 th8:=th4"or3"(c,a,b,th6(c,a,b,o)):or3(a,b,c)
 303 -or3
 304 d@[o:or3(a,b,c)][i:imp(a,d)][j:imp(b,d)][k:imp(c,d)]
 305 or3app:=orapp(or(b,c),d,o,i,[x:or(b,c)]orapp(b,c,d,x,j,k)):d
 306 c@and3:=and(a,and(b,c)):'prop'
 307 [a1:and3(a,b,c)]
 308 and3e1:=ande1(and(b,c),a1):a
 309 and3e2:=ande1(b,c,ande2(and(b,c),a1)):b
 310 and3e3:=ande2(b,c,ande2(and(b,c),a1)):c
 311 c@[a1:a][b1:b][c1:c]
 312 and3i:=andi(a,and(b,c),a1,andi(b,c,b1,c1)):and3(a,b,c)
 313 +and3
 314 c@[a1:and3(a,b,c)]
 315 th1:=and3i(b,c,a,and3e2(a1),and3e3(a1),and3e1(a1)):and3(b,c,a)
 316 th2:=th1(b,c,a,th1):and3(c,a,b)
 317 th3:=andi(and3e1(a1),and3e2(a1)):and(a,b)
 318 th4:=ande2(and(b,c),a1):and(b,c)
 319 th5:=th3(c,a,b,th2):and(c,a)
 320 th6:=and3i(c,b,a,and3e3(a1),and3e2(a1),and3e1(a1)):and3(c,b,a)
 321 -and3
 322 c@ec3:=and3(ec,ec(b,c),ec(c,a)):'prop'
 323 [e:ec3(a,b,c)]
 324 +ec3
 325 th1:=and3e1(ec,ec(b,c),ec(c,a),e):ec(a,b)
 326 th2:=and3e2(ec,ec(b,c),ec(c,a),e):ec(b,c)
 327 th3:=and3e3(ec,ec(b,c),ec(c,a),e):ec(c,a)
 328 th4:=th1"l.and3"(ec,ec(b,c),ec(c,a),e):ec3(b,c,a)
 329 th5:=th4(b,c,a,th4):ec3(c,a,b)
 330 th5a:=and3i(ec(c,b),ec(b,a),ec(a,c),comec(b,c,th2(e)),comec(a,b,th1(e)),comec(c,a,th3(e))):ec3(c,b,a)
 331 -ec3
 332 [a1:a]
 333 ec3e12:=ece1(th1".ec3",a1):not(b)
 334 ec3e13:=ece2(c,a,th3".ec3",a1):not(c)
 335 e@[b1:b]
 336 ec3e23:=ec3e12(b,c,a,th4".ec3",b1):not(c)
 337 ec3e21:=ec3e13(b,c,a,th4".ec3",b1):not(a)
 338 e@[c1:c]
 339 ec3e31:=ec3e12(c,a,b,th5".ec3",c1):not(a)
 340 ec3e32:=ec3e13(c,a,b,th5".ec3",c1):not(b)
 341 +*ec3
 342 c@[e:ec(a,b)][f:ec(b,c)][g:ec(c,a)]
 343 th6:=and3i(ec,ec(b,c),ec(c,a),e,f,g):ec3(a,b,c)
 344 c@[e:ec3(a,b,c)][o:or(a,b)]
 345 th7:=orapp(not(c),o,[x:a]ece2(c,a,th3"ec3"(e),x),[x:b]ece1(b,c,th2"ec3"(e),x)):not(c)
 346 e@[o:or(b,c)]
 347 th8:=th7(b,c,a,th4"ec3"(e),o):not(a)
 348 e@[o:or(c,a)]
 349 th9:=th7(c,a,b,th5"ec3"(e),o):not(b)
 350 -ec3
 351 c@[n:not(a)][m:not(b)]
 352 ec3i1:=th6".ec3"(eci1(n),eci1(b,c,m),eci2(c,a,n)):ec3(a,b,c)
 353 c@[n:not(b)][m:not(c)]
 354 ec3i2:=th6".ec3"(eci2(n),eci1(b,c,n),eci1(c,a,m)):ec3(a,b,c)
 355 c@[n:not(c)][m:not(a)]
 356 ec3i3:=th6".ec3"(eci1(m),eci2(b,c,n),eci1(c,a,n)):ec3(a,b,c)
 357 +*ec3
 358 d@[e:'prop'][f:'prop'][o1:or3(a,b,c)][p1:ec3(d,e,f)][i:imp(a,d)][j:imp(b,e)][k:imp(c,f)][d1:d]
 359 t1:=[x:b]mp(e,con,<x>j,ec3e12(d,e,f,p1,d1)):not(b)
 360 t2:=[x:c]mp(f,con,<x>k,ec3e13(d,e,f,p1,d1)):not(c)
 361 th10:=or3e1(o1,t1,t2):a
 362 k@[e1:e]
 363 th11:=th10(b,c,a,e,f,d,th4"l.or3"(o1),th4"ec3"(d,e,f,p1),j,k,i,e1):b
 364 k@[f1:f]
 365 th12:=th10(c,a,b,f,d,e,th5"l.or3"(o1),th5"ec3"(d,e,f,p1),k,i,j,f1):c
 366 -ec3
 367 c@orec3:=and(or3(a,b,c),ec3(a,b,c)):'prop'
 368 [o:orec3(a,b,c)]
 369 orec3e1:=ande1(or3(a,b,c),ec3(a,b,c),o):or3(a,b,c)
 370 orec3e2:=ande2(or3(a,b,c),ec3(a,b,c),o):ec3(a,b,c)
 371 c@[o:or3(a,b,c)][e:ec3(a,b,c)]
 372 orec3i:=andi(or3(a,b,c),ec3(a,b,c),o,e):orec3(a,b,c)
 373 +orec3
 374 c@[o:orec3(a,b,c)]
 375 th1:=orec3i(b,c,a,th4"l.or3"(orec3e1(o)),th4"l.ec3"(orec3e2(o))):orec3(b,c,a)
 376 th2:=orec3i(c,a,b,th5"l.or3"(orec3e1(o)),th5"l.ec3"(orec3e2(o))):orec3(c,a,b)
 377 -orec3
 378 +e
 379 sigma@[s:sigma][t:sigma]
 380 is:='prim':'prop'
 381 s@refis:='prim':is(s,s)
 382 p@[s:sigma][t:sigma][sp:<s>p][i:is(s,t)]
 383 isp:='prim':<t>p
 384 sigma@[s:sigma][t:sigma][i:is(s,t)]
 385 symis:=isp([x:sigma]is(x,s),s,t,refis(s),i):is(t,s)
 386 t@[u:sigma][i:is(s,t)][j:is(t,u)]
 387 tris:=isp([x:sigma]is(x,u),t,s,j,symis(i)):is(s,u)
 388 u@[i:is(u,s)][j:is(u,t)]
 389 tris1:=tris(s,u,t,symis(u,s,i),j):is(s,t)
 390 u@[i:is(s,u)][j:is(t,u)]
 391 tris2:=tris(s,u,t,i,symis(t,u,j)):is(s,t)
 392 sp@[i:is(t,s)]
 393 isp1:=isp(symis(t,s,i)):<t>p
 394 t@[n:not(is(s,t))]
 395 symnotis:=th3"l.imp"(is(t,s),is(s,t),n,[x:is(t,s)]symis(t,s,x)):not(is(t,s))
 396 +notis
 397 u@[n:not(is(s,t))][i:is(s,u)]
 398 th1:=isp([x:sigma]not(is(x,t)),s,u,n,i):not(is(u,t))
 399 n@[i:is(u,s)]
 400 th2:=th1(symis(u,s,i)):not(is(u,t))
 401 n@[i:is(t,u)]
 402 th3:=isp([x:sigma]not(is(s,x)),t,u,n,i):not(is(s,u))
 403 n@[i:is(u,t)]
 404 th4:=th3(symis(u,t,i)):not(is(s,u))
 405 u@[v:sigma][n:not(is(s,t))][i:is(s,u)][j:is(t,v)]
 406 th5:=th3(u,t,v,th1(n,i),j):not(is(u,v))
 407 -notis
 408 u@[v:sigma][i:is(s,t)][j:is(t,u)][k:is(u,v)]
 409 tr3is:=tris(s,u,v,tris(i,j),k):is(s,v)
 410 v@[w:sigma][i:is(s,t)][j:is(t,u)][k:is(u,v)][l:is(v,w)]
 411 tr4is:=tris(s,v,w,tr3is(i,j,k),l):is(s,w)
 412 p@amone:=[x:sigma][y:sigma][u:<x>p][v:<y>p]is(x,y):'prop'
 413 [a1:amone(sigma,p)][s:sigma][t:sigma][sp:<s>p][tp:<t>p]
 414 amonee:=<tp><sp><t><s>a1:is(s,t)
 415 p@one:=and(amone(sigma,p),some(sigma,p)):'prop'
 416 [a1:amone(sigma,p)][s:some(sigma,p)]
 417 onei:=andi(amone(sigma,p),some(sigma,p),a1,s):one(sigma,p)
 418 p@[o1:one(sigma,p)]
 419 onee1:=ande1(amone(sigma,p),some(sigma,p),o1):amone(sigma,p)
 420 onee2:=ande2(amone(sigma,p),some(sigma,p),o1):some(sigma,p)
 421 ind:='prim':sigma
 422 oneax:='prim':<ind>p
 423 +one
 424 [s:sigma][sp:<s>p]
 425 th1:=amonee(onee1,ind,s,oneax,sp):is(ind,s)
 426 -one
 427 sigma@[tau:'type'][f:[x:sigma]tau][s:sigma][t:sigma][i:is(s,t)]
 428 isf:=isp(sigma,[x:sigma]is(tau,<s>f,<x>f),s,t,refis(tau,<s>f),i):is(tau,<s>f,<t>f)
 429 f@injective:=all([x:sigma]all([y:sigma]imp(is(tau,<x>f,<y>f),is(x,y)))):'prop'
 430 [i:injective(f)][s:sigma][t:sigma][j:is(tau,<s>f,<t>f)]
 431 isfe:=<j><t><s>i:is(s,t)
 432 f@[t0:tau]
 433 image:=some([x:sigma]is(tau,t0,<x>f)):'prop'
 434 f@[s:sigma]
 435 tofs:=<s>f:tau
 436 imagei:=somei([x:sigma]is(tau,tofs,<x>f),s,refis(tau,tofs)):image(tofs)
 437 +inj
 438 i@[ta:tau][tb:tau][j:is(tau,ta,tb)][sa:sigma][sb:sigma][ja:is(tau,ta,tofs(sa))][jb:is(tau,tb,tofs(sb))]
 439 t1:=tr3is(tau,tofs(sa),ta,tb,tofs(sb),symis(tau,ta,tofs(sa),ja),j,jb):is(tau,tofs(sa),tofs(sb))
 440 th1:=isfe(sa,sb,t1):is(sa,sb)
 441 i@[t0:tau]
 442 th2:=[x:sigma][y:sigma][u:is(tau,t0,<x>f)][v:is(tau,t0,<y>f)]th1(t0,t0,refis(tau,t0),x,y,u,v):amone([x:sigma]is(tau,t0,<x>f))
 443 [j:image(f,t0)]
 444 th3:=onei([x:sigma]is(tau,t0,<x>f),th2,j):one([x:sigma]is(tau,t0,<x>f))
 445 -inj
 446 i@[t0:tau][j:image(f,t0)]
 447 soft:=ind([x:sigma]is(tau,t0,<x>f),th3".inj"(t0,j)):sigma
 448 i@inverse:=[x:tau][u:image(f,x)]soft(x,u):[x:tau][u:image(f,x)]sigma
 449 j@ists1:=oneax([x:sigma]is(tau,t0,<x>f),th3".inj"(t0,j)):is(tau,t0,tofs(soft(t0,j)))
 450 ists2:=symis(tau,t0,tofs(soft(t0,j)),ists1):is(tau,tofs(soft(t0,j)),t0)
 451 i@[ta:tau][ja:image(ta)][tb:tau][jb:image(tb)][j:is(tau,ta,tb)]
 452 isinv:=th1".inj"(ta,tb,j,soft(ta,ja),soft(tb,jb),ists1(ta,ja),ists1(tb,jb)):is(soft(ta,ja),soft(tb,jb))
 453 jb@[j:is(soft(ta,ja),soft(tb,jb))]
 454 isinve:=tr3is(tau,ta,tofs(soft(ta,ja)),tofs(soft(tb,jb)),tb,ists1(ta,ja),isf(soft(ta,ja),soft(tb,jb),j),ists2(tb,jb)):is(tau,ta,tb)
 455 i@[s:sigma]
 456 isst1:=isfe(s,soft(tofs(s),imagei(s)),ists1(tofs(s),imagei(s))):is(s,soft(tofs(s),imagei(s)))
 457 isst2:=symis(s,soft(tofs(s),imagei(s)),isst1):is(soft(tofs(s),imagei(s)),s)
 458 f@surjective:=all(tau,[x:tau]image(x)):'prop'
 459 bijective:=and(injective,surjective):'prop'
 460 [b:bijective(f)]
 461 +*inj
 462 b@t2:=ande1(injective,surjective,b):injective(f)
 463 t3:=ande2(injective,surjective,b):surjective(f)
 464 [t:tau]
 465 so:=soft(t2,t,<t>t3):sigma
 466 -inj
 467 b@invf:=[x:tau]so".inj"(x):[x:tau]sigma
 468 [s:sigma]
 469 thinvf1:=tris(s,soft(t2".inj",tofs(s),imagei(s)),<<s>f>invf,isst1(t2".inj",s),isinv(t2".inj",tofs(s),imagei(s),tofs(s),<tofs(s)>t3".inj",refis(tau,tofs(s)))):is(s,<<s>f>invf)
 470 b@[t:tau]
 471 thinvf2:=ists1(t2".inj",t,<t>t3".inj"):is(tau,t,<<t>invf>f)
 472 tau@[upsilon:'type'][f:[x:sigma]tau][g:[x:tau]upsilon]
 473 +*inj
 474 g@[if:injective(sigma,tau,f)][ig:injective(tau,upsilon,g)]
 475 h:=[x:sigma]<<x>f>g:[x:sigma]upsilon
 476 [s:sigma][t:sigma][i:is(upsilon,<s>h,<t>h)]
 477 t4:=isfe(tau,upsilon,g,ig,<s>f,<t>f,i):is(tau,<s>f,<t>f)
 478 t5:=isfe(f,if,s,t,t4):is(s,t)
 479 ig@th4:=[x:sigma][y:sigma][v:is(upsilon,<x>h,<y>h)]t5(x,y,v):injective(sigma,upsilon,[x:sigma]<<x>f>g)
 480 -inj
 481 +surj
 482 g@[sf:surjective(sigma,tau,f)][sg:surjective(tau,upsilon,g)]
 483 h:=[x:sigma]<<x>f>g:[x:sigma]upsilon
 484 [u:upsilon]
 485 t1:=<u>sg:image(tau,upsilon,g,u)
 486 [t:tau][i:is(upsilon,u,<t>g)]
 487 t2:=<t>sf:image(sigma,tau,f,t)
 488 [s:sigma][j:is(tau,t,<s>f)]
 489 t3:=tris(upsilon,u,<t>g,<s>h,i,isf(tau,upsilon,g,t,<s>f,j)):is(upsilon,u,<s>h)
 490 t4:=somei([x:sigma]is(upsilon,u,<x>h),s,t3):image(sigma,upsilon,h,u)
 491 i@t5:=someapp([x:sigma]is(tau,t,<x>f),t2,image(sigma,upsilon,h,u),[x:sigma][v:is(tau,t,<x>f)]t4(x,v)):image(sigma,upsilon,h,u)
 492 u@t6:=someapp(tau,[x:tau]is(upsilon,u,<x>g),t1,image(sigma,upsilon,h,u),[x:tau][v:is(upsilon,u,<x>g)]t5(x,v)):image(sigma,upsilon,h,u)
 493 sg@th1:=[x:upsilon]t6(x):surjective(sigma,upsilon,[x:sigma]<<x>f>g)
 494 -surj
 495 +bij
 496 g@[bf:bijective(sigma,tau,f)][bg:bijective(tau,upsilon,g)]
 497 h:=[x:sigma]<<x>f>g:[x:sigma]upsilon
 498 t1:=th4"e.inj"(f,g,ande1(injective(f),surjective(f),bf),ande1(injective(tau,upsilon,g),surjective(tau,upsilon,g),bg)):injective(sigma,upsilon,h)
 499 t2:=th1"e.surj"(f,g,ande2(injective(f),surjective(f),bf),ande2(injective(tau,upsilon,g),surjective(tau,upsilon,g),bg)):surjective(sigma,upsilon,h)
 500 th1:=andi(injective(sigma,upsilon,h),surjective(sigma,upsilon,h),t1,t2):bijective(sigma,upsilon,[x:sigma]<<x>f>g)
 501 -bij
 502 tau@[f:[x:sigma]tau][g:[x:sigma]tau][i:is([x:sigma]tau,f,g)][s:sigma]
 503 fise:=isp([x:sigma]tau,[y:[x:sigma]tau]is(tau,<s>f,<s>y),f,g,refis(tau,<s>f),i):is(tau,<s>f,<s>g)
 504 g@[i:[x:sigma]is(tau,<x>f,<x>g)]
 505 fisi:='prim':is([x:sigma]tau,f,g)
 506 +fis
 507 g@[i:is([x:sigma]tau,f,g)][s:sigma][t:sigma][j:is(s,t)]
 508 th1:=tris(tau,<s>f,<t>f,<t>g,isf(f,s,t,j),fise(i,t)):is(tau,<s>f,<t>g)
 509 -fis
 510 p@ot:='prim':'type'
 511 [o1:ot]
 512 in:='prim':sigma
 513 inp:='prim':<in>p
 514 p@otax1:='prim':injective(ot,sigma,[x:ot]in(x))
 515 [s:sigma][sp:<s>p]
 516 otax2:='prim':image(ot,sigma,[x:ot]in(x),s)
 517 o1@[o2:ot][i:is(ot,o1,o2)]
 518 isini:=isf(ot,sigma,[x:ot]in(x),o1,o2,i):is(in(o1),in(o2))
 519 o2@[i:is(in(o1),in(o2))]
 520 isine:=isfe(ot,sigma,[x:ot]in(x),otax1,o1,o2,i):is(ot,o1,o2)
 521 sp@out:=soft(ot,sigma,[x:ot]in(x),otax1,s,otax2):ot
 522 [t:sigma][tp:<t>p][i:is(s,t)]
 523 isouti:=isinv(ot,sigma,[x:ot]in(x),otax1,s,otax2,t,otax2(t,tp),i):is(ot,out(s,sp),out(t,tp))
 524 tp@[i:is(ot,out(s,sp),out(t,tp))]
 525 isoute:=isinve(ot,sigma,[x:ot]in(x),otax1,s,otax2,t,otax2(t,tp),i):is(s,t)
 526 o1@isoutin:=tris(ot,o1,soft(ot,sigma,[x:ot]in(x),otax1,in(o1),imagei(ot,sigma,[x:ot]in(x),o1)),out(in(o1),inp(o1)),isst1(ot,sigma,[x:ot]in(x),otax1,o1),isinv(ot,sigma,[x:ot]in(x),otax1,in(o1),imagei(ot,sigma,[x:ot]in(x),o1),in(o1),otax2(in(o1),inp(o1)),refis(sigma,in(o1)))):is(ot,o1,out(in(o1),inp(o1)))
 527 sp@isinout:=ists1(ot,sigma,[x:ot]in(x),otax1,s,otax2):is(s,in(out(s,sp)))
 528 tau@pairtype:='prim':'type'
 529 [s:sigma][t:tau]
 530 pair:='prim':pairtype
 531 tau@[p1:pairtype]
 532 first:='prim':sigma
 533 second:='prim':tau
 534 pairis1:='prim':is(pairtype,pair(first,second),p1)
 535 pairis2:=symis(pairtype,pair(first,second),p1,pairis1):is(pairtype,p1,pair(first,second))
 536 t@firstis1:='prim':is(sigma,first(pair),s)
 537 firstis2:=symis(sigma,first(pair),s,firstis1):is(sigma,s,first(pair))
 538 secondis1:='prim':is(tau,second(pair),t)
 539 secondis2:=symis(tau,second(pair),t,secondis1):is(tau,t,second(pair))
 540 a@[ksi:'type'][x:ksi][y:ksi]
 541 +ite
 542 [z:ksi]
 543 prop1:=and(imp(a,is(ksi,z,x)),imp(not(a),is(ksi,z,y))):'prop'
 544 y@[a1:a][x1:ksi][y1:ksi][px1:prop1(x1)][py1:prop1(y1)]
 545 t1:=ande1(imp(a,is(ksi,x1,x)),imp(not(a),is(ksi,x1,y)),px1):imp(a,is(ksi,x1,x))
 546 t2:=mp(a,is(ksi,x1,x),a1,t1):is(ksi,x1,x)
 547 t3:=t2(y1,x1,py1,px1):is(ksi,y1,x)
 548 t4:=tris2(ksi,x1,y1,x,t2,t3):is(ksi,x1,y1)
 549 a1@t5:=[s:ksi][t:ksi][ps:prop1(s)][pt:prop1(t)]t4(s,t,ps,pt):amone(ksi,[t:ksi]prop1(t))
 550 t6:=[x1:a]refis(ksi,x):imp(a,is(ksi,x,x))
 551 t7:=th2"l.imp"(not(a),is(ksi,x,y),weli(a,a1)):imp(not(a),is(ksi,x,y))
 552 t8:=andi(imp(a,is(ksi,x,x)),imp(not(a),is(ksi,x,y)),t6,t7):prop1(x)
 553 t9:=somei(ksi,[t:ksi]prop1(t),x,t8):some(ksi,[t:ksi]prop1(t))
 554 t10:=onei(ksi,[t:ksi]prop1(t),t5,t9):one(ksi,[t:ksi]prop1(t))
 555 y@[n:not(a)][x1:ksi][y1:ksi][px1:prop1(x1)][py1:prop1(y1)]
 556 t11:=ande2(imp(a,is(ksi,x1,x)),imp(not(a),is(ksi,x1,y)),px1):imp(not(a),is(ksi,x1,y))
 557 t12:=mp(not(a),is(ksi,x1,y),n,t11):is(ksi,x1,y)
 558 t13:=t12(y1,x1,py1,px1):is(ksi,y1,y)
 559 t14:=tris2(ksi,x1,y1,y,t12,t13):is(ksi,x1,y1)
 560 n@t15:=[s:ksi][t:ksi][ps:prop1(s)][pt:prop1(t)]t14(s,t,ps,pt):amone(ksi,[t:ksi]prop1(t))
 561 t16:=[x1:not(a)]refis(ksi,y):imp(not(a),is(ksi,y,y))
 562 t17:=th2"l.imp"(a,is(ksi,y,x),n):imp(a,is(ksi,y,x))
 563 t18:=andi(imp(a,is(ksi,y,x)),imp(not(a),is(ksi,y,y)),t17,t16):prop1(y)
 564 t19:=somei(ksi,[t:ksi]prop1(t),y,t18):some(ksi,[t:ksi]prop1(t))
 565 t20:=onei(ksi,[t:ksi]prop1(t),t15,t19):one(ksi,[t:ksi]prop1(t))
 566 y@t21:=th1"l.imp"(a,one(ksi,[t:ksi]prop1(t)),[t:a]t10(t),[t:not(a)]t20(t)):one(ksi,[t:ksi]prop1(t))
 567 -ite
 568 ite:=ind(ksi,[t:ksi]prop1".ite"(t),t21".ite"):ksi
 569 +*ite
 570 y@t22:=oneax(ksi,[t:ksi]prop1(t),t21):prop1(ite)
 571 t23:=ande1(imp(a,is(ksi,ite,x)),imp(not(a),is(ksi,ite,y)),t22):imp(a,is(ksi,ite,x))
 572 t24:=ande2(imp(a,is(ksi,ite,x)),imp(not(a),is(ksi,ite,y)),t22):imp(not(a),is(ksi,ite,y))
 573 -ite
 574 y@[a1:a]
 575 itet:=mp(a,is(ksi,ite,x),a1,t23".ite"):is(ksi,ite,x)
 576 y@[n:not(a)]
 577 itef:=mp(not(a),is(ksi,ite,y),n,t24".ite"):is(ksi,ite,y)
 578 sigma@[s0:sigma][t0:sigma]
 579 +wissel
 580 [s:sigma]
 581 wa:=ite(is(s,s0),sigma,t0,s):sigma
 582 [i:is(s,s0)]
 583 t1:=itet(is(s,s0),sigma,t0,s,i):is(wa,t0)
 584 s@[n:not(is(s,s0))]
 585 t2:=itef(is(s,s0),sigma,t0,s,n):is(wa,s)
 586 s@wb:=ite(is(s,t0),sigma,s0,wa):sigma
 587 [i:is(s,t0)]
 588 t3:=itet(is(s,t0),sigma,s0,wa,i):is(wb,s0)
 589 s@[n:not(is(s,t0))]
 590 t4:=itef(is(s,t0),sigma,s0,wa,n):is(wb,wa)
 591 s@[i:is(s,s0)][j:is(s0,t0)]
 592 t5:=tris(wb,s0,t0,t3(tris(s,s0,t0,i,j)),j):is(wb,t0)
 593 i@[n:not(is(s0,t0))]
 594 t6:=tris(wb,wa,t0,t4(th2"e.notis"(s0,t0,s,n,i)),t1(i)):is(wb,t0)
 595 i@t7:=th1"l.imp"(is(s0,t0),is(wb,t0),[t:is(s0,t0)]t5(t),[t:not(is(s0,t0))]t6(t)):is(wb,t0)
 596 s@[n:not(is(s,s0))][o:not(is(s,t0))]
 597 t8:=tris(wb,wa,s,t4(o),t2(n)):is(wb,s)
 598 -wissel
 599 wissel:=[x:sigma]wb".wissel"(x):[x:sigma]sigma
 600 [s:sigma][i:is(s,s0)]
 601 iswissel1:=t7".wissel"(s,i):is(<s>wissel,t0)
 602 s@[i:is(s,t0)]
 603 iswissel2:=t3".wissel"(s,i):is(<s>wissel,s0)
 604 s@[n:not(is(s,s0))][o:not(is(s,t0))]
 605 iswissel3:=t8".wissel"(s,n,o):is(<s>wissel,s)
 606 +*wissel
 607 s@[t:sigma][i:is(wb(s),wb(t))][n:not(is(s,t))][j:is(s,s0)]
 608 t9:=symnotis(s0,t,th1"e.notis"(s,t,s0,n,j)):not(is(t,s0))
 609 [k:is(s0,t0)]
 610 t10:=th3"e.notis"(t,s0,t0,t9,k):not(is(t,t0))
 611 t11:=tris(wb(s),wb(t),t,i,t8(t,t9,t10)):is(wb(s),t)
 612 t12:=<tris1(t,t0,wb(s),t11,t7(j))>t10:con
 613 j@t13:=[v:is(s0,t0)]t12(v):not(is(s0,t0))
 614 [k:is(t,t0)]
 615 t14:=tris(wb(s),wb(t),s0,i,t3(t,k)):is(wb(s),s0)
 616 t15:=t12(tris1(s0,t0,wb(s),t14,t7(j))):con
 617 j@t16:=[v:is(t,t0)]t15(v):not(is(t,t0))
 618 t17:=tris(wb(s),wb(t),t,i,t8(t,t9,t16)):is(wb(s),t)
 619 t18:=t15(tris1(t,t0,wb(s),t17,t7(j))):con
 620 n@t19:=[v:is(s,s0)]t18(v):not(is(s,s0))
 621 t20:=t19(t,s,symis(wb(s),wb(t),i),symnotis(s,t,n)):not(is(t,s0))
 622 [j:is(s,t0)]
 623 t21:=symnotis(t0,t,th1"e.notis"(s,t,t0,n,j)):not(is(t,t0))
 624 t22:=tris(wb(s),wb(t),t,i,t8(t,t20,t21)):is(wb(s),t)
 625 t23:=<tris1(t,s0,wb(s),t22,t3(j))>t20:con
 626 n@t24:=[v:is(s,t0)]t23(v):not(is(s,t0))
 627 t25:=t24(t,s,symis(wb(s),wb(t),i),symnotis(s,t,n)):not(is(t,t0))
 628 t26:=tris(wb(s),wb(t),t,i,t8(t,t20,t25)):is(wb(s),t)
 629 t27:=<tris1(s,t,wb(s),t8(t19,t24),t26)>n:con
 630 i@t28:=et(is(s,t),[v:not(is(s,t))]t27(v)):is(s,t)
 631 t0@th1:=[x:sigma][y:sigma][v:is(wb(x),wb(y))]t28(x,y,v):injective(sigma,sigma,wissel)
 632 s@[i:is(s,s0)]
 633 t29:=tris2(s,wb(t0),s0,i,t3(t0,refis(t0))):is(s,wb(t0))
 634 t30:=somei(sigma,[x:sigma]is(s,wb(x)),t0,t29):image(sigma,sigma,wissel,s)
 635 s@[i:is(s,t0)]
 636 t31:=tris2(s,wb(s0),t0,i,t7(s0,refis(s0))):is(s,wb(s0))
 637 t32:=somei(sigma,[x:sigma]is(s,wb(x)),s0,t31):image(sigma,sigma,wissel,s)
 638 s@[n:not(is(s,s0))][o:not(is(s,t0))]
 639 t33:=symis(wb(s),s,t8(n,o)):is(s,wb(s))
 640 t34:=somei(sigma,[x:sigma]is(s,wb(x)),s,t33):image(sigma,sigma,wissel,s)
 641 n@t35:=th1"l.imp"(is(s,t0),image(sigma,sigma,wissel,s),[v:is(s,t0)]t32(v),[v:not(is(s,t0))]t34(v)):image(sigma,sigma,wissel,s)
 642 s@t36:=th1"l.imp"(is(s,s0),image(sigma,sigma,wissel,s),[v:is(s,s0)]t30(v),[v:not(is(s,s0))]t35(v)):image(sigma,sigma,wissel,s)
 643 t0@th2:=[x:sigma]t36(x):surjective(sigma,sigma,wissel)
 644 th3:=andi(injective(sigma,sigma,wissel),surjective(sigma,sigma,wissel),th1,th2):bijective(sigma,sigma,wissel)
 645 -wissel
 646 tau@[f:[x:sigma]tau][s0:sigma][t0:sigma]
 647 changef:=[x:sigma]<<x>wissel(s0,t0)>f:[x:sigma]tau
 648 [s:sigma][i:is(s,s0)]
 649 changef1:=isf(sigma,tau,f,<s>wissel(s0,t0),t0,iswissel1(s0,t0,s,i)):is(tau,<s>changef,<t0>f)
 650 s@[i:is(s,t0)]
 651 changef2:=isf(sigma,tau,f,<s>wissel(s0,t0),s0,iswissel2(s0,t0,s,i)):is(tau,<s>changef,<s0>f)
 652 s@[n:not(is(s,s0))][o:not(is(s,t0))]
 653 changef3:=isf(sigma,tau,f,<s>wissel(s0,t0),s,iswissel3(s0,t0,s,n,o)):is(tau,<s>changef,<s>f)
 654 +*wissel
 655 t0@[i:injective(f)]
 656 th4:=th4"e.inj"(sigma,sigma,tau,wissel(s0,t0),f,th1(s0,t0),i):injective(changef)
 657 t0@[s:surjective(f)]
 658 th5:=th1"e.surj"(sigma,sigma,tau,wissel(s0,t0),f,th2(s0,t0),s):surjective(changef)
 659 t0@[b:bijective(f)]
 660 th6:=th1"e.bij"(sigma,sigma,tau,wissel(s0,t0),f,th3(s0,t0),b):bijective(changef)
 661 -wissel
 662 -e
 663 +r
 664 a@[b:[x:a]'prop']
 665 imp:=b:'prop'
 666 [a1:a][i:imp(a,b)]
 667 mp:=<a1>i:<a1>b
 668 +imp
 669 b@[n:not(a)]
 670 %set etared
 671 th2:=[x:a]cone(<x>b,mp"l"(a,con,x,n)):imp(a,b)
 672 %reset etared
 673 -imp
 674 b@ec:=[x:a]not(<x>b):'prop'
 675 [n:not(a)]
 676 eci1:=[x:a]cone(not(<x>b),mp"l"(a,con,x,n)):ec(a,b)
 677 b@[a1:and(a,b)]
 678 ande2:=<ande1(a,b,a1)>ande2"l"(a,b,a1):<ande1(a,b,a1)>b
 679 a@[ksi:'type']
 680 +ite
 681 [x1:ksi][y1:ksi]
 682 is:=is"l.e"(ksi,x1,y1):'prop'
 683 -ite
 684 [x:[t:a]ksi][y:[t:not(a)]ksi][i:[t:a][u:a]is".ite"(<t>x,<u>x)][j:[t:not(a)][u:not(a)]is".ite"(<t>y,<u>y)]
 685 +*ite
 686 j@[z:ksi]
 687 prop1:=and(imp(a,[t:a]is(z,<t>x)),imp(not(a),[t:not(a)]is(z,<t>y))):'prop'
 688 j@[a1:a][x1:ksi][y1:ksi][px1:prop1(x1)][py1:prop1(y1)]
 689 t1:=ande1"l"(imp(a,[t:a]is(x1,<t>x)),imp(not(a),[t:not(a)]is(x1,<t>y)),px1):imp(a,[t:a]is(x1,<t>x))
 690 t2:=mp(a,[t:a]is(x1,<t>x),a1,t1):is(x1,<a1>x)
 691 t3:=t2(y1,x1,py1,px1):is(y1,<a1>x)
 692 t4:=tris2"l.e"(ksi,x1,y1,<a1>x,t2,t3):is(x1,y1)
 693 a1@t5:=[s:ksi][t:ksi][ps:prop1(s)][pt:prop1(t)]t4(s,t,ps,pt):amone"l.e"(ksi,[t:ksi]prop1(t))
 694 t6:=<a1>i:imp(a,[t:a]is(<a1>x,<t>x))
 695 t7:=th2"r.imp"(not(a),[t:not(a)]is(<a1>x,<t>y),weli(a,a1)):imp(not(a),[t:not(a)]is(<a1>x,<t>y))
 696 t8:=andi(imp(a,[t:a]is(<a1>x,<t>x)),imp(not(a),[t:not(a)]is(<a1>x,<t>y)),t6,t7):prop1(<a1>x)
 697 t9:=somei(ksi,[t:ksi]prop1(t),<a1>x,t8):some(ksi,[t:ksi]prop1(t))
 698 t10:=onei"l.e"(ksi,[t:ksi]prop1(t),t5,t9):one"l.e"(ksi,[t:ksi]prop1(t))
 699 j@[n:not(a)][x1:ksi][y1:ksi][px1:prop1(x1)][py1:prop1(y1)]
 700 t11:=ande2"l"(imp(a,[t:a]is(x1,<t>x)),imp(not(a),[t:not(a)]is(x1,<t>y)),px1):imp(not(a),[t:not(a)]is(x1,<t>y))
 701 t12:=mp(not(a),[t:not(a)]is(x1,<t>y),n,t11):is(x1,<n>y)
 702 t13:=t12(y1,x1,py1,px1):is(y1,<n>y)
 703 t14:=tris2"l.e"(ksi,x1,y1,<n>y,t12,t13):is(x1,y1)
 704 n@t15:=[s:ksi][t:ksi][ps:prop1(s)][pt:prop1(t)]t14(s,t,ps,pt):amone"l.e"(ksi,[t:ksi]prop1(t))
 705 t16:=<n>j:imp(not(a),[t:not(a)]is(<n>y,<t>y))
 706 t17:=th2"r.imp"(a,[t:a]is(<n>y,<t>x),n):imp(a,[t:a]is(<n>y,<t>x))
 707 t18:=andi"l"(imp(a,[t:a]is(<n>y,<t>x)),imp(not(a),[t:not(a)]is(<n>y,<t>y)),t17,t16):prop1(<n>y)
 708 t19:=somei(ksi,[t:ksi]prop1(t),<n>y,t18):some(ksi,[t:ksi]prop1(t))
 709 t20:=onei"l.e"(ksi,[t:ksi]prop1(t),t15,t19):one"l.e"(ksi,[t:ksi]prop1(t))
 710 j@t21:=th1"l.imp"(a,one"l.e"(ksi,[t:ksi]prop1(t)),[t:a]t10(t),[t:not(a)]t20(t)):one"l.e"(ksi,[t:ksi]prop1(t))
 711 -ite
 712 j@ite:=ind"l.e"(ksi,[t:ksi]prop1".ite"(t),t21".ite"):ksi
 713 +*ite
 714 j@t22:=oneax"l.e"(ksi,[t:ksi]prop1(t),t21):prop1(ite)
 715 t23:=ande1"l"(imp(a,[t:a]is(ite,<t>x)),imp(not(a),[t:not(a)]is(ite,<t>y)),t22):imp(a,[t:a]is(ite,<t>x))
 716 t24:=ande2"l"(imp(a,[t:a]is(ite,<t>x)),imp(not(a),[t:not(a)]is(ite,<t>y)),t22):imp(not(a),[t:not(a)]is(ite,<t>y))
 717 -ite
 718 j@[a1:a]
 719 itet:=mp(a,[t:a]is".ite"(ite,<t>x),a1,t23".ite"):is".ite"(ksi,ite,<a1>x)
 720 j@[n:not(a)]
 721 itef:=mp(not(a),[t:not(a)]is".ite"(ite,<t>y),n,t24".ite"):is".ite"(ksi,ite,<n>y)
 722 -r
 723 +*e
 724 +st
 725 sigma@set:='prim':'type'
 726 [s:sigma][s0:set]
 727 esti:='prim':'prop'
 728 p@setof:='prim':set
 729 [s:sigma][sp:<s>p]
 730 estii:='prim':esti(s,setof(p))
 731 s@[e:esti(s,setof(p))]
 732 estie:='prim':<s>p
 733 sigma@[s0:set]
 734 empty:=non([x:sigma]esti(x,s0)):'prop'
 735 nonempty:=some([x:sigma]esti(x,s0)):'prop'
 736 [n:[x:sigma]not(esti(x,s0))]
 737 emptyi:=n:empty(s0)
 738 s0@[e:empty(s0)][s:sigma]
 739 emptye:=<s>e:not(esti(s,s0))
 740 s0@[s:sigma][ses0:esti(s,s0)]
 741 nonemptyi:=somei([x:sigma]esti(x,s0),s,ses0):nonempty(s0)
 742 s0@[n:nonempty(s0)][x:'prop'][x1:[y:sigma][z:esti(y,s0)]x]
 743 nonemptyapp:=someapp([y:sigma]esti(y,s0),n,x,x1):x
 744 s0@[t0:set]
 745 incl:=all([x:sigma]imp(esti(x,s0),esti(x,t0))):'prop'
 746 [e:[x:sigma][y:esti(x,s0)]esti(x,t0)]
 747 incli:=e:incl(s0,t0)
 748 t0@[i:incl(s0,t0)][s:sigma][ses0:esti(s,s0)]
 749 incle:=<ses0><s>i:esti(s,t0)
 750 s0@refincl:=[x:sigma][y:esti(x,s0)]y:incl(s0,s0)
 751 t0@disj:=all([x:sigma]ec(esti(x,s0),esti(x,t0))):'prop'
 752 [n:[x:sigma][y:esti(x,s0)]not(esti(x,t0))]
 753 disji1:=n:disj(s0,t0)
 754 t0@[n:[x:sigma][y:esti(x,t0)]not(esti(x,s0))]
 755 disji2:=[x:sigma]th2"l.ec"(esti(x,s0),esti(x,t0),<x>n):disj(s0,t0)
 756 t0@[d:disj(s0,t0)][s:sigma][ses0:esti(s,s0)]
 757 disje1:=ece1(esti(s,s0),esti(s,t0),<s>d,ses0):not(esti(s,t0))
 758 s@[set0:esti(s,t0)]
 759 disje2:=ece2(esti(s,s0),esti(s,t0),<s>d,set0):not(esti(s,s0))
 760 t0@[d:disj(s0,t0)]
 761 symdisj:=[x:sigma][y:esti(x,t0)]disje2(d,x,y):disj(t0,s0)
 762 +disj
 763 t0@[s:sigma][ses0:esti(s,s0)][set0:esti(s,t0)]
 764 th1:=th1"l.all"([x:sigma]ec(esti(x,s0),esti(x,t0)),s,th4"l.imp"(esti(s,s0),not(esti(s,t0)),ses0,weli(esti(s,t0),set0))):not(disj(s0,t0))
 765 th2:=th1(t0,s0,s,set0,ses0):not(disj(t0,s0))
 766 -disj
 767 t0@[i:is(set,s0,t0)][s:sigma][ses0:esti(s,s0)]
 768 issete1:=isp(set,[x:set]esti(s,x),s0,t0,ses0,i):esti(s,t0)
 769 s@[set0:esti(s,t0)]
 770 issete2:=isp1(set,[x:set]esti(s,x),t0,s0,set0,i):esti(s,s0)
 771 +isset
 772 i@th1:=[x:sigma][y:esti(x,s0)]issete1(x,y):incl(s0,t0)
 773 th2:=[x:sigma][y:esti(x,t0)]issete2(x,y):incl(t0,s0)
 774 -isset
 775 t0@[i:incl(s0,t0)][j:incl(t0,s0)]
 776 isseti:='prim':is(set,s0,t0)
 777 +*isset
 778 t0@[s:sigma][ses0:esti(s,s0)][n:not(esti(s,t0))]
 779 th3:=th3"l.imp"(is(set,s0,t0),esti(s,t0),n,[t:is(set,s0,t0)]issete1(t,s,ses0)):not(is(set,s0,t0))
 780 th4:=symnotis(set,s0,t0,th3):not(is(set,t0,s0))
 781 s@nissetprop:=and(esti(s,s0),not(esti(s,t0))):'prop'
 782 [n:not(nissetprop(s0,t0,s))][e:esti(s,s0)]
 783 t1:=et(esti(s,t0),th3"l.and"(esti(s,s0),not(esti(s,t0)),n,e)):esti(s,t0)
 784 t0@[n:not(is(set,s0,t0))][m:not(some([x:sigma]nissetprop(s0,t0,x)))][l:non([x:sigma]nissetprop(t0,s0,x))][s:sigma]
 785 t2:=th4"l.some"([x:sigma]nissetprop(s0,t0,x),m,s):not(nissetprop(s0,t0,s))
 786 t3:=<s>l:not(nissetprop(t0,s0,s))
 787 l@t4:=isseti(s0,t0,[x:sigma][y:esti(x,s0)]t1(s0,t0,x,t2(x),y),[x:sigma][y:esti(x,t0)]t1(t0,s0,x,t3(x),y)):is(set,s0,t0)
 788 m@t5:=th3"l.imp"(non([x:sigma]nissetprop(t0,s0,x)),is(set,s0,t0),n,[y:non([x:sigma]nissetprop(t0,s0,x))]t4(y)):some([x:sigma]nissetprop(t0,s0,x))
 789 n@th5:=th1"l.or"(some([x:sigma]nissetprop(s0,t0,x)),some([x:sigma]nissetprop(t0,s0,x)),[y:not(some([x:sigma]nissetprop(s0,t0,x)))]t5(y)):or(some([x:sigma]nissetprop(s0,t0,x)),some([x:sigma]nissetprop(t0,s0,x)))
 790 -isset
 791 sigma@[alpha:'type'][sa:[x:alpha]set]
 792 unmore:=setof([x:sigma]some(alpha,[y:alpha]esti(x,<y>sa))):set
 793 [s:sigma][a:alpha][seasa:esti(s,<a>sa)]
 794 eunmore1:=estii([x:sigma]some(alpha,[y:alpha]esti(x,<y>sa)),s,somei(alpha,[y:alpha]esti(s,<y>sa),a,seasa)):esti(s,unmore(sa))
 795 s@[seun:esti(s,unmore(sa))][x:'prop'][x1:[y:alpha][z:esti(s,<y>sa)]x]
 796 unmoreapp:=someapp(alpha,[y:alpha]esti(s,<y>sa),estie([z:sigma]some(alpha,[y:alpha]esti(z,<y>sa)),s,seun),x,x1):x
 797 +eq
 798 sigma@[r:[x:sigma][y:sigma]'prop'][refr1:[x:sigma]<x><x>r][symr1:[x:sigma][y:sigma][t:<y><x>r]<x><y>r][trr1:[x:sigma][y:sigma][z:sigma][t:<y><x>r][u:<z><y>r]<z><x>r][s:sigma]
 799 refr:=<s>refr1:<s><s>r
 800 [t:sigma][tsr:<t><s>r]
 801 symr:=<tsr><t><s>symr1:<s><t>r
 802 t@[u:sigma][tsr:<t><s>r][utr:<u><t>r]
 803 trr:=<utr><tsr><u><t><s>trr1:<u><s>r
 804 u@[sur:<s><u>r][tur:<t><u>r]
 805 tr1r:=trr(s,u,t,symr(u,s,sur),tur):<t><s>r
 806 u@[usr:<u><s>r][utr:<u><t>r]
 807 tr2r:=trr(s,u,t,usr,symr(t,u,utr)):<t><s>r
 808 s@ecelt:=setof(<s>r):set
 809 +1
 810 th1:=estii(<s>r,s,refr):esti(s,ecelt(s))
 811 t@[tsr:<t><s>r]
 812 th2:=estii(<s>r,t,tsr):esti(t,ecelt(s))
 813 t@[e:esti(t,ecelt(s))]
 814 th3:=estie(<s>r,t,e):<t><s>r
 815 tsr@[u:sigma][e:esti(u,ecelt(s))]
 816 t1:=th2(t,u,tr1r(t,u,s,tsr,th3(u,e))):esti(u,ecelt(t))
 817 tsr@th4:=isseti(ecelt(s),ecelt(t),[x:sigma][y:esti(x,ecelt(s))]t1(x,y),[x:sigma][y:esti(x,ecelt(t))]t1(t,s,symr(tsr),x,y)):is(set,ecelt(s),ecelt(t))
 818 t@[n:not(<t><s>r)][u:sigma][e:esti(u,ecelt(s))]
 819 t2:=th3"l.imp"(esti(u,ecelt(t)),<t><s>r,n,[x:esti(u,ecelt(t))]tr2r(s,t,u,th3(u,e),th3(t,u,x))):not(esti(u,ecelt(t)))
 820 n@th5:=[x:sigma][y:esti(x,ecelt(s))]t2(x,y):disj(ecelt(s),ecelt(t))
 821 s@th6:=nonemptyi(ecelt(s),s,th1):nonempty(ecelt(s))
 822 -1
 823 trr1@[s0:set][s:sigma]
 824 ecp:=is(set,s0,ecelt(s)):'prop'
 825 s0@anec:=some([x:sigma]ecp(x)):'prop'
 826 +2
 827 trr1@[s:sigma]
 828 th1:=somei([x:sigma]ecp(ecelt(s),x),s,refis(set,ecelt(s))):anec(ecelt(s))
 829 -2
 830 [ecs0:anec(s0)]
 831 +*2
 832 ecs0@[s:sigma][ses0:esti(s,s0)][t:sigma][e:ecp(s0,t)]
 833 t1:=issete1(s0,ecelt(t),e,s,ses0):esti(s,ecelt(t))
 834 t2:=th4"eq.1"(t,s,th3"eq.1"(t,s,t1)):is(set,ecelt(t),ecelt(s))
 835 t3:=tris(set,s0,ecelt(t),ecelt(s),e,t2):is(set,s0,ecelt(s))
 836 ses0@th2:=someapp([x:sigma]ecp(x),ecs0,is(set,s0,ecelt(s)),[x:sigma][y:ecp(x)]t3(x,y)):is(set,s0,ecelt(s))
 837 [t:sigma][tes0:esti(t,s0)]
 838 th3:=th3"eq.1"(s,t,issete1(s0,ecelt(s),th2,t,tes0)):<t><s>r
 839 t@[tsr:<t><s>r]
 840 th4:=issete2(s0,ecelt(s),th2,t,th2"eq.1"(s,t,tsr)):esti(t,s0)
 841 ecs0@[s:sigma][e:ecp(s0,s)]
 842 t4:=isp(set,[x:set]nonempty(x),ecelt(s),s0,th6"eq.1"(s),symis(set,s0,ecelt(s),e)):nonempty(s0)
 843 ecs0@th5:=someapp([x:sigma]ecp(x),ecs0,nonempty(s0),[x:sigma][y:ecp(x)]t4(x,y)):nonempty(s0)
 844 -2
 845 +3
 846 ecs0@[t0:set][ect0:anec(t0)][s:sigma][ses0:esti(s,s0)][t:sigma][tet0:esti(t,t0)][tsr:<t><s>r]
 847 th1:=tr3is(set,s0,ecelt(s),ecelt(t),t0,th2"eq.2"(s0,ecs0,s,ses0),th4"eq.1"(s,t,tsr),symis(set,t0,ecelt(t),th2"eq.2"(t0,ect0,t,tet0))):is(set,s0,t0)
 848 tet0@[n:not(<t><s>r)]
 849 t1:=isp1(set,[x:set]disj(x,ecelt(t)),ecelt(s),s0,th5"eq.1"(s,t,n),th2"eq.2"(s0,ecs0,s,ses0)):disj(s0,ecelt(t))
 850 th2:=isp1(set,[x:set]disj(s0,x),ecelt(t),t0,t1,th2"eq.2"(t0,ect0,t,tet0)):disj(s0,t0)
 851 t0@[i:is(set,s0,t0)][s:sigma][ses0:esti(s,s0)]
 852 t2:=issete1(s0,t0,i,s,ses0):esti(s,t0)
 853 t3:=th1"st.disj"(s0,t0,s,ses0,t2):not(disj(s0,t0))
 854 i@th3:=nonemptyapp(s0,th5"eq.2"(s0,ecs0),not(disj(s0,t0)),[x:sigma][y:esti(x,s0)]t3(x,y)):not(disj(s0,t0))
 855 -3
 856 trr1@ect:=ot(set,[x:set]anec(x)):'type'
 857 ecs0@ectset:=out(set,[x:set]anec(x),s0,ecs0):ect
 858 trr1@[s:sigma]
 859 ectelt:=ectset(ecelt(s),th1".2"(s)):ect
 860 trr1@[e:ect]
 861 ecect:=in(set,[x:set]anec(x),e):set
 862 +4
 863 th1:=inp(set,[x:set]anec(x),e):anec(ecect(e))
 864 th2:=th5"eq.2"(ecect(e),th1):nonempty(ecect(e))
 865 [x:'prop'][x1:[y:sigma][z:esti(y,ecect(e))]x]
 866 th3:=nonemptyapp(ecect(e),th2,x,x1):x
 867 s@th4:=isinout(set,[x:set]anec(x),ecelt(s),th1"eq.2"(s)):is(set,ecelt(s),ecect(ectelt(s)))
 868 th5:=issete1(ecelt(s),ecect(ectelt(s)),th4,s,th1"eq.1"(s)):esti(s,ecect(ectelt(s)))
 869 th6:=eunmore1(ect,[x:ect]ecect(x),s,ectelt(s),th5):esti(s,unmore(ect,[x:ect]ecect(x)))
 870 e@[s:sigma][see:esti(s,ecect(e))][t:sigma][tee:esti(t,ecect(e))]
 871 th7:=th3"eq.2"(ecect(e),th1,s,see,t,tee):<t><s>r
 872 t@[tsr:<t><s>r]
 873 th8:=th4"eq.2"(ecect(e),th1,s,see,t,tsr):esti(t,ecect(e))
 874 -4
 875 +5
 876 [f:ect][i:is(ect,e,f)]
 877 th1:=isini(set,[x:set]anec(x),e,f,i):is(set,ecect(e),ecect(f))
 878 f@[i:is(set,ecect(e),ecect(f))]
 879 th2:=isine(set,[x:set]anec(x),e,f,i):is(ect,e,f)
 880 f@[s:sigma][see:esti(s,ecect(e))][t:sigma][tef:esti(t,ecect(f))][tsr:<t><s>r]
 881 th3:=th2(th1"eq.3"(ecect(e),th1"eq.4"(e),ecect(f),th1"eq.4"(f),s,see,t,tef,tsr)):is(ect,e,f)
 882 see@[i:is(ect,e,f)]
 883 th4:=issete1(ecect(e),ecect(f),th1(i),s,see):esti(s,ecect(f))
 884 tef@[i:is(ect,e,f)]
 885 th5:=th3"eq.2"(ecect(f),th1"eq.4"(f),s,th4(i),t,tef):<t><s>r
 886 trr1@[s:sigma][t:sigma][tsr:<t><s>r]
 887 th6:=isouti(set,[x:set]anec(x),ecelt(s),th1"eq.2"(s),ecelt(t),th1"eq.2"(t),th4"eq.1"(s,t,tsr)):is(ect,ectelt(s),ectelt(t))
 888 -5
 889 trr1@[alpha:'type'][fu:[x:sigma]alpha]
 890 fixfu:=[x:sigma][y:sigma][z:<y><x>r]is(alpha,<x>fu,<y>fu):'prop'
 891 +10
 892 [ff:fixfu][e:ect][a1:alpha][s:sigma]
 893 prop1:=and(esti(s,ecect(e)),is(alpha,<s>fu,a1)):'prop'
 894 a1@prop2:=some([x:sigma]prop1(x)):'prop'
 895 e@[s:sigma][see:esti(s,ecect(e))]
 896 t1:=andi(esti(s,ecect(e)),is(alpha,<s>fu,<s>fu),see,refis(alpha,<s>fu)):prop1(<s>fu,s)
 897 t2:=somei([x:sigma]prop1(<s>fu,x),s,t1):prop2(<s>fu)
 898 t3:=somei(alpha,[x:alpha]prop2(x),<s>fu,t2):some(alpha,[x:alpha]prop2(x))
 899 e@t4:=th3"eq.4"(e,some(alpha,[x:alpha]prop2(x)),[x:sigma][y:esti(x,ecect(e))]t3(x,y)):some(alpha,[x:alpha]prop2(x))
 900 a1@[b1:alpha][pa1:prop2(a1)][pb1:prop2(b1)][s:sigma][pa1s:prop1(a1,s)][t:sigma][pb1t:prop1(b1,t)]
 901 t5:=ande1(esti(s,ecect(e)),is(alpha,<s>fu,a1),pa1s):esti(s,ecect(e))
 902 t6:=ande1(esti(t,ecect(e)),is(alpha,<t>fu,b1),pb1t):esti(t,ecect(e))
 903 t7:=th7"eq.4"(e,s,t5,t,t6):<t><s>r
 904 t8:=ande2(esti(s,ecect(e)),is(alpha,<s>fu,a1),pa1s):is(alpha,<s>fu,a1)
 905 t9:=ande2(esti(t,ecect(e)),is(alpha,<t>fu,b1),pb1t):is(alpha,<t>fu,b1)
 906 t10:=tr3is(alpha,a1,<s>fu,<t>fu,b1,symis(alpha,<s>fu,a1,t8),<t7><t><s>ff,t9):is(alpha,a1,b1)
 907 pa1s@t11:=someapp([x:sigma]prop1(b1,x),pb1,is(alpha,a1,b1),[x:sigma][y:prop1(b1,x)]t10(x,y)):is(alpha,a1,b1)
 908 pb1@t12:=someapp([x:sigma]prop1(a1,x),pa1,is(alpha,a1,b1),[x:sigma][y:prop1(a1,x)]t11(x,y)):is(alpha,a1,b1)
 909 e@t13:=[x:alpha][y:alpha][u:prop2(x)][v:prop2(y)]t12(x,y,u,v):amone(alpha,[x:alpha]prop2(x))
 910 t14:=onei(alpha,[x:alpha]prop2(x),t13,t4):one(alpha,[x:alpha]prop2(x))
 911 -10
 912 e".10"@indeq:=ind(alpha,[x:alpha]prop2".10"(x),t14".10"):alpha
 913 +*10
 914 e@th1:=oneax(alpha,[x:alpha]prop2(x),t14):some([x:sigma]and(esti(x,ecect(e)),is(alpha,<x>fu,indeq)))
 915 [s:sigma][see:esti(s,ecect(e))]
 916 th2:=t12(<s>fu,indeq,t2(s,see),th1):is(alpha,<s>fu,indeq)
 917 ff@[s:sigma]
 918 th3:=th2(ectelt(s),s,th5"eq.4"(s)):is(alpha,<s>fu,indeq(ectelt(s)))
 919 -10
 920 alpha@[fu2:[x:sigma][y:sigma]alpha]
 921 fixfu2:=[x:sigma][y:sigma][z:sigma][u:sigma][v:<y><x>r][w:<u><z>r]is(alpha,<z><x>fu2,<u><y>fu2):'prop'
 922 +11
 923 [ff2:fixfu2][s:sigma][t:sigma][tsr:<t><s>r][u:sigma]
 924 t1:=<refr(u)><tsr><u><u><t><s>ff2:is(alpha,<u><s>fu2,<u><t>fu2)
 925 tsr@t2:=fisi(sigma,alpha,<s>fu2,<t>fu2,[x:sigma]t1(x)):is([x:sigma]alpha,<s>fu2,<t>fu2)
 926 ff2@[e:ect]
 927 i:=indeq([x:sigma]alpha,fu2,[x:sigma][y:sigma][z:<y><x>r]t2(x,y,z),e):[x:sigma]alpha
 928 [s:sigma][t:sigma][tsr:<t><s>r][u:sigma][uee:esti(u,ecect(e))]
 929 t3:=th2"eq.10"([x:sigma]alpha,fu2,[x:sigma][y:sigma][z:<y><x>r]t2(x,y,z),e,u,uee):is([x:sigma]alpha,<u>fu2,i)
 930 t4:=fise(alpha,<u>fu2,i,t3,s):is(alpha,<s><u>fu2,<s>i)
 931 t5:=fise(alpha,<u>fu2,i,t3,t):is(alpha,<t><u>fu2,<t>i)
 932 t6:=<tsr><refr(u)><t><s><u><u>ff2:is(alpha,<s><u>fu2,<t><u>fu2)
 933 t7:=tr3is(alpha,<s>i,<s><u>fu2,<t><u>fu2,<t>i,symis(alpha,<s><u>fu2,<s>i,t4),t6,t5):is(alpha,<s>i,<t>i)
 934 tsr@t8:=th3"eq.4"(e,is(alpha,<s>i,<t>i),[x:sigma][y:esti(x,ecect(e))]t7(x,y)):is(alpha,<s>i,<t>i)
 935 -11
 936 e".11"@[f:ect]
 937 indeq2:=indeq(i".11",[x:sigma][y:sigma][z:<y><x>r]t8".11"(x,y,z),f):alpha
 938 +*11
 939 f@[s:sigma][see:esti(s,ecect(e))][t:sigma][tef:esti(t,ecect(f))]
 940 t9:=th2"eq.10"(i,[x:sigma][y:sigma][z:<y><x>r]t8(x,y,z),f,t,tef):is(alpha,<t>i,indeq2(e,f))
 941 t10:=th2"eq.10"([x:sigma]alpha,fu2,[x:sigma][y:sigma][z:<y><x>r]t2(x,y,z),e,s,see):is([x:sigma]alpha,<s>fu2,i)
 942 t11:=fise(alpha,<s>fu2,i,t10,t):is(alpha,<t><s>fu2,<t>i)
 943 th1:=tris(alpha,<t><s>fu2,<t>i,indeq2,t11,t9):is(alpha,<t><s>fu2,indeq2(e,f))
 944 ff2@[s:sigma][t:sigma]
 945 th2:=th1(ectelt(s),ectelt(t),s,th5"eq.4"(s),t,th5"eq.4"(t)):is(alpha,<t><s>fu2,indeq2(ectelt(s),ectelt(t)))
 946 -11
 947 +landau
 948 +n
 949 @nat:='prim':'type'
 950 [x:nat][y:nat]
 951 is:=is"e"(nat,x,y):'prop'
 952 nis:=not(is(x,y)):'prop'
 953 x@[s:set(nat)]
 954 in:=esti(nat,x,s):'prop'
 955 @[p:[x:nat]'prop']
 956 some:=some"l"(nat,p):'prop'
 957 all:=all"l"(nat,p):'prop'
 958 one:=one"e"(nat,p):'prop'
 959 @1:='prim':nat
 960 suc:='prim':[x:nat]nat
 961 [x:nat][y:nat][i:is(x,y)]
 962 ax2:=isf(nat,nat,suc,x,y,i):is(<x>suc,<y>suc)
 963 @ax3:='prim':[x:nat]nis(<x>suc,1)
 964 ax4:='prim':[x:nat][y:nat][u:is(<x>suc,<y>suc)]is(x,y)
 965 [s:set(nat)]
 966 cond1:=in(1,s):'prop'
 967 cond2:=all([x:nat]imp(in(x,s),in(<x>suc,s))):'prop'
 968 @ax5:='prim':[s:set(nat)][u:cond1(s)][v:cond2(s)][x:nat]in(x,s)
 969 [p:[x:nat]'prop'][1p:<1>p][xsp:[x:nat][y:<x>p]<<x>suc>p][x:nat]
 970 +i1
 971 s:=setof(nat,p):set(nat)
 972 t1:=estii(nat,p,1,1p):cond1(s)
 973 [y:nat][yes:in(y,s)]
 974 t2:=estie(nat,p,y,yes):<y>p
 975 t3:=estii(nat,p,<y>suc,<t2><y>xsp):in(<y>suc,s)
 976 x@t4:=<x><[y:nat][u:in(y,s)]t3(y,u)><t1><s>ax5:in(x,s)
 977 -i1
 978 induction:=estie(nat,p,x,t4".i1"):<x>p
 979 @[x:nat][y:nat][n:nis(x,y)]
 980 +21
 981 [i:is(<x>suc,<y>suc)]
 982 t1:=<i><y><x>ax4:is(x,y)
 983 -21
 984 satz1:=th3"l.imp"(is(<x>suc,<y>suc),is(x,y),n,[u:is(<x>suc,<y>suc)]t1".21"(u)):nis(<x>suc,<y>suc)
 985 +22
 986 x@prop1:=nis(<x>suc,x):'prop'
 987 @t1:=<1>ax3:prop1(1)
 988 x@[p:prop1(x)]
 989 t2:=satz1(<x>suc,x,p):prop1(<x>suc)
 990 -22
 991 x@satz2:=induction([y:nat]prop1".22"(y),t1".22",[y:nat][u:prop1".22"(y)]t2".22"(y,u),x):nis(<x>suc,x)
 992 +23
 993 prop1:=or(is(x,1),some([u:nat]is(x,<u>suc))):'prop'
 994 @t1:=ori1(is(1,1),some([u:nat]is(1,<u>suc)),refis(nat,1)):prop1(1)
 995 x@t2:=somei(nat,[u:nat]is(<x>suc,<u>suc),x,refis(nat,<x>suc)):some([u:nat]is(<x>suc,<u>suc))
 996 t3:=ori2(is(<x>suc,1),some([u:nat]is(<x>suc,<u>suc)),t2):prop1(<x>suc)
 997 t4:=induction([y:nat]prop1(y),t1,[y:nat][u:prop1(y)]t3(y),x):prop1(x)
 998 -23
 999 [n:nis(x,1)]
1000 satz3:=ore2(is(x,1),some([u:nat]is(x,<u>suc)),t4".23",n):some([u:nat]is(x,<u>suc))
1001 y@[z:nat][i:is(x,<y>suc)][j:is(x,<z>suc)]
1002 +*23
1003 j@t5:=<tris1(nat,<y>suc,<z>suc,x,i,j)><z><y>ax4:is(y,z)
1004 x@t6:=[y:nat][z:nat][u:is(x,<y>suc)][v:is(x,<z>suc)]t5(y,z,u,v):amone(nat,[u:nat]is(x,<u>suc))
1005 -23
1006 n@satz3a:=onei(nat,[u:nat]is(x,<u>suc),t6".23",satz3):one([u:nat]is(x,<u>suc))
1007 +24
1008 x@[f:[y:nat]nat]
1009 prop1:=all([y:nat]is(<<y>suc>f,<<y>f>suc)):'prop'
1010 prop2:=and(is(<1>f,<x>suc),prop1):'prop'
1011 x@[a:[y:nat]nat][b:[y:nat]nat][pa:prop2(a)][pb:prop2(b)][y:nat]
1012 prop3:=is(<y>a,<y>b):'prop'
1013 pb@t1:=ande1(is(<1>a,<x>suc),prop1(a),pa):is(<1>a,<x>suc)
1014 t2:=ande1(is(<1>b,<x>suc),prop1(b),pb):is(<1>b,<x>suc)
1015 t3:=tris2(nat,<1>a,<1>b,<x>suc,t1,t2):prop3(1)
1016 y@[p:prop3(y)]
1017 t4:=ax2(<y>a,<y>b,p):is(<<y>a>suc,<<y>b>suc)
1018 t5:=ande2(is(<1>a,<x>suc),prop1(a),pa):prop1(a)
1019 t6:=ande2(is(<1>b,<x>suc),prop1(b),pb):prop1(b)
1020 t7:=<y>t5:is(<<y>suc>a,<<y>a>suc)
1021 t8:=<y>t6:is(<<y>suc>b,<<y>b>suc)
1022 t9:=tr3is(nat,<<y>suc>a,<<y>a>suc,<<y>b>suc,<<y>suc>b,t7,t4,symis"e"(nat,<<y>suc>b,<<y>b>suc,t8)):prop3(<y>suc)
1023 y@t10:=induction([z:nat]prop3(z),t3,[z:nat][u:prop3(z)]t9(z,u),y):prop3(y)
1024 pb@t11:=fisi(nat,nat,a,b,[y:nat]t10(y)):is"e"([y:nat]nat,a,b)
1025 x@aa:=[z:[y:nat]nat][u:[y:nat]nat][v:prop2(z)][w:prop2(u)]t11(z,u,v,w):amone([y:nat]nat,[z:[y:nat]nat]prop2(z))
1026 prop4:=some"l"([y:nat]nat,[z:[y:nat]nat]prop2(z)):'prop'
1027 @t12:=[x:nat]refis(nat,<<x>suc>suc):prop1(1,suc)
1028 t13:=andi(is(<1>suc,<1>suc),prop1(1,suc),refis(nat,<1>suc),t12):prop2(1,suc)
1029 t14:=somei([y:nat]nat,[z:[y:nat]nat]prop2(1,z),suc,t13):prop4(1)
1030 x@[p:prop4(x)][f:[y:nat]nat][pf:prop2(f)]
1031 g:=[y:nat]<<y>f>suc:[y:nat]nat
1032 [y:nat]
1033 t15:=refis(nat,<y>g):is(<y>g,<<y>f>suc)
1034 pf@t16:=ande1(is(<1>f,<x>suc),prop1(f),pf):is(<1>f,<x>suc)
1035 t17:=tris(nat,<1>g,<<1>f>suc,<<x>suc>suc,t15(1),ax2(<1>f,<x>suc,t16)):is(<1>g,<<x>suc>suc)
1036 y@t18:=ande2(is(<1>f,<x>suc),prop1(f),pf):prop1(f)
1037 t19:=<y>t18:is(<<y>suc>f,<<y>f>suc)
1038 t20:=tris2(nat,<<y>suc>f,<y>g,<<y>f>suc,t19,t15):is(<<y>suc>f,<y>g)
1039 t21:=tris(nat,<<y>suc>g,<<<y>suc>f>suc,<<y>g>suc,t15(<y>suc),ax2(<<y>suc>f,<y>g,t20)):is(<<y>suc>g,<<y>g>suc)
1040 pf@t22:=[y:nat]t21(y):prop1(<x>suc,g)
1041 t23:=andi(is(<1>g,<<x>suc>suc),prop1(<x>suc,g),t17,t22):prop2(<x>suc,g)
1042 t24:=somei([y:nat]nat,[z:[y:nat]nat]prop2(<x>suc,z),g,t23):prop4(<x>suc)
1043 p@t25:=someapp([y:nat]nat,[z:[y:nat]nat]prop2(z),p,prop4(<x>suc),[z:[y:nat]nat][u:prop2(z)]t24(z,u)):prop4(<x>suc)
1044 x@bb:=induction([y:nat]prop4(y),t14,[y:nat][u:prop4(y)]t25(y,u),x):prop4(x)
1045 -24
1046 x@satz4:=onei([y:nat]nat,[z:[y:nat]nat]prop2".24"(z),aa".24",bb".24"):one"e"([y:nat]nat,[z:[y:nat]nat]and(is(<1>z,<x>suc),all([y:nat]is(<<y>suc>z,<<y>z>suc))))
1047 plus:=ind([y:nat]nat,[z:[y:nat]nat]prop2".24"(z),satz4):[y:nat]nat
1048 y@pl:=<y>plus:nat
1049 +*24
1050 x@t26:=oneax([y:nat]nat,[z:[y:nat]nat]prop2(z),satz4):prop2(plus)
1051 -24
1052 x@satz4a:=ande1(is(<1>plus,<x>suc),prop1".24"(plus),t26".24"):is(pl(x,1),<x>suc)
1053 +*24
1054 x@t27:=ande2(is(<1>plus,<x>suc),prop1(plus),t26):prop1(plus)
1055 -24
1056 y@satz4b:=<y>t27".24":is(pl(x,<y>suc),<pl(x,y)>suc)
1057 +*24
1058 @t28:=t11(1,plus(1),suc,t26(1),t13):is"e"([y:nat]nat,plus(1),suc)
1059 -24
1060 x@satz4c:=fise(nat,nat,plus(1),suc,t28".24",x):is(pl(1,x),<x>suc)
1061 +*24
1062 x@t29:=t11(<x>suc,plus(<x>suc),[y:nat]<<y>plus>suc,t26(<x>suc),t23(bb,plus,t26)):is"e"([y:nat]nat,plus(<x>suc),[y:nat]<<y>plus>suc)
1063 -24
1064 y@satz4d:=fise(nat,nat,plus(<x>suc),[z:nat]<<z>plus>suc,t29".24",y):is(pl(<x>suc,y),<pl(x,y)>suc)
1065 x@satz4e:=symis(nat,pl(x,1),<x>suc,satz4a):is(<x>suc,pl(x,1))
1066 y@satz4f:=symis(nat,pl(x,<y>suc),<pl(x,y)>suc,satz4b):is(<pl(x,y)>suc,pl(x,<y>suc))
1067 x@satz4g:=symis(nat,pl(1,x),<x>suc,satz4c):is(<x>suc,pl(1,x))
1068 y@satz4h:=symis(nat,pl(<x>suc,y),<pl(x,y)>suc,satz4d):is(<pl(x,y)>suc,pl(<x>suc,y))
1069 z@[i:is(x,y)]
1070 ispl1:=isf(nat,nat,[u:nat]pl(u,z),x,y,i):is(pl(x,z),pl(y,z))
1071 ispl2:=isf(nat,nat,[u:nat]pl(z,u),x,y,i):is(pl(z,x),pl(z,y))
1072 z@[u:nat][i:is(x,y)][j:is(z,u)]
1073 ispl12:=tris(nat,pl(x,z),pl(y,z),pl(y,u),ispl1(i),ispl2(z,u,y,j)):is(pl(x,z),pl(y,u))
1074 +25
1075 z@prop1:=is(pl(pl(x,y),z),pl(x,pl(y,z))):'prop'
1076 y@t1:=tr3is(nat,pl(pl(x,y),1),<pl(x,y)>suc,pl(x,<y>suc),pl(x,pl(y,1)),satz4a(pl(x,y)),satz4f,ispl2(<y>suc,pl(y,1),x,satz4e(y))):prop1(1)
1077 z@[p:prop1(z)]
1078 t2:=ax2(pl(pl(x,y),z),pl(x,pl(y,z)),p):is(<pl(pl(x,y),z)>suc,<pl(x,pl(y,z))>suc)
1079 t3:=tr4is(nat,pl(pl(x,y),<z>suc),<pl(pl(x,y),z)>suc,<pl(x,pl(y,z))>suc,pl(x,<pl(y,z)>suc),pl(x,pl(y,<z>suc)),satz4b(pl(x,y),z),t2,satz4f(x,pl(y,z)),ispl2(<pl(y,z)>suc,pl(y,<z>suc),x,satz4f(y,z))):prop1(<z>suc)
1080 -25
1081 z@satz5:=induction([u:nat]prop1".25"(u),t1".25",[u:nat][v:prop1".25"(u)]t3".25"(u,v),z):is(pl(pl(x,y),z),pl(x,pl(y,z)))
1082 asspl1:=satz5:is(pl(pl(x,y),z),pl(x,pl(y,z)))
1083 asspl2:=symis(nat,pl(pl(x,y),z),pl(x,pl(y,z)),satz5):is(pl(x,pl(y,z)),pl(pl(x,y),z))
1084 +26
1085 y@prop1:=is(pl(x,y),pl(y,x)):'prop'
1086 t1:=satz4a(y):is(pl(y,1),<y>suc)
1087 t2:=satz4c(y):is(pl(1,y),<y>suc)
1088 t3:=tris2(nat,pl(1,y),pl(y,1),<y>suc,t2,t1):prop1(1,y)
1089 [p:prop1(x,y)]
1090 t4:=tris(nat,<pl(x,y)>suc,<pl(y,x)>suc,pl(y,<x>suc),ax2(pl(x,y),pl(y,x),p),satz4f(y,x)):is(<pl(x,y)>suc,pl(y,<x>suc))
1091 t5:=satz4d:is(pl(<x>suc,y),<pl(x,y)>suc)
1092 t6:=tris(nat,pl(<x>suc,y),<pl(x,y)>suc,pl(y,<x>suc),t5,t4):prop1(<x>suc,y)
1093 -26
1094 y@satz6:=induction([z:nat]prop1".26"(z,y),t3".26",[z:nat][u:prop1".26"(z,y)]t6".26"(z,y,u),x):is(pl(x,y),pl(y,x))
1095 compl:=satz6:is(pl(x,y),pl(y,x))
1096 +*26
1097 x@t7:=tris(nat,pl(x,1),<x>suc,pl(1,x),satz4a,satz4g):prop1(1)
1098 y@[p:prop1(y)]
1099 t8:=tr3is(nat,pl(x,<y>suc),<pl(x,y)>suc,<pl(y,x)>suc,pl(<y>suc,x),satz4b,ax2(pl(x,y),pl(y,x),p),satz4h(y,x)):prop1(<y>suc)
1100 y@anders:=induction([z:nat]prop1(z),t7,[z:nat][u:prop1(z)]t8(z,u),y):is(pl(x,y),pl(y,x))
1101 -26
1102 +27
1103 y@prop1:=nis(y,pl(x,y)):'prop'
1104 x@t1:=symnotis(nat,<x>suc,1,<x>ax3):nis(1,<x>suc)
1105 t2:=th4"e.notis"(nat,1,<x>suc,pl(x,1),t1,satz4a):prop1(1)
1106 y@[p:prop1(y)]
1107 t3:=satz1(y,pl(x,y),p):nis(<y>suc,<pl(x,y)>suc)
1108 t4:=th4"e.notis"(nat,<y>suc,<pl(x,y)>suc,pl(x,<y>suc),t3,satz4b):prop1(<y>suc)
1109 -27
1110 y@satz7:=induction([z:nat]prop1".27"(z),t2".27",[z:nat][u:prop1".27"(z)]t4".27"(z,u),y):nis(y,pl(x,y))
1111 z@[n:nis(y,z)]
1112 +28
1113 prop1:=nis(pl(x,y),pl(x,z)):'prop'
1114 t1:=satz1(y,z,n):nis(<y>suc,<z>suc)
1115 t2:=th5"e.notis"(nat,<y>suc,<z>suc,pl(1,y),pl(1,z),t1,satz4g(y),satz4g(z)):prop1(1,y,z,n)
1116 [p:prop1(x,y,z,n)]
1117 t3:=satz1(pl(x,y),pl(x,z),p):nis(<pl(x,y)>suc,<pl(x,z)>suc)
1118 t4:=th5"e.notis"(nat,<pl(x,y)>suc,<pl(x,z)>suc,pl(<x>suc,y),pl(<x>suc,z),t3,satz4h,satz4h(z)):prop1(<x>suc,y,z,n)
1119 -28
1120 satz8:=induction([u:nat]prop1".28"(u,y,z,n),t2".28",[u:nat][v:prop1".28"(u,y,z,n)]t4".28"(u,y,z,n,v),x):nis(pl(x,y),pl(x,z))
1121 z@[i:is(pl(x,y),pl(x,z))]
1122 satz8a:=th7"l.imp"(is(y,z),nis(pl(x,y),pl(x,z)),weli(is(pl(x,y),pl(x,z)),i),[u:nis(y,z)]satz8(u)):is(y,z)
1123 z@diffprop:=is(x,pl(y,z)):'prop'
1124 +*28
1125 y@[u:nat][v:nat][du:diffprop(u)][dv:diffprop(v)]
1126 t5:=satz8a(y,u,v,tris1(nat,pl(y,u),pl(y,v),x,du,dv)):is(u,v)
1127 -28
1128 y@satz8b:=[u:nat][v:nat][du:diffprop(u)][dv:diffprop(v)]t5".28"(u,v,du,dv):amone(nat,[z:nat]is(x,pl(y,z)))
1129 +29
1130 i:=is(x,y):'prop'
1131 ii:=some([u:nat]diffprop(x,y,u)):'prop'
1132 iii:=some([v:nat]diffprop(y,x,v)):'prop'
1133 [one1:i][u:nat]
1134 t1:=tris(nat,pl(u,x),pl(x,u),pl(y,u),compl(u,x),ispl1(u,one1)):is(pl(u,x),pl(y,u))
1135 t2:=th3"e.notis"(nat,x,pl(u,x),pl(y,u),satz7(u,x),t1):nis(x,pl(y,u))
1136 one1@t3:=th5"l.some"(nat,[u:nat]diffprop(u),[u:nat]t2(u)):not(ii)
1137 y@t4:=th1"l.ec"(i,ii,[z:i]t3(z)):ec(i,ii)
1138 one1@t5:=t3(y,x,symis(nat,x,y,one1)):not(iii)
1139 y@t6:=th2"l.ec"(iii,i,[z:i]t5(z)):ec(iii,i)
1140 [two1:ii][three1:iii][u:nat][du:diffprop(x,y,u)][v:nat][dv:diffprop(y,x,v)]
1141 t6a:=tr4is(nat,x,pl(y,u),pl(pl(x,v),u),pl(x,pl(v,u)),pl(pl(v,u),x),du,ispl1(y,pl(x,v),u,dv),asspl1(x,v,u),compl(x,pl(v,u))):is(x,pl(pl(v,u),x))
1142 t7:=mp(is(x,pl(pl(v,u),x)),con,t6a,satz7(pl(v,u),x)):con
1143 du@t8:=someapp(nat,[v:nat]diffprop(y,x,v),three1,con,[v:nat][dv:diffprop(y,x,v)]t7(v,dv)):con
1144 three1@t9:=someapp(nat,[u:nat]diffprop(u),two1,con,[u:nat][du:diffprop(u)]t8(u,du)):con
1145 two1@t10:=[z:iii]t9(z):not(iii)
1146 y@t11:=th1"l.ec"(ii,iii,[z:ii]t10(z)):ec(ii,iii)
1147 a:=th6"l.ec3"(i,ii,iii,t4,t11,t6):ec3(i,ii,iii)
1148 prop1:=or3(i,ii,iii):'prop'
1149 x@[j:is(x,1)]
1150 t12:=or3i1(i(1),ii(1),iii(1),j):prop1(1)
1151 x@[n:nis(x,1)][u:nat][j:is(x,<u>suc)]
1152 t13:=tris(nat,x,<u>suc,pl(1,u),j,satz4g(u)):is(x,pl(1,u))
1153 t14:=somei(nat,[z:nat]diffprop(x,1,z),u,t13):ii(1)
1154 t15:=someapp(nat,[u1:nat]is(x,<u1>suc),satz3(x,n),ii(1),[u1:nat][z:is(x,<u1>suc)]t14(u1,z)):ii(1)
1155 t16:=or3i2(i(1),ii(1),iii(1),t15):prop1(1)
1156 n@t16a:=someapp(nat,[u:nat]is(x,<u>suc),satz3(x,n),prop1(1),[u:nat][v:is(x,<u>suc)]t16(u,v)):prop1(1)
1157 x@t17:=th1"l.imp"(is(x,1),prop1(1),[z:is(x,1)]t12(z),[z:nis(x,1)]t16a(z)):prop1(1)
1158 y@[p:prop1(y)][one1:i(y)]
1159 t18:=tris(nat,<y>suc,pl(y,1),pl(x,1),satz4e(y),ispl1(y,x,1,symis(nat,x,y,one1))):is(<y>suc,pl(x,1))
1160 t19:=somei(nat,[z:nat]diffprop(<y>suc,x,z),1,t18):iii(<y>suc)
1161 t20:=or3i3(i(<y>suc),ii(<y>suc),iii(<y>suc),t19):prop1(<y>suc)
1162 p@[two1:ii(y)][u:nat][du:diffprop(x,y,u)][j:is(u,1)]
1163 t21:=tr3is(nat,x,pl(y,u),pl(y,1),<y>suc,du,ispl2(u,1,y,j),satz4a(y)):is(x,<y>suc)
1164 t22:=or3i1(i(<y>suc),ii(<y>suc),iii(<y>suc),t21):prop1(<y>suc)
1165 du@[n:nis(u,1)][w:nat][j:is(u,<w>suc)]
1166 t23:=tris(nat,u,<w>suc,pl(1,w),j,satz4g(w)):is(u,pl(1,w))
1167 t24:=tr4is(nat,x,pl(y,u),pl(y,pl(1,w)),pl(pl(y,1),w),pl(<y>suc,w),du,ispl2(u,pl(1,w),y,t23),asspl2(y,1,w),ispl1(pl(y,1),<y>suc,w,satz4a(y))):is(x,pl(<y>suc,w))
1168 t25:=somei(nat,[z:nat]diffprop(x,<y>suc,z),w,t24):ii(<y>suc)
1169 n@t26:=someapp(nat,[z:nat]is(u,<z>suc),satz3(u,n),ii(<y>suc),[z:nat][v:is(u,<z>suc)]t25(z,v)):ii(<y>suc)
1170 t27:=or3i2(i(<y>suc),ii(<y>suc),iii(<y>suc),t26):prop1(<y>suc)
1171 du@t28:=th1"l.imp"(is(u,1),prop1(<y>suc),[z:is(u,1)]t22(z),[z:nis(u,1)]t27(z)):prop1(<y>suc)
1172 two1@t28a:=someapp(nat,[u:nat]diffprop(u),two1,prop1(<y>suc),[u:nat][du:diffprop(u)]t28(u,du)):prop1(<y>suc)
1173 p@[three1:iii(y)][v:nat][dv:diffprop(y,x,v)]
1174 t29:=tris(nat,<y>suc,<pl(x,v)>suc,pl(x,<v>suc),ax2(y,pl(x,v),dv),satz4f(x,v)):is(<y>suc,pl(x,<v>suc))
1175 t30:=somei(nat,[z:nat]diffprop(<y>suc,x,z),<v>suc,t29):iii(<y>suc)
1176 three1@t31:=someapp(nat,[v:nat]diffprop(y,x,v),three1,iii(<y>suc),[v:nat][dv:diffprop(y,x,v)]t30(v,dv)):iii(<y>suc)
1177 t32:=or3i3(i(<y>suc),ii(<y>suc),iii(<y>suc),t31):prop1(<y>suc)
1178 p@t33:=or3app(i(y),ii(y),iii(y),prop1(<y>suc),p,[z:i(y)]t20(z),[z:ii(y)]t28a(z),[z:iii(y)]t32(z)):prop1(<y>suc)
1179 y@b:=induction([z:nat]prop1(z),t17,[z:nat][u:prop1(z)]t33(z,u),y):or3(i,ii,iii)
1180 -29
1181 satz9:=orec3i(i".29",ii".29",iii".29",b".29",a".29"):orec3(is(x,y),some([u:nat]is(x,pl(y,u))),some([v:nat]is(y,pl(x,v))))
1182 satz9a:=b".29":or3(is(x,y),some([u:nat]diffprop(x,y,u)),some([v:nat]diffprop(y,x,v)))
1183 satz9b:=a".29":ec3(is(x,y),some([u:nat]diffprop(x,y,u)),some([v:nat]diffprop(y,x,v)))
1184 more:=some([u:nat]diffprop(x,y,u)):'prop'
1185 less:=some([v:nat]diffprop(y,x,v)):'prop'
1186 satz10:=satz9:orec3(is(x,y),more(x,y),less(x,y))
1187 satz10a:=satz9a:or3(is(x,y),more(x,y),less(x,y))
1188 satz10b:=satz9b:ec3(is(x,y),more(x,y),less(x,y))
1189 [m:more(x,y)]
1190 satz11:=m:less(y,x)
1191 y@[l:less(x,y)]
1192 satz12:=l:more(y,x)
1193 y@moreis:=or(more,is(x,y)):'prop'
1194 lessis:=or(less,is(x,y)):'prop'
1195 [m:moreis(x,y)]
1196 satz13:=th9"l.or"(more,is(x,y),less(y,x),is(y,x),m,[z:more]satz11(z),[z:is(x,y)]symis(nat,x,y,z)):lessis(y,x)
1197 y@[l:lessis(x,y)]
1198 satz14:=th9"l.or"(less,is(x,y),more(y,x),is(y,x),l,[z:less]satz12(z),[z:is(x,y)]symis(nat,x,y,z)):moreis(y,x)
1199 z@[i:is(x,y)][m:more(x,z)]
1200 ismore1:=isp(nat,[u:nat]more(u,z),x,y,m,i):more(y,z)
1201 i@[m:more(z,x)]
1202 ismore2:=isp(nat,[u:nat]more(z,u),x,y,m,i):more(z,y)
1203 i@[l:less(x,z)]
1204 isless1:=isp(nat,[u:nat]less(u,z),x,y,l,i):less(y,z)
1205 i@[l:less(z,x)]
1206 isless2:=isp(nat,[u:nat]less(z,u),x,y,l,i):less(z,y)
1207 i@[m:moreis(x,z)]
1208 ismoreis1:=isp(nat,[u:nat]moreis(u,z),x,y,m,i):moreis(y,z)
1209 i@[m:moreis(z,x)]
1210 ismoreis2:=isp(nat,[u:nat]moreis(z,u),x,y,m,i):moreis(z,y)
1211 i@[l:lessis(x,z)]
1212 islessis1:=isp(nat,[u:nat]lessis(u,z),x,y,l,i):lessis(y,z)
1213 i@[l:lessis(z,x)]
1214 islessis2:=isp(nat,[u:nat]lessis(z,u),x,y,l,i):lessis(z,y)
1215 y@[i:is(x,y)]
1216 moreisi2:=ori2(more(x,y),is(x,y),i):moreis(x,y)
1217 lessisi2:=ori2(less(x,y),is(x,y),i):lessis(x,y)
1218 y@[m:more(x,y)]
1219 moreisi1:=ori1(more(x,y),is(x,y),m):moreis(x,y)
1220 y@[l:less(x,y)]
1221 lessisi1:=ori1(less(x,y),is(x,y),l):lessis(x,y)
1222 z@[u:nat][i:is(x,y)][j:is(z,u)][m:more(x,z)]
1223 ismore12:=ismore2(z,u,y,j,ismore1(x,y,z,i,m)):more(y,u)
1224 j@[l:less(x,z)]
1225 isless12:=isless2(z,u,y,j,isless1(x,y,z,i,l)):less(y,u)
1226 j@[m:moreis(x,z)]
1227 ismoreis12:=ismoreis2(z,u,y,j,ismoreis1(x,y,z,i,m)):moreis(y,u)
1228 j@[l:lessis(x,z)]
1229 islessis12:=islessis2(z,u,y,j,islessis1(x,y,z,i,l)):lessis(y,u)
1230 y@[m:moreis(x,y)]
1231 satz10c:=th7"l.ec3"(is(x,y),more(x,y),less(x,y),satz10b,comor(more(x,y),is(x,y),m)):not(less(x,y))
1232 y@[l:lessis(x,y)]
1233 satz10d:=th9"l.ec3"(is(x,y),more(x,y),less(x,y),satz10b,l):not(more(x,y))
1234 y@[n:not(more(x,y))]
1235 satz10e:=th2"l.or3"(is(x,y),more(x,y),less(x,y),satz10a,n):lessis(x,y)
1236 y@[n:not(less(x,y))]
1237 satz10f:=comor(is(x,y),more(x,y),th3"l.or3"(is(x,y),more(x,y),less(x,y),satz10a,n)):moreis(x,y)
1238 y@[m:more(x,y)]
1239 satz10g:=th3"l.or"(less(x,y),is(x,y),ec3e23(is(x,y),more(x,y),less(x,y),satz10b,m),ec3e21(is(x,y),more(x,y),less(x,y),satz10b,m)):not(lessis(x,y))
1240 y@[l:less(x,y)]
1241 satz10h:=th3"l.or"(more(x,y),is(x,y),ec3e32(is(x,y),more(x,y),less(x,y),satz10b,l),ec3e31(is(x,y),more(x,y),less(x,y),satz10b,l)):not(moreis(x,y))
1242 y@[n:not(moreis(x,y))]
1243 satz10j:=or3e3(is(x,y),more(x,y),less(x,y),satz10a,th5"l.or"(more(x,y),is(x,y),n),th4"l.or"(more(x,y),is(x,y),n)):less(x,y)
1244 y@[n:not(lessis(x,y))]
1245 satz10k:=or3e2(is(x,y),more(x,y),less(x,y),satz10a,th4"l.or"(less(x,y),is(x,y),n),th5"l.or"(less(x,y),is(x,y),n)):more(x,y)
1246 z@[l:less(x,y)][k:less(y,z)]
1247 +315
1248 [v:nat][dv:diffprop(y,x,v)][w:nat][dw:diffprop(z,y,w)]
1249 t1:=tr3is(nat,z,pl(y,w),pl(pl(x,v),w),pl(x,pl(v,w)),dw,ispl1(y,pl(x,v),w,dv),asspl1(x,v,w)):is(z,pl(x,pl(v,w)))
1250 t2:=somei(nat,[u:nat]diffprop(z,x,u),pl(v,w),t1):less(x,z)
1251 dv@t3:=someapp(nat,[w:nat]diffprop(z,y,w),k,less(x,z),[w:nat][dw:diffprop(z,y,w)]t2(w,dw)):less(x,z)
1252 -315
1253 satz15:=someapp(nat,[v:nat]diffprop(y,x,v),l,less(x,z),[v:nat][dv:diffprop(y,x,v)]t3".315"(v,dv)):less(x,z)
1254 trless:=satz15:less(x,z)
1255 z@[m:more(x,y)][n:more(y,z)]
1256 trmore:=satz15(z,y,x,n,m):more(x,z)
1257 +*315
1258 n@anders:=satz12(z,x,satz15(z,y,x,satz11(y,z,n),satz11(m))):more(x,z)
1259 -315
1260 z@[l:lessis(x,y)][k:less(y,z)]
1261 satz16a:=orapp(less(x,y),is(x,y),less(x,z),l,[u:less(x,y)]trless(u,k),[u:is(x,y)]isless1(y,x,z,symis(nat,x,y,u),k)):less(x,z)
1262 z@[l:less(x,y)][k:lessis(y,z)]
1263 satz16b:=orapp(less(y,z),is(y,z),less(x,z),k,[u:less(y,z)]trless(l,u),[u:is(y,z)]isless2(y,z,x,u,l)):less(x,z)
1264 z@[m:moreis(x,y)][n:more(y,z)]
1265 satz16c:=satz16b(z,y,x,n,satz13(m)):more(x,z)
1266 z@[m:more(x,y)][n:moreis(y,z)]
1267 satz16d:=satz16a(z,y,x,satz13(y,z,n),m):more(x,z)
1268 z@[l:lessis(x,y)][k:lessis(y,z)]
1269 +317
1270 [i:is(x,y)][j:is(y,z)]
1271 t1:=lessisi2(x,z,tris(nat,x,y,z,i,j)):lessis(x,z)
1272 i@[j:less(y,z)]
1273 t2:=lessisi1(x,z,satz16a(l,j)):lessis(x,z)
1274 i@t3:=orapp(less(y,z),is(y,z),lessis(x,z),k,[u:less(y,z)]t2(u),[u:is(y,z)]t1(u)):lessis(x,z)
1275 k@[j:less(x,y)]
1276 t4:=lessisi1(x,z,satz16b(j,k)):lessis(x,z)
1277 -317
1278 satz17:=orapp(less(x,y),is(x,y),lessis(x,z),l,[u:less(x,y)]t4".317"(u),[u:is(x,y)]t3".317"(u)):lessis(x,z)
1279 +*317
1280 k@[j:less(x,y)]
1281 t5:=lessisi1(x,z,satz16b(j,k)):lessis(x,z)
1282 k@[i:is(x,y)]
1283 t6:=islessis1(y,x,z,symis(nat,x,y,i),k):lessis(x,z)
1284 k@anders:=orapp(less(x,y),is(x,y),lessis(x,z),l,[u:less(x,y)]t5(u),[u:is(x,y)]t6(u)):lessis(x,z)
1285 -317
1286 k@trlessis:=satz17:lessis(x,z)
1287 z@[m:moreis(x,y)][n:moreis(y,z)]
1288 trmoreis:=satz14(z,x,satz17(z,y,x,satz13(y,z,n),satz13(m))):moreis(x,z)
1289 y@satz18:=somei(nat,[u:nat]diffprop(pl(x,y),x,u),y,refis(nat,pl(x,y))):more(pl(x,y),x)
1290 satz18a:=satz18:less(x,pl(x,y))
1291 x@satz18b:=ismore1(pl(x,1),<x>suc,x,satz4a,satz18(1)):more(<x>suc,x)
1292 satz18c:=satz18b:less(x,<x>suc)
1293 z@[m:more(x,y)]
1294 +319
1295 [u:nat][du:diffprop(u)]
1296 t1:=tris(nat,x,pl(y,u),pl(u,y),du,compl(y,u)):is(x,pl(u,y))
1297 t2:=tr3is(nat,pl(x,z),pl(pl(u,y),z),pl(u,pl(y,z)),pl(pl(y,z),u),ispl1(x,pl(u,y),z,t1),asspl1(u,y,z),compl(u,pl(y,z))):is(pl(x,z),pl(pl(y,z),u))
1298 t3:=somei(nat,[v:nat]diffprop(pl(x,z),pl(y,z),v),u,t2):more(pl(x,z),pl(y,z))
1299 -319
1300 satz19a:=someapp(nat,[u:nat]diffprop(u),m,more(pl(x,z),pl(y,z)),[u:nat][v:diffprop(u)]t3".319"(u,v)):more(pl(x,z),pl(y,z))
1301 z@[i:is(x,y)]
1302 satz19b:=ispl1(x,y,z,i):is(pl(x,z),pl(y,z))
1303 z@[l:less(x,y)]
1304 satz19c:=satz11(pl(y,z),pl(x,z),satz19a(y,x,z,satz12(x,y,l))):less(pl(x,z),pl(y,z))
1305 +*319
1306 l@anders1:=satz19a(y,x,z,l):less(pl(x,z),pl(y,z))
1307 -319
1308 m@satz19d:=ismore12(pl(x,z),pl(z,x),pl(y,z),pl(z,y),compl(x,z),compl(y,z),satz19a):more(pl(z,x),pl(z,y))
1309 i@satz19e:=ispl2(x,y,z,i):is(pl(z,x),pl(z,y))
1310 l@satz19f:=isless12(pl(x,z),pl(z,x),pl(y,z),pl(z,y),compl(x,z),compl(y,z),satz19c):less(pl(z,x),pl(z,y))
1311 +*319
1312 l@anders2:=satz19d(y,x,z,l):less(pl(z,x),pl(z,y))
1313 -319
1314 z@[u:nat][i:is(x,y)][m:more(z,u)]
1315 satz19g:=ismore2(pl(x,u),pl(y,u),pl(x,z),ispl1(x,y,u,i),satz19d(z,u,x,m)):more(pl(x,z),pl(y,u))
1316 satz19h:=ismore12(pl(x,z),pl(z,x),pl(y,u),pl(u,y),compl(x,z),compl(y,u),satz19g):more(pl(z,x),pl(u,y))
1317 i@[l:less(z,u)]
1318 satz19j:=isless2(pl(x,u),pl(y,u),pl(x,z),ispl1(x,y,u,i),satz19f(z,u,x,l)):less(pl(x,z),pl(y,u))
1319 satz19k:=isless12(pl(x,z),pl(z,x),pl(y,u),pl(u,y),compl(x,z),compl(y,u),satz19j):less(pl(z,x),pl(u,y))
1320 z@[m:moreis(x,y)]
1321 +*319
1322 m@[n:more(x,y)]
1323 t4:=moreisi1(pl(x,z),pl(y,z),satz19a(n)):moreis(pl(x,z),pl(y,z))
1324 m@[i:is(x,y)]
1325 t5:=moreisi2(pl(x,z),pl(y,z),ispl1(x,y,z,i)):moreis(pl(x,z),pl(y,z))
1326 -319
1327 m@satz19l:=orapp(more(x,y),is(x,y),moreis(pl(x,z),pl(y,z)),m,[u:more(x,y)]t4".319"(u),[u:is(x,y)]t5".319"(u)):moreis(pl(x,z),pl(y,z))
1328 satz19m:=ismoreis12(pl(x,z),pl(z,x),pl(y,z),pl(z,y),compl(x,z),compl(y,z),satz19l):moreis(pl(z,x),pl(z,y))
1329 z@[l:lessis(x,y)]
1330 satz19n:=satz13(pl(y,z),pl(x,z),satz19l(y,x,z,satz14(l))):lessis(pl(x,z),pl(y,z))
1331 satz19o:=satz13(pl(z,y),pl(z,x),satz19m(y,x,z,satz14(l))):lessis(pl(z,x),pl(z,y))
1332 +320
1333 z@t1:=satz10a(x,y):or3(is(x,y),more(x,y),less(x,y))
1334 t2:=satz10b(pl(x,z),pl(y,z)):ec3(is(pl(x,z),pl(y,z)),more(pl(x,z),pl(y,z)),less(pl(x,z),pl(y,z)))
1335 -320
1336 z@[m:more(pl(x,z),pl(y,z))]
1337 satz20a:=th11"l.ec3"(is(x,y),more(x,y),less(x,y),is(pl(x,z),pl(y,z)),more(pl(x,z),pl(y,z)),less(pl(x,z),pl(y,z)),t1".320",t2".320",[u:is(x,y)]satz19b(x,y,z,u),[u:more(x,y)]satz19a(x,y,z,u),[u:less(x,y)]satz19c(x,y,z,u),m):more(x,y)
1338 z@[i:is(pl(x,z),pl(y,z))]
1339 satz20b:=th10"l.ec3"(is(x,y),more(x,y),less(x,y),is(pl(x,z),pl(y,z)),more(pl(x,z),pl(y,z)),less(pl(x,z),pl(y,z)),t1".320",t2".320",[u:is(x,y)]satz19b(x,y,z,u),[u:more(x,y)]satz19a(x,y,z,u),[u:less(x,y)]satz19c(x,y,z,u),i):is(x,y)
1340 z@[l:less(pl(x,z),pl(y,z))]
1341 satz20c:=th12"l.ec3"(is(x,y),more(x,y),less(x,y),is(pl(x,z),pl(y,z)),more(pl(x,z),pl(y,z)),less(pl(x,z),pl(y,z)),t1".320",t2".320",[u:is(x,y)]satz19b(x,y,z,u),[u:more(x,y)]satz19a(x,y,z,u),[u:less(x,y)]satz19c(x,y,z,u),l):less(x,y)
1342 +*320
1343 i@t3:=tr3is(nat,pl(z,x),pl(x,z),pl(y,z),pl(z,y),compl(z,x),i,compl(y,z)):is(pl(z,x),pl(z,y))
1344 andersb:=satz8a(z,x,y,t3):is(x,y)
1345 l@andersc:=satz20a(y,x,z,l):less(x,y)
1346 -320
1347 z@[m:more(pl(z,x),pl(z,y))]
1348 satz20d:=satz20a(ismore12(pl(z,x),pl(x,z),pl(z,y),pl(y,z),compl(z,x),compl(z,y),m)):more(x,y)
1349 z@[i:is(pl(z,x),pl(z,y))]
1350 satz20e:=satz20b(tr3is(nat,pl(x,z),pl(z,x),pl(z,y),pl(y,z),compl(x,z),i,compl(z,y))):is(x,y)
1351 z@[l:less(pl(z,x),pl(z,y))]
1352 satz20f:=satz20c(isless12(pl(z,x),pl(x,z),pl(z,y),pl(y,z),compl(z,x),compl(z,y),l)):less(x,y)
1353 u@[m:more(x,y)][n:more(z,u)]
1354 +321
1355 t1:=satz19a(x,y,z,m):more(pl(x,z),pl(y,z))
1356 t2:=ismore12(pl(z,y),pl(y,z),pl(u,y),pl(y,u),compl(z,y),compl(u,y),satz19a(z,u,y,n)):more(pl(y,z),pl(y,u))
1357 -321
1358 satz21:=trmore(pl(x,z),pl(y,z),pl(y,u),t1".321",t2".321"):more(pl(x,z),pl(y,u))
1359 +*321
1360 n@anders:=trmore(pl(x,z),pl(y,z),pl(y,u),satz19a(x,y,z,m),satz19d(z,u,y,n)):more(pl(x,z),pl(y,u))
1361 -321
1362 u@[l:less(x,y)][k:less(z,u)]
1363 satz21a:=satz21(y,x,u,z,l,k):less(pl(x,z),pl(y,u))
1364 +*321
1365 k@andersa:=satz11(pl(y,u),pl(x,z),satz21(y,x,u,z,satz12(x,y,l),satz12(z,u,k))):less(pl(x,z),pl(y,u))
1366 -321
1367 u@[m:moreis(x,y)][n:more(z,u)]
1368 satz22a:=orapp(more(x,y),is(x,y),more(pl(x,z),pl(y,u)),m,[v:more(x,y)]satz21(v,n),[v:is(x,y)]satz19g(u,v,n)):more(pl(x,z),pl(y,u))
1369 u@[m:more(x,y)][n:moreis(z,u)]
1370 satz22b:=orapp(more(z,u),is(z,u),more(pl(x,z),pl(y,u)),n,[v:more(z,u)]satz21(m,v),[v:is(z,u)]satz19h(z,u,x,y,v,m)):more(pl(x,z),pl(y,u))
1371 u@[l:lessis(x,y)][k:less(z,u)]
1372 satz22c:=satz22a(y,x,u,z,satz14(x,y,l),k):less(pl(x,z),pl(y,u))
1373 u@[l:less(x,y)][k:lessis(z,u)]
1374 satz22d:=satz22b(y,x,u,z,l,satz14(z,u,k)):less(pl(x,z),pl(y,u))
1375 u@[m:moreis(x,y)][n:moreis(z,u)]
1376 +323
1377 [i:is(x,y)][j:is(z,u)]
1378 t1:=moreisi2(pl(x,z),pl(y,u),tris(nat,pl(x,z),pl(y,z),pl(y,u),ispl1(x,y,z,i),ispl2(z,u,y,j))):moreis(pl(x,z),pl(y,u))
1379 i@[o:more(z,u)]
1380 t2:=moreisi1(pl(x,z),pl(y,u),satz22a(m,o)):moreis(pl(x,z),pl(y,u))
1381 i@t3:=orapp(more(z,u),is(z,u),moreis(pl(x,z),pl(y,u)),n,[v:more(z,u)]t2(v),[v:is(z,u)]t1(v)):moreis(pl(x,z),pl(y,u))
1382 n@[o:more(x,y)]
1383 t4:=moreisi1(pl(x,z),pl(y,u),satz22b(o,n)):moreis(pl(x,z),pl(y,u))
1384 -323
1385 satz23:=orapp(more(x,y),is(x,y),moreis(pl(x,z),pl(y,u)),m,[v:more(x,y)]t4".323"(v),[v:is(x,y)]t3".323"(v)):moreis(pl(x,z),pl(y,u))
1386 +*323
1387 n@[o:more(x,y)]
1388 t5:=moreisi1(pl(x,z),pl(y,u),satz22b(o,n)):moreis(pl(x,z),pl(y,u))
1389 n@[i:is(x,y)]
1390 t6:=ismoreis2(pl(x,u),pl(y,u),pl(x,z),ispl1(u,i),satz19m(z,u,x,n)):moreis(pl(x,z),pl(y,u))
1391 n@anders:=orapp(more(x,y),is(x,y),moreis(pl(x,z),pl(y,u)),m,[v:more(x,y)]t5(v),[v:is(x,y)]t6(v)):moreis(pl(x,z),pl(y,u))
1392 -323
1393 u@[l:lessis(x,y)][k:lessis(z,u)]
1394 satz23a:=satz13(pl(y,u),pl(x,z),satz23(y,x,u,z,satz14(l),satz14(z,u,k))):lessis(pl(x,z),pl(y,u))
1395 +324
1396 x@[n:nis(x,1)][u:nat][i:is(x,<u>suc)]
1397 t1:=tris(nat,x,<u>suc,pl(1,u),i,satz4g(u)):is(x,pl(1,u))
1398 t2:=ismore1(pl(1,u),x,1,symis(nat,x,pl(1,u),t1),satz18(1,u)):more(x,1)
1399 n@t3:=someapp(nat,[u:nat]is(x,<u>suc),satz3(x,n),more(x,1),[u:nat][v:is(x,<u>suc)]t2(u,v)):more(x,1)
1400 -324
1401 x@satz24:=th2"l.or"(more(x,1),is(x,1),[u:nis(x,1)]t3".324"(u)):moreis(x,1)
1402 satz24a:=satz13(x,1,satz24):lessis(1,x)
1403 satz24b:=t3".324"(<x>suc,<x>ax3):more(<x>suc,1)
1404 satz24c:=satz24b:less(1,<x>suc)
1405 y@[m:more(y,x)]
1406 +325
1407 [u:nat][du:diffprop(y,x,u)]
1408 t1:=satz19m(u,1,x,satz24(u)):moreis(pl(x,u),pl(x,1))
1409 t2:=ismoreis1(pl(x,u),y,pl(x,1),symis(nat,y,pl(x,u),du),t1):moreis(y,pl(x,1))
1410 -325
1411 satz25:=someapp(nat,[u:nat]diffprop(y,x,u),m,moreis(y,pl(x,1)),[u:nat][v:diffprop(y,x,u)]t2".325"(u,v)):moreis(y,pl(x,1))
1412 satz25a:=ismoreis2(pl(x,1),<x>suc,y,satz4a,satz25):moreis(y,<x>suc)
1413 y@[l:less(y,x)]
1414 satz25b:=satz13(x,pl(y,1),satz25(y,x,l)):lessis(pl(y,1),x)
1415 satz25c:=islessis1(pl(y,1),<y>suc,x,satz4a(y),satz25b):lessis(<y>suc,x)
1416 y@[l:less(y,pl(x,1))]
1417 +326
1418 [m:more(y,x)]
1419 t1:=satz25(m):moreis(y,pl(x,1))
1420 l@t2:=th3"l.imp"(more(y,x),moreis(y,pl(x,1)),satz10h(y,pl(x,1),l),[v:more(y,x)]t1(v)):not(more(y,x))
1421 -326
1422 satz26:=satz10e(y,x,t2".326"):lessis(y,x)
1423 y@[l:less(y,<x>suc)]
1424 satz26a:=satz26(isless2(<x>suc,pl(x,1),y,satz4e,l)):lessis(y,x)
1425 y@[m:more(pl(y,1),x)]
1426 satz26b:=satz14(x,y,satz26(y,x,m)):moreis(y,x)
1427 y@[m:more(<y>suc,x)]
1428 satz26c:=satz26b(ismore1(<y>suc,pl(y,1),x,satz4e(y),m)):moreis(y,x)
1429 @[p:[x:nat]'prop'][n:nat]
1430 +327
1431 [m:nat]
1432 lbprop:=imp(<m>p,lessis(n,m)):'prop'
1433 -327
1434 lb:=all([x:nat]lbprop".327"(x)):'prop'
1435 min:=and(lb,<n>p):'prop'
1436 p@[s:some(p)]
1437 +*327
1438 s@[n:nat]
1439 t1:=[x:<n>p]satz24a(n):lbprop(1,n)
1440 s@t2:=[x:nat]t1(x):lb(1)
1441 [l:[x:nat]lb(x)][y:nat][yp:<y>p]
1442 t3:=satz18(y,1):more(pl(y,1),y)
1443 t4:=satz10g(pl(y,1),y,t3):not(lessis(pl(y,1),y))
1444 t5:=th4"l.imp"(<y>p,lessis(pl(y,1),y),yp,t4):not(lbprop(pl(y,1),y))
1445 t6:=th1"l.all"(nat,[x:nat]lbprop(pl(y,1),x),y,t5):not(lb(pl(y,1)))
1446 t7:=mp(lb(pl(y,1)),con,<pl(y,1)>l,t6):con
1447 l@t8:=someapp(nat,p,s,con,[x:nat][y:<x>p]t7(x,y)):con
1448 s@[n:non(nat,[x:nat]and(lb(x),not(lb(pl(x,1)))))][m:nat][l:lb(m)]
1449 t9:=<m>n:not(and(lb(m),not(lb(pl(m,1)))))
1450 t10:=et(lb(pl(m,1)),th3"l.and"(lb(m),not(lb(pl(m,1))),t9,l)):lb(pl(m,1))
1451 t11:=isp(nat,[x:nat]lb(x),pl(m,1),<m>suc,t10,satz4a(m)):lb(<m>suc)
1452 n@t12:=[x:nat]induction([y:nat]lb(y),t2,[y:nat][z:lb(y)]t11(y,z),x):[x:nat]lb(x)
1453 s@t13:=[x:non(nat,[x:nat]and(lb(x),not(lb(pl(x,1)))))]t8(t12(x)):some([x:nat]and(lb(x),not(lb(pl(x,1)))))
1454 [m:nat][a:and(lb(m),not(lb(pl(m,1))))]
1455 t14:=ande1(lb(m),not(lb(pl(m,1))),a):lb(m)
1456 t15:=ande2(lb(m),not(lb(pl(m,1))),a):not(lb(pl(m,1)))
1457 [nmp:not(<m>p)][n:nat][np:<n>p]
1458 t16:=mp(<n>p,lessis(m,n),np,<n>t14):lessis(m,n)
1459 t17:=th3"l.imp"(is(m,n),<m>p,nmp,[x:is(m,n)]isp(nat,p,n,m,np,symis(nat,m,n,x))):not(is(m,n))
1460 t18:=ore1(less(m,n),is(m,n),t16,t17):less(m,n)
1461 t19:=satz25b(n,m,t18):lessis(pl(m,1),n)
1462 nmp@t20:=[x:nat][y:<x>p]t19(x,y):lb(pl(m,1))
1463 t21:=mp(lb(pl(m,1)),con,t20,t15):con
1464 a@t22:=et(<m>p,[x:not(<m>p)]t21(x)):<m>p
1465 t23:=andi(lb(m),<m>p,t14,t22):min(m)
1466 -327
1467 s@satz27:=th6"l.some"(nat,[x:nat]and(lb(x),not(lb(pl(x,1)))),[x:nat]min(x),t13".327",[x:nat][y:and(lb(x),not(lb(pl(x,1))))]t23".327"(x,y)):some([x:nat]min(p,x))
1468 +*327
1469 p@[n:non(nat,[x:nat]min(x))][u:nat]
1470 t24:=[x:<u>p]satz24a(u):lbprop(1,u)
1471 n@t25:=[x:nat]t24(x):lb(1)
1472 u@[l:lb(u)]
1473 t26:=<u>n:not(min(u))
1474 t27:=th3"l.and"(lb(u),<u>p,t26,l):not(<u>p)
1475 [v:nat][vp:<v>p]
1476 t28:=th3"l.imp"(is(u,v),<u>p,t27,[x:is(u,v)]isp1(nat,p,v,u,vp,x)):nis(u,v)
1477 t29:=mp(<v>p,lessis(u,v),vp,<v>l):lessis(u,v)
1478 t30:=ore1(less(u,v),is(u,v),t29,t28):less(u,v)
1479 t31:=satz25c(v,u,t30):lessis(<u>suc,v)
1480 v@t32:=[x:<v>p]t31(x):lbprop(<u>suc,v)
1481 l@t33:=[x:nat]t32(x):lb(<u>suc)
1482 u@t34:=induction([x:nat]lb(x),t25,[x:nat][y:lb(x)]t33(x,y),u):lb(u)
1483 p@[s:some(p)][u:nat][up:<u>p]
1484 t35:=satz10g(<u>suc,u,satz18b(u)):not(lessis(<u>suc,u))
1485 t36:=th4"l.imp"(<u>p,lessis(<u>suc,u),up,t35):not(lbprop(<u>suc,u))
1486 t37:=th1"l.all"(nat,[x:nat]lbprop(<u>suc,x),u,t36):not(lb(<u>suc))
1487 t38:=[y:non(nat,[x:nat]min(x))]mp(lb(<u>suc),con,t34(y,<u>suc),t37):some([x:nat]min(x))
1488 s@anders:=someapp(nat,p,s,some([x:nat]min(x)),[x:nat][y:<x>p]t38(x,y)):some([x:nat]min(x))
1489 -327
1490 +*327
1491 p@[n:nat][m:nat][mn:min(p,n)][mm:min(p,m)]
1492 t39:=ande1(lb(n),<n>p,mn):lb(n)
1493 t40:=ande1(lb(m),<m>p,mm):lb(m)
1494 t41:=ande2(lb(n),<n>p,mn):<n>p
1495 t42:=ande2(lb(m),<m>p,mm):<m>p
1496 t43:=<m>t39:lbprop(n,m)
1497 t44:=<n>t40:lbprop(m,n)
1498 t45:=mp(<m>p,lessis(n,m),t42,t43):lessis(n,m)
1499 t46:=mp(<n>p,lessis(m,n),t41,t44):lessis(m,n)
1500 t47:=ore2(more(n,m),is(n,m),satz14(m,n,t46),satz10d(n,m,t45)):is(n,m)
1501 p@t48:=[x:nat][y:nat][u:min(x)][v:min(y)]t47(x,y,u,v):amone(nat,[x:nat]min(p,x))
1502 -327
1503 s@satz27a:=onei(nat,[x:nat]min(p,x),t48".327",satz27):one([x:nat]min(p,x))
1504 +428
1505 x@[f:[y:nat]nat]
1506 prop1:=all([y:nat]is(<<y>suc>f,pl(<y>f,x))):'prop'
1507 prop2:=and(is(<1>f,x),prop1):'prop'
1508 x@[a:[y:nat]nat][b:[y:nat]nat][pa:prop2(a)][pb:prop2(b)][y:nat]
1509 prop3:=is(<y>a,<y>b):'prop'
1510 pb@t1:=ande1(is(<1>a,x),prop1(a),pa):is(<1>a,x)
1511 t2:=ande1(is(<1>b,x),prop1(b),pb):is(<1>b,x)
1512 t3:=tris2(nat,<1>a,<1>b,x,t1,t2):prop3(1)
1513 y@[p:prop3(y)]
1514 t4:=ispl1(<y>a,<y>b,x,p):is(pl(<y>a,x),pl(<y>b,x))
1515 t5:=ande2(is(<1>a,x),prop1(a),pa):prop1(a)
1516 t6:=ande2(is(<1>b,x),prop1(b),pb):prop1(b)
1517 t7:=<y>t5:is(<<y>suc>a,pl(<y>a,x))
1518 t8:=<y>t6:is(<<y>suc>b,pl(<y>b,x))
1519 t9:=tr3is(nat,<<y>suc>a,pl(<y>a,x),pl(<y>b,x),<<y>suc>b,t7,t4,symis(nat,<<y>suc>b,pl(<y>b,x),t8)):prop3(<y>suc)
1520 y@t10:=induction([z:nat]prop3(z),t3,[z:nat][u:prop3(z)]t9(z,u),y):prop3(y)
1521 pb@t11:=fisi(nat,nat,a,b,[y:nat]t10(y)):is"e"([y:nat]nat,a,b)
1522 x@a1:=[z:[y:nat]nat][u:[y:nat]nat][v:prop2(z)][w:prop2(u)]t11(z,u,v,w):amone([y:nat]nat,[z:[y:nat]nat]prop2(z))
1523 prop4:=some"l"([y:nat]nat,[z:[y:nat]nat]prop2(z)):'prop'
1524 @id:=[y:nat]y:[y:nat]nat
1525 t12:=[x:nat]satz4e(x):prop1(1,id)
1526 t13:=andi(is(<1>id,1),prop1(1,id),refis(nat,1),t12):prop2(1,id)
1527 t14:=somei([y:nat]nat,[z:[y:nat]nat]prop2(1,z),id,t13):prop4(1)
1528 x@[p:prop4(x)][f:[y:nat]nat][pf:prop2(f)]
1529 g:=[y:nat]pl(<y>f,y):[y:nat]nat
1530 t15:=ande1(is(<1>f,x),prop1(f),pf):is(<1>f,x)
1531 t16:=tris(nat,<1>g,pl(x,1),<x>suc,ispl1(<1>f,x,1,t15),satz4a(x)):is(<1>g,<x>suc)
1532 [y:nat]
1533 t17:=ande2(is(<1>f,x),prop1(f),pf):prop1(f)
1534 t18:=<y>t17:is(<<y>suc>f,pl(<y>f,x))
1535 t19:=tris(nat,<<y>suc>g,pl(pl(<y>f,x),<y>suc),pl(<y>f,pl(x,<y>suc)),ispl1(<<y>suc>f,pl(<y>f,x),<y>suc,t18),asspl1(<y>f,x,<y>suc)):is(<<y>suc>g,pl(<y>f,pl(x,<y>suc)))
1536 t20:=tr3is(nat,pl(x,<y>suc),<pl(x,y)>suc,pl(<x>suc,y),pl(y,<x>suc),satz4b(x,y),satz4h(x,y),compl(<x>suc,y)):is(pl(x,<y>suc),pl(y,<x>suc))
1537 t21:=tr3is(nat,<<y>suc>g,pl(<y>f,pl(x,<y>suc)),pl(<y>f,pl(y,<x>suc)),pl(<y>g,<x>suc),t19,ispl2(pl(x,<y>suc),pl(y,<x>suc),<y>f,t20),asspl2(<y>f,y,<x>suc)):is(<<y>suc>g,pl(<y>g,<x>suc))
1538 pf@t22:=[y:nat]t21(y):prop1(<x>suc,g)
1539 t23:=andi(is(<1>g,<x>suc),prop1(<x>suc,g),t16,t22):prop2(<x>suc,g)
1540 t24:=somei([y:nat]nat,[z:[y:nat]nat]prop2(<x>suc,z),g,t23):prop4(<x>suc)
1541 p@t25:=someapp([y:nat]nat,[z:[y:nat]nat]prop2(z),p,prop4(<x>suc),[z:[y:nat]nat][u:prop2(z)]t24(z,u)):prop4(<x>suc)
1542 x@b1:=induction([y:nat]prop4(y),t14,[y:nat][u:prop4(y)]t25(y,u),x):prop4(x)
1543 -428
1544 x@satz28:=onei([y:nat]nat,[z:[y:nat]nat]prop2".428"(z),a1".428",b1".428"):one"e"([y:nat]nat,[z:[y:nat]nat]and(is(<1>z,x),all([y:nat]is(<<y>suc>z,pl(<y>z,x)))))
1545 times:=ind([y:nat]nat,[z:[y:nat]nat]prop2".428"(z),satz28):[y:nat]nat
1546 y@ts:=<y>times:nat
1547 +*428
1548 x@t26:=oneax([y:nat]nat,[z:[y:nat]nat]prop2(z),satz28):prop2(times)
1549 -428
1550 x@satz28a:=ande1(is(<1>times,x),prop1".428"(times),t26".428"):is(ts(x,1),x)
1551 +*428
1552 x@t27:=ande2(is(<1>times,x),prop1(times),t26):prop1(times)
1553 -428
1554 y@satz28b:=<y>t27".428":is(ts(x,<y>suc),pl(ts(x,y),x))
1555 +*428
1556 @t28:=t11(1,times(1),id,t26(1),t13):is"e"([y:nat]nat,times(1),id)
1557 -428
1558 x@satz28c:=fise(nat,nat,times(1),id".428",t28".428",x):is(ts(1,x),x)
1559 +*428
1560 x@t29:=t11(<x>suc,times(<x>suc),[y:nat]pl(<y>times,y),t26(<x>suc),t23(b1,times,t26)):is"e"([y:nat]nat,times(<x>suc),[y:nat]pl(<y>times,y))
1561 -428
1562 y@satz28d:=fise(nat,nat,times(<x>suc),[z:nat]pl(<z>times,z),t29".428",y):is(ts(<x>suc,y),pl(ts(x,y),y))
1563 x@satz28e:=symis(nat,ts(x,1),x,satz28a):is(x,ts(x,1))
1564 y@satz28f:=symis(nat,ts(x,<y>suc),pl(ts(x,y),x),satz28b):is(pl(ts(x,y),x),ts(x,<y>suc))
1565 x@satz28g:=symis(nat,ts(1,x),x,satz28c):is(x,ts(1,x))
1566 y@satz28h:=symis(nat,ts(<x>suc,y),pl(ts(x,y),y),satz28d):is(pl(ts(x,y),y),ts(<x>suc,y))
1567 z@[i:is(x,y)]
1568 ists1:=isf(nat,nat,[u:nat]ts(u,z),x,y,i):is(ts(x,z),ts(y,z))
1569 ists2:=isf(nat,nat,[u:nat]ts(z,u),x,y,i):is(ts(z,x),ts(z,y))
1570 z@[u:nat][i:is(x,y)][j:is(z,u)]
1571 ists12:=tris(nat,ts(x,z),ts(y,z),ts(y,u),ists1(i),ists2(z,u,y,j)):is(ts(x,z),ts(y,u))
1572 +429
1573 y@prop1:=is(ts(x,y),ts(y,x)):'prop'
1574 t1:=satz28a(y):is(ts(y,1),y)
1575 t2:=satz28c(y):is(ts(1,y),y)
1576 t3:=tris2(nat,ts(1,y),ts(y,1),y,t2,t1):prop1(1,y)
1577 [p:prop1(x,y)]
1578 t4:=tris(nat,pl(ts(x,y),y),pl(ts(y,x),y),ts(y,<x>suc),ispl1(ts(x,y),ts(y,x),y,p),satz28f(y,x)):is(pl(ts(x,y),y),ts(y,<x>suc))
1579 t5:=satz28d:is(ts(<x>suc,y),pl(ts(x,y),y))
1580 t6:=tris(nat,ts(<x>suc,y),pl(ts(x,y),y),ts(y,<x>suc),t5,t4):prop1(<x>suc,y)
1581 -429
1582 y@satz29:=induction([z:nat]prop1".429"(z,y),t3".429",[z:nat][u:prop1".429"(z,y)]t6".429"(z,y,u),x):is(ts(x,y),ts(y,x))
1583 comts:=satz29:is(ts(x,y),ts(y,x))
1584 +*429
1585 x@t7:=tris(nat,ts(x,1),x,ts(1,x),satz28a,satz28g):prop1(1)
1586 y@[p:prop1(y)]
1587 t8:=tr3is(nat,ts(x,<y>suc),pl(ts(x,y),x),pl(ts(y,x),x),ts(<y>suc,x),satz28b(x,y),ispl1(ts(x,y),ts(y,x),x,p),satz28h(y,x)):prop1(<y>suc)
1588 y@anders:=induction([z:nat]prop1(z),t7,[z:nat][u:prop1(z)]t8(z,u),y):is(ts(x,y),ts(y,x))
1589 -429
1590 +430
1591 z@prop1:=is(ts(x,pl(y,z)),pl(ts(x,y),ts(x,z))):'prop'
1592 y@t1:=tr3is(nat,ts(x,pl(y,1)),ts(x,<y>suc),pl(ts(x,y),x),pl(ts(x,y),ts(x,1)),ists2(pl(y,1),<y>suc,x,satz4a(y)),satz28b,ispl2(x,ts(x,1),ts(x,y),satz28e(x))):prop1(1)
1593 z@[p:prop1(z)]
1594 t2:=tr3is(nat,ts(x,pl(y,<z>suc)),ts(x,<pl(y,z)>suc),pl(ts(x,pl(y,z)),x),pl(pl(ts(x,y),ts(x,z)),x),ists2(pl(y,<z>suc),<pl(y,z)>suc,x,satz4b(y,z)),satz28b(x,pl(y,z)),ispl1(ts(x,pl(y,z)),pl(ts(x,y),ts(x,z)),x,p)):is(ts(x,pl(y,<z>suc)),pl(pl(ts(x,y),ts(x,z)),x))
1595 t3:=tr3is(nat,ts(x,pl(y,<z>suc)),pl(pl(ts(x,y),ts(x,z)),x),pl(ts(x,y),pl(ts(x,z),x)),pl(ts(x,y),ts(x,<z>suc)),t2,asspl1(ts(x,y),ts(x,z),x),ispl2(pl(ts(x,z),x),ts(x,<z>suc),ts(x,y),satz28f(x,z))):prop1(<z>suc)
1596 -430
1597 z@satz30:=induction([u:nat]prop1".430"(u),t1".430",[u:nat][v:prop1".430"(u)]t3".430"(u,v),z):is(ts(x,pl(y,z)),pl(ts(x,y),ts(x,z)))
1598 disttp1:=tr3is(nat,ts(pl(x,y),z),ts(z,pl(x,y)),pl(ts(z,x),ts(z,y)),pl(ts(x,z),ts(y,z)),comts(pl(x,y),z),satz30(z,x,y),ispl12(ts(z,x),ts(x,z),ts(z,y),ts(y,z),comts(z,x),comts(z,y))):is(ts(pl(x,y),z),pl(ts(x,z),ts(y,z)))
1599 disttp2:=satz30:is(ts(x,pl(y,z)),pl(ts(x,y),ts(x,z)))
1600 distpt1:=symis(nat,ts(pl(x,y),z),pl(ts(x,z),ts(y,z)),disttp1):is(pl(ts(x,z),ts(y,z)),ts(pl(x,y),z))
1601 distpt2:=symis(nat,ts(x,pl(y,z)),pl(ts(x,y),ts(x,z)),disttp2):is(pl(ts(x,y),ts(x,z)),ts(x,pl(y,z)))
1602 +431
1603 prop1:=is(ts(ts(x,y),z),ts(x,ts(y,z))):'prop'
1604 y@t1:=tris(nat,ts(ts(x,y),1),ts(x,y),ts(x,ts(y,1)),satz28a(ts(x,y)),ists2(y,ts(y,1),x,satz28e(y))):prop1(1)
1605 z@[p:prop1(z)]
1606 t2:=tr4is(nat,ts(ts(x,y),<z>suc),pl(ts(ts(x,y),z),ts(x,y)),pl(ts(x,ts(y,z)),ts(x,y)),ts(x,pl(ts(y,z),y)),ts(x,ts(y,<z>suc)),satz28b(ts(x,y),z),ispl1(ts(ts(x,y),z),ts(x,ts(y,z)),ts(x,y),p),distpt2(x,ts(y,z),y),ists2(pl(ts(y,z),y),ts(y,<z>suc),x,satz28f(y,z))):prop1(<z>suc)
1607 -431
1608 satz31:=induction([u:nat]prop1".431"(u),t1".431",[u:nat][v:prop1".431"(u)]t2".431"(u,v),z):is(ts(ts(x,y),z),ts(x,ts(y,z)))
1609 assts1:=satz31:is(ts(ts(x,y),z),ts(x,ts(y,z)))
1610 assts2:=symis(nat,ts(ts(x,y),z),ts(x,ts(y,z)),assts1):is(ts(x,ts(y,z)),ts(ts(x,y),z))
1611 [m:more(x,y)]
1612 +432
1613 [u:nat][du:diffprop(u)]
1614 t1:=tris(nat,ts(x,z),ts(pl(y,u),z),pl(ts(y,z),ts(u,z)),ists1(x,pl(y,u),z,du),disttp1(y,u,z)):is(ts(x,z),pl(ts(y,z),ts(u,z)))
1615 t2:=somei(nat,[v:nat]diffprop(ts(x,z),ts(y,z),v),ts(u,z),t1):more(ts(x,z),ts(y,z))
1616 -432
1617 satz32a:=someapp(nat,[u:nat]diffprop(u),m,more(ts(x,z),ts(y,z)),[u:nat][v:diffprop(u)]t2".432"(u,v)):more(ts(x,z),ts(y,z))
1618 z@[i:is(x,y)]
1619 satz32b:=ists1(x,y,z,i):is(ts(x,z),ts(y,z))
1620 z@[l:less(x,y)]
1621 satz32c:=satz11(ts(y,z),ts(x,z),satz32a(y,x,z,satz12(x,y,l))):less(ts(x,z),ts(y,z))
1622 +*432
1623 l@anders1:=satz32a(y,x,z,l):less(ts(x,z),ts(y,z))
1624 -432
1625 m@satz32d:=ismore12(ts(x,z),ts(z,x),ts(y,z),ts(z,y),comts(x,z),comts(y,z),satz32a):more(ts(z,x),ts(z,y))
1626 i@satz32e:=ists2(x,y,z,i):is(ts(z,x),ts(z,y))
1627 l@satz32f:=isless12(ts(x,z),ts(z,x),ts(y,z),ts(z,y),comts(x,z),comts(y,z),satz32c):less(ts(z,x),ts(z,y))
1628 +*432
1629 l@anders2:=satz32d(y,x,z,l):less(ts(z,x),ts(z,y))
1630 -432
1631 z@[u:nat][i:is(x,y)][m:more(z,u)]
1632 satz32g:=ismore2(ts(x,u),ts(y,u),ts(x,z),ists1(x,y,u,i),satz32d(z,u,x,m)):more(ts(x,z),ts(y,u))
1633 satz32h:=ismore12(ts(x,z),ts(z,x),ts(y,u),ts(u,y),comts(x,z),comts(y,u),satz32g):more(ts(z,x),ts(u,y))
1634 i@[l:less(z,u)]
1635 satz32j:=isless2(ts(x,u),ts(y,u),ts(x,z),ists1(x,y,u,i),satz32f(z,u,x,l)):less(ts(x,z),ts(y,u))
1636 satz32k:=isless12(ts(x,z),ts(z,x),ts(y,u),ts(u,y),comts(x,z),comts(y,u),satz32j):less(ts(z,x),ts(u,y))
1637 z@[m:moreis(x,y)]
1638 +*432
1639 m@[n:more(x,y)]
1640 t3:=moreisi1(ts(x,z),ts(y,z),satz32a(n)):moreis(ts(x,z),ts(y,z))
1641 m@[i:is(x,y)]
1642 t4:=moreisi2(ts(x,z),ts(y,z),ists1(x,y,z,i)):moreis(ts(x,z),ts(y,z))
1643 -432
1644 m@satz32l:=orapp(more(x,y),is(x,y),moreis(ts(x,z),ts(y,z)),m,[u:more(x,y)]t3".432"(u),[u:is(x,y)]t4".432"(u)):moreis(ts(x,z),ts(y,z))
1645 satz32m:=ismoreis12(ts(x,z),ts(z,x),ts(y,z),ts(z,y),comts(x,z),comts(y,z),satz32l):moreis(ts(z,x),ts(z,y))
1646 z@[l:lessis(x,y)]
1647 satz32n:=satz13(ts(y,z),ts(x,z),satz32l(y,x,z,satz14(l))):lessis(ts(x,z),ts(y,z))
1648 satz32o:=satz13(ts(z,y),ts(z,x),satz32m(y,x,z,satz14(l))):lessis(ts(z,x),ts(z,y))
1649 +433
1650 z@t1:=satz10a(x,y):or3(is(x,y),more(x,y),less(x,y))
1651 t2:=satz10b(ts(x,z),ts(y,z)):ec3(is(ts(x,z),ts(y,z)),more(ts(x,z),ts(y,z)),less(ts(x,z),ts(y,z)))
1652 -433
1653 z@[m:more(ts(x,z),ts(y,z))]
1654 satz33a:=th11"l.ec3"(is(x,y),more(x,y),less(x,y),is(ts(x,z),ts(y,z)),more(ts(x,z),ts(y,z)),less(ts(x,z),ts(y,z)),t1".433",t2".433",[u:is(x,y)]satz32b(x,y,z,u),[u:more(x,y)]satz32a(x,y,z,u),[u:less(x,y)]satz32c(x,y,z,u),m):more(x,y)
1655 z@[i:is(ts(x,z),ts(y,z))]
1656 satz33b:=th10"l.ec3"(is(x,y),more(x,y),less(x,y),is(ts(x,z),ts(y,z)),more(ts(x,z),ts(y,z)),less(ts(x,z),ts(y,z)),t1".433",t2".433",[u:is(x,y)]satz32b(x,y,z,u),[u:more(x,y)]satz32a(x,y,z,u),[u:less(x,y)]satz32c(x,y,z,u),i):is(x,y)
1657 z@[l:less(ts(x,z),ts(y,z))]
1658 satz33c:=th12"l.ec3"(is(x,y),more(x,y),less(x,y),is(ts(x,z),ts(y,z)),more(ts(x,z),ts(y,z)),less(ts(x,z),ts(y,z)),t1".433",t2".433",[u:is(x,y)]satz32b(x,y,z,u),[u:more(x,y)]satz32a(x,y,z,u),[u:less(x,y)]satz32c(x,y,z,u),l):less(x,y)
1659 +*433
1660 l@anders:=satz33a(y,x,z,l):less(x,y)
1661 -433
1662 u@[m:more(x,y)][n:more(z,u)]
1663 +434
1664 t1:=satz32a(x,y,z,m):more(ts(x,z),ts(y,z))
1665 t2:=ismore12(ts(z,y),ts(y,z),ts(u,y),ts(y,u),comts(z,y),comts(u,y),satz32a(z,u,y,n)):more(ts(y,z),ts(y,u))
1666 -434
1667 satz34:=trmore(ts(x,z),ts(y,z),ts(y,u),t1".434",t2".434"):more(ts(x,z),ts(y,u))
1668 +*434
1669 n@anders:=trmore(ts(x,z),ts(y,z),ts(y,u),satz32a(x,y,z,m),satz32d(z,u,y,n)):more(ts(x,z),ts(y,u))
1670 -434
1671 u@[l:less(x,y)][k:less(z,u)]
1672 satz34a:=satz34(y,x,u,z,l,k):less(ts(x,z),ts(y,u))
1673 +*434
1674 k@andersa:=satz11(ts(y,u),ts(x,z),satz34(y,x,u,z,satz12(x,y,l),satz12(z,u,k))):less(ts(x,z),ts(y,u))
1675 -434
1676 u@[m:moreis(x,y)][n:more(z,u)]
1677 satz35a:=orapp(more(x,y),is(x,y),more(ts(x,z),ts(y,u)),m,[v:more(x,y)]satz34(v,n),[v:is(x,y)]satz32g(u,v,n)):more(ts(x,z),ts(y,u))
1678 u@[m:more(x,y)][n:moreis(z,u)]
1679 satz35b:=orapp(more(z,u),is(z,u),more(ts(x,z),ts(y,u)),n,[v:more(z,u)]satz34(m,v),[v:is(z,u)]satz32h(z,u,x,y,v,m)):more(ts(x,z),ts(y,u))
1680 u@[l:lessis(x,y)][k:less(z,u)]
1681 satz35c:=satz35a(y,x,u,z,satz14(x,y,l),k):less(ts(x,z),ts(y,u))
1682 u@[l:less(x,y)][k:lessis(z,u)]
1683 satz35d:=satz35b(y,x,u,z,l,satz14(z,u,k)):less(ts(x,z),ts(y,u))
1684 u@[m:moreis(x,y)][n:moreis(z,u)]
1685 +436
1686 [i:is(x,y)][j:is(z,u)]
1687 t1:=moreisi2(ts(x,z),ts(y,u),tris(nat,ts(x,z),ts(y,z),ts(y,u),ists1(x,y,z,i),ists2(z,u,y,j))):moreis(ts(x,z),ts(y,u))
1688 i@[o:more(z,u)]
1689 t2:=moreisi1(ts(x,z),ts(y,u),satz35a(m,o)):moreis(ts(x,z),ts(y,u))
1690 i@t3:=orapp(more(z,u),is(z,u),moreis(ts(x,z),ts(y,u)),n,[v:more(z,u)]t2(v),[v:is(z,u)]t1(v)):moreis(ts(x,z),ts(y,u))
1691 n@[o:more(x,y)]
1692 t4:=moreisi1(ts(x,z),ts(y,u),satz35b(o,n)):moreis(ts(x,z),ts(y,u))
1693 -436
1694 satz36:=orapp(more(x,y),is(x,y),moreis(ts(x,z),ts(y,u)),m,[v:more(x,y)]t4".436"(v),[v:is(x,y)]t3".436"(v)):moreis(ts(x,z),ts(y,u))
1695 +*436
1696 n@[o:more(x,y)]
1697 t5:=moreisi1(ts(x,z),ts(y,u),satz35b(o,n)):moreis(ts(x,z),ts(y,u))
1698 n@[i:is(x,y)]
1699 t6:=ismoreis2(ts(x,u),ts(y,u),ts(x,z),ists1(u,i),satz32m(z,u,x,n)):moreis(ts(x,z),ts(y,u))
1700 n@anders:=orapp(more(x,y),is(x,y),moreis(ts(x,z),ts(y,u)),m,[v:more(x,y)]t5(v),[v:is(x,y)]t6(v)):moreis(ts(x,z),ts(y,u))
1701 -436
1702 u@[l:lessis(x,y)][k:lessis(z,u)]
1703 satz36a:=satz13(ts(y,u),ts(x,z),satz36(y,x,u,z,satz14(l),satz14(z,u,k))):lessis(ts(x,z),ts(y,u))
1704 y@[m:more(x,y)]
1705 +mn
1706 t1:=onei(nat,[z:nat]diffprop(x,y,z),satz8b(x,y),m):one([z:nat]diffprop(x,y,z))
1707 -mn
1708 mn:=ind(nat,[z:nat]diffprop(x,y,z),t1".mn"):nat
1709 +*mn
1710 m@th1a:=oneax(nat,[z:nat]diffprop(x,y,z),t1):is(x,pl(y,mn(x,y,m)))
1711 th1b:=symis(nat,x,pl(y,mn(x,y,m)),th1a):is(pl(y,mn(x,y,m)),x)
1712 th1c:=tris(nat,x,pl(y,mn(x,y,m)),pl(mn(x,y,m),y),th1a,compl(y,mn(x,y,m))):is(x,pl(mn(x,y,m),y))
1713 th1d:=symis(nat,x,pl(mn(x,y,m),y),th1c):is(pl(mn(x,y,m),y),x)
1714 y@[z:nat][m:more(x,y)][i:is(pl(y,z),x)]
1715 th1e:=<th1a(m)><symis(nat,pl(y,z),x,i)><mn(x,y,m)><z>satz8b(x,y):is(z,mn(x,y,m))
1716 -mn
1717 z@[u:nat][m:more(x,z)][n:more(y,u)][i:is(x,y)][j:is(z,u)]
1718 +*mn
1719 j@t2:=tr3is(nat,pl(u,mn(x,z,m)),pl(z,mn(x,z,m)),x,y,ispl1(u,z,mn(x,z,m),symis(nat,z,u,j)),th1b(x,z,m),i):is(pl(u,mn(x,z,m)),y)
1720 -mn
1721 j@ismn12:=th1e".mn"(y,u,mn(x,z,m),n,t2".mn"):is(mn(x,z,m),mn(y,u,n))
1722 @[n:nat]
1723 1to:=ot(nat,[x:nat]lessis(x,n)):'type'
1724 [x:nat][l:lessis(x,n)]
1725 outn:=out(nat,[y:nat]lessis(y,n),x,l):1to(n)
1726 n@[xn:1to(n)]
1727 inn:=in"e"(nat,[y:nat]lessis(y,n),xn):nat
1728 1top:=inp(nat,[y:nat]lessis(y,n),xn):lessis(inn,n)
1729 l@[y:nat][k:lessis(y,n)][i:is(x,y)]
1730 isoutni:=isouti(nat,[z:nat]lessis(z,n),x,l,y,k,i):is"e"(1to(n),outn(x,l),outn(y,k))
1731 k@[i:is"e"(1to(n),outn(x,l),outn(y,k))]
1732 isoutne:=isoute(nat,[z:nat]lessis(z,n),x,l,y,k,i):is(x,y)
1733 xn@[yn:1to(n)][i:is"e"(1to(n),xn,yn)]
1734 isinni:=isini(nat,[z:nat]lessis(z,n),xn,yn,i):is(inn(xn),inn(yn))
1735 yn@[i:is(inn(xn),inn(yn))]
1736 isinne:=isine(nat,[z:nat]lessis(z,n),xn,yn,i):is"e"(1to(n),xn,yn)
1737 xn@isoutinn:=isoutin(nat,[y:nat]lessis(y,n),xn):is"e"(1to(n),xn,outn(inn(xn),1top(xn)))
1738 l@isinoutn:=isinout(nat,[y:nat]lessis(y,n),x,l):is(x,inn(outn(x,l)))
1739 @1o:=outn(1,1,lessisi2(1,1,refis(nat,1))):1to(1)
1740 [u:1to(1)]
1741 +singlet
1742 u0:=inn(1,u):nat
1743 t1:=1top(1,u):lessis(u0,1)
1744 t2:=ore2(more(u0,1),is(u0,1),satz24(u0),satz10d(u0,1,t1)):is(u0,1)
1745 th1:=tris(1to(1),u,outn(1,u0,t1),1o,isoutinn(1,u),isoutni(1,u0,t1,1,lessisi2(1,1,refis(nat,1)),t2)):is"e"(1to(1),u,1o)
1746 -singlet
1747 @2:=pl(1,1):nat
1748 [x:nat]
1749 +pair
1750 [l:lessis(x,2)][n:nis(x,2)]
1751 t1:=satz26(1,x,ore1(less(x,2),is(x,2),l,n)):lessis(x,1)
1752 t2:=ore2(more(x,1),is(x,1),satz24(x),satz10d(x,1,t1)):is(x,1)
1753 l@th1:=th2"l.or"(is(x,1),is(x,2),[t:nis(x,2)]t2(t)):or(is(x,1),is(x,2))
1754 @th2:=th1"e.notis"(nat,<1>suc,1,2,<1>ax3,satz4e(1)):nis(2,1)
1755 -pair
1756 @1t:=outn(2,1,satz24a(2)):1to(2)
1757 2t:=outn(2,2,lessisi2(2,2,refis(nat,2))):1to(2)
1758 +*pair
1759 @[u:1to(2)]
1760 u0:=inn(2,u):nat
1761 t3:=1top(2,u):lessis(u0,2)
1762 [i:is(u0,1)]
1763 t4:=isoutni(2,u0,t3,1,satz24a(2),i):is"e"(1to(2),outn(2,u0,t3),1t)
1764 t5:=tris(1to(2),u,outn(2,u0,t3),1t,isoutinn(2,u),t4):is"e"(1to(2),u,1t)
1765 u@[i:is(u0,2)]
1766 t6:=isoutni(2,u0,t3,2,lessisi2(2,2,refis(nat,2)),i):is"e"(1to(2),outn(2,u0,t3),2t)
1767 t7:=tris(1to(2),u,outn(2,u0,t3),2t,isoutinn(2,u),t6):is"e"(1to(2),u,2t)
1768 u@th3:=th9"l.or"(is(u0,1),is(u0,2),is"e"(1to(2),u,1t),is"e"(1to(2),u,2t),th1(u0,t3),[t:is(u0,1)]t5(t),[t:is(u0,2)]t7(t)):or(is"e"(1to(2),u,1t),is"e"(1to(2),u,2t))
1769 @[i:is"e"(1to(2),2t,1t)]
1770 t9:=isini(nat,[x:nat]lessis(x,2),2t,1t,i):is(u0(2t),u0(1t))
1771 t10:=tr3is(nat,2,u0(2t),u0(1t),1,isinoutn(2,2,lessisi2(2,2,refis(nat,2))),t9,symis(nat,1,u0(1t),isinoutn(2,1,satz24a(2)))):is(2,1)
1772 @th4:=th3"l.imp"(is"e"(1to(2),2t,1t),is(2,1),th2,[t:is"e"(1to(2),2t,1t)]t10(t)):not(is"e"(1to(2),2t,1t))
1773 -pair
1774 @[alpha:'type']
1775 pair1type:=[x:1to(2)]alpha:'type'
1776 [a:alpha][b:alpha]
1777 pair1:=[x:1to(2)]ite(is"e"(1to(2),x,1t),alpha,a,b):pair1type
1778 alpha@[p:pair1type]
1779 first1:=<1t>p:alpha
1780 second1:=<2t>p:alpha
1781 b@first1is1:=itet(is"e"(1to(2),1t,1t),alpha,a,b,refis(1to(2),1t)):is"e"(alpha,first1(pair1),a)
1782 first1is2:=symis(alpha,first1(pair1),a,first1is1):is"e"(alpha,a,first1(pair1))
1783 second1is1:=itef(is"e"(1to(2),2t,1t),alpha,a,b,th4".pair"):is"e"(alpha,second1(pair1),b)
1784 second1is2:=symis(alpha,second1(pair1),b,second1is1):is"e"(alpha,b,second1(pair1))
1785 +*pair
1786 p@[q:pair1type][i:is"e"(alpha,first1(p),first1(q))][j:is"e"(alpha,second1(p),second1(q))][u:1to(2)][u1:is"e"(1to(2),u,1t)]
1787 t11:=isf(1to(2),alpha,p,u,1t,u1):is"e"(alpha,<u>p,first1(p))
1788 t12:=symis(alpha,<u>q,first1(q),isf(1to(2),alpha,q,u,1t,u1)):is"e"(alpha,first1(q),<u>q)
1789 t13:=tr3is(alpha,<u>p,first1(p),first1(q),<u>q,t11,i,t12):is"e"(alpha,<u>p,<u>q)
1790 u@[u2:is"e"(1to(2),u,2t)]
1791 t14:=isf(1to(2),alpha,p,u,2t,u2):is"e"(alpha,<u>p,second1(p))
1792 t15:=symis(alpha,<u>q,second1(q),isf(1to(2),alpha,q,u,2t,u2)):is"e"(alpha,second1(q),<u>q)
1793 t16:=tr3is(alpha,<u>p,second1(p),second1(q),<u>q,t14,j,t15):is"e"(alpha,<u>p,<u>q)
1794 u@t17:=orapp(is"e"(1to(2),u,1t),is"e"(1to(2),u,2t),is"e"(alpha,<u>p,<u>q),th3(u),[t:is"e"(1to(2),u,1t)]t13(t),[t:is"e"(1to(2),u,2t)]t16(t)):is"e"(alpha,<u>p,<u>q)
1795 j@th5:=fisi(1to(2),alpha,p,q,[t:1to(2)]t17(t)):is"e"(pair1type,p,q)
1796 p@q0:=pair1(first1,second1):pair1type
1797 t18:=first1is1(first1(p),second1):is"e"(alpha,first1(q0),first1(p))
1798 t19:=second1is1(first1,second1):is"e"(alpha,second1(q0),second1(p))
1799 -pair
1800 p@pair1is1:=th5".pair"(q0".pair",p,t18".pair",t19".pair"):is"e"(pair1type,pair1(first1,second1),p)
1801 pair1is2:=symis(pair1type,pair1(first1,second1),p,pair1is1):is"e"(pair1type,p,pair1(first1,second1))
1802 @[x:nat]
1803 lessisi3:=lessisi2(x,x,refis(nat,x)):lessis(x,x)
1804 1out:=outn(x,1,satz24a(x)):1to(x)
1805 xout:=outn(x,x,lessisi3(x)):1to(x)
1806 [y:nat][l:lessis(y,x)][u:1to(y)]
1807 +left
1808 ui:=inn(y,u):nat
1809 t1:=1top(y,u):lessis(ui,y)
1810 t2:=trlessis(ui,y,x,t1,l):lessis(ui,x)
1811 -left
1812 left1to:=outn(x,ui".left",t2".left"):1to(x)
1813 [v:1to(y)][i:is"e"(1to(x),left1to(u),left1to(v))]
1814 +*left
1815 i@t3:=isoutne(x,ui,t2,ui(v),t2(v),i):is(ui,ui(v))
1816 -left
1817 i@thleft1:=isinne(y,u,v,t3".left"):is"e"(1to(y),u,v)
1818 l@thleft2:=[u:1to(y)][v:1to(y)][t:is"e"(1to(x),left1to(u),left1to(v))]thleft1(u,v,t):injective(1to(y),1to(x),[t:1to(y)]left1to(t))
1819 y@[u:1to(y)]
1820 +right
1821 ui:=inn(y,u):nat
1822 t4:=1top(y,u):lessis(ui,y)
1823 t5:=satz19o(ui,y,x,t4):lessis(pl(x,ui),pl(x,y))
1824 -right
1825 right1to:=outn(pl(x,y),pl(x,ui".right"),t5".right"):1to(pl(x,y))
1826 [v:1to(y)][i:is"e"(1to(pl(x,y)),right1to(u),right1to(v))]
1827 +*right
1828 i@t6:=isoutne(pl(x,y),pl(x,ui(u)),t5(u),pl(x,ui(v)),t5(v),i):is(pl(x,ui(u)),pl(x,ui(v)))
1829 t7:=satz20e(ui(u),ui(v),x,t6):is(ui(u),ui(v))
1830 -right
1831 i@thright1:=isinne(y,u,v,t7".right"):is"e"(1to(y),u,v)
1832 @[alpha:'type'][x:nat][y:nat][l:lessis(y,x)][f:[t:1to(x)]alpha]
1833 left:=[t:1to(y)]<left1to(x,y,l,t)>f:[t:1to(y)]alpha
1834 y@[f:[t:1to(pl(x,y))]alpha]
1835 right:=[t:1to(y)]<right1to(x,y,t)>f:[t:1to(y)]alpha
1836 y@[i:is(y,x)][f:[t:1to(y)]alpha]
1837 +*left
1838 f@t4:=lessisi2(y,x,i):lessis(y,x)
1839 t5:=lessisi2(x,y,symis(nat,y,x,i)):lessis(x,y)
1840 f1:=left(y,x,t5,f):[t:1to(x)]alpha
1841 f2:=left(t4,f1):[t:1to(y)]alpha
1842 [u:1to(y)]
1843 t6:=isinoutn(x,inn(y,u),trlessis(inn(y,u),y,x,1top(y,u),t4)):is(inn(y,u),inn(x,left1to(x,y,t4,u)))
1844 t7:=tris(1to(y),u,outn(y,inn(y,u),1top(y,u)),left1to(y,x,t5,left1to(x,y,t4,u)),isoutinn(y,u),isoutni(y,inn(y,u),1top(y,u),inn(x,left1to(x,y,t4,u)),trlessis(inn(x,left1to(x,y,t4,u)),x,y,1top(x,left1to(x,y,t4,u)),t5),t6)):is"e"(1to(y),u,left1to(y,x,t5,left1to(x,y,t4,u)))
1845 t8:=isf(1to(y),alpha,f,u,left1to(y,x,t5,left1to(x,y,t4,u)),t7):is"e"(alpha,<u>f,<u>f2)
1846 -left
1847 f@thleft:=fisi(1to(y),alpha,f,f2".left",[t:1to(y)]t8".left"(t)):is"e"([t:1to(y)]alpha,f,left(x,y,lessisi2(y,x,i),left(y,x,lessisi2(x,y,symis(nat,y,x,i)),f)))
1848 @frac:=pair1type(nat):'type'
1849 [x1:nat][x2:nat]
1850 fr:=pair1(nat,x1,x2):frac
1851 @[x:frac]
1852 num:=first1(nat,x):nat
1853 den:=second1(nat,x):nat
1854 x2@numis:=first1is1(nat,x1,x2):is(num(fr(x1,x2)),x1)
1855 isnum:=first1is2(nat,x1,x2):is(x1,num(fr(x1,x2)))
1856 denis:=second1is1(nat,x1,x2):is(den(fr(x1,x2)),x2)
1857 isden:=second1is2(nat,x1,x2):is(x2,den(fr(x1,x2)))
1858 x@1x:=num(x):nat
1859 2x:=den(x):nat
1860 fris:=pair1is1(nat,x):is"e"(frac,fr(1x,2x),x)
1861 isfr:=pair1is2(nat,x):is"e"(frac,x,fr(1x,2x))
1862 x2@[y1:nat][y2:nat]
1863 12isnd:=ists12(x1,num(fr(x1,x2)),y2,den(fr(y1,y2)),isnum(x1,x2),isden(y1,y2)):is(ts(x1,y2),ts(num(fr(x1,x2)),den(fr(y1,y2))))
1864 ndis12:=symis(nat,ts(x1,y2),ts(num(fr(x1,x2)),den(fr(y1,y2))),12isnd):is(ts(num(fr(x1,x2)),den(fr(y1,y2))),ts(x1,y2))
1865 x@[n1:nat][n2:nat]
1866 1disnd:=ists1(n1,num(fr(n1,n2)),2x,isnum(n1,n2)):is(ts(n1,2x),ts(num(fr(n1,n2)),2x))
1867 ndis1d:=symis(nat,ts(n1,2x),ts(num(fr(n1,n2)),2x),1disnd):is(ts(num(fr(n1,n2)),2x),ts(n1,2x))
1868 n2isnd:=ists2(n2,den(fr(n1,n2)),1x,isden(n1,n2)):is(ts(1x,n2),ts(1x,den(fr(n1,n2))))
1869 ndisn2:=symis(nat,ts(1x,n2),ts(1x,den(fr(n1,n2))),n2isnd):is(ts(1x,den(fr(n1,n2))),ts(1x,n2))
1870 x2@[n:nat][i:is(x1,n)]
1871 isn:=isf(nat,frac,[t:nat]fr(t,x2),x1,n,i):is"e"(frac,fr(x1,x2),fr(n,x2))
1872 n@[i:is(x2,n)]
1873 isd:=isf(nat,frac,[t:nat]fr(x1,t),x2,n,i):is"e"(frac,fr(x1,x2),fr(x1,n))
1874 y2@[i:is(x1,y1)][j:is(x2,y2)]
1875 isnd:=tris(frac,fr(x1,x2),fr(y1,x2),fr(y1,y2),isn(x1,x2,y1,i),isd(y1,x2,y2,j)):is"e"(frac,fr(x1,x2),fr(y1,y2))
1876 x@[y:frac]
1877 1y:=num(y):nat
1878 2y:=den(y):nat
1879 eq:=is(ts(1x,2y),ts(1y,2x)):'prop'
1880 y2@[i:is(ts(x1,y2),ts(y1,x2))]
1881 eqi12:=tr3is(nat,ts(num(fr(x1,x2)),den(fr(y1,y2))),ts(x1,y2),ts(y1,x2),ts(num(fr(y1,y2)),den(fr(x1,x2))),ndis12(x1,x2,y1,y2),i,12isnd(y1,y2,x1,x2)):eq(fr(x1,x2),fr(y1,y2))
1882 n2@[i:is(ts(1x,n2),ts(n1,2x))]
1883 eqi1:=isp(frac,[t:frac]eq(t,fr(n1,n2)),fr(1x,2x),x,eqi12(1x,2x,n1,n2,i),fris):eq(x,fr(n1,n2))
1884 n2@[i:is(ts(n1,2x),ts(1x,n2))]
1885 eqi2:=isp(frac,[t:frac]eq(fr(n1,n2),t),fr(1x,2x),x,eqi12(n1,n2,1x,2x,i),fris):eq(fr(n1,n2),x)
1886 x@satz37:=refis(nat,ts(1x,2x)):eq(x,x)
1887 refeq:=satz37:eq(x,x)
1888 y@[i:is"e"(frac,x,y)]
1889 refeq1:=isp(frac,[t:frac]eq(x,t),x,y,refeq,i):eq(x,y)
1890 refeq2:=isp(frac,[t:frac]eq(t,x),x,y,refeq,i):eq(y,x)
1891 y2@[i:is(x1,y1)][j:is(x2,y2)]
1892 eqnd:=refeq1(fr(x1,x2),fr(y1,y2),isnd(i,j)):eq(fr(x1,x2),fr(y1,y2))
1893 x2@[n:nat][i:is(x1,n)]
1894 eqn:=refeq1(fr(x1,x2),fr(n,x2),isn(n,i)):eq(fr(x1,x2),fr(n,x2))
1895 n@[i:is(x2,n)]
1896 eqd:=refeq1(fr(x1,x2),fr(x1,n),isd(n,i)):eq(fr(x1,x2),fr(x1,n))
1897 y@[e:eq(x,y)]
1898 satz38:=symis(nat,ts(1x,2y),ts(1y,2x),e):eq(y,x)
1899 symeq:=satz38:eq(y,x)
1900 @[a:nat][b:nat][c:nat][d:nat]
1901 +ii1
1902 t1:=tris(nat,ts(b,ts(c,d)),ts(ts(b,c),d),ts(d,ts(b,c)),assts2(b,c,d),comts(ts(b,c),d)):is(ts(b,ts(c,d)),ts(d,ts(b,c)))
1903 -ii1
1904 stets:=tr4is(nat,ts(ts(a,b),ts(c,d)),ts(a,ts(b,ts(c,d))),ts(a,ts(d,ts(b,c))),ts(ts(a,d),ts(b,c)),ts(ts(a,d),ts(c,b)),assts1(a,b,ts(c,d)),ists2(ts(b,ts(c,d)),ts(d,ts(b,c)),a,t1".ii1"),assts2(a,d,ts(b,c)),ists2(ts(b,c),ts(c,b),ts(a,d),comts(b,c))):is(ts(ts(a,b),ts(c,d)),ts(ts(a,d),ts(c,b)))
1905 +*ii1
1906 d@t2:=tr3is(nat,ts(b,ts(c,d)),ts(ts(c,d),b),ts(ts(d,c),b),ts(d,ts(c,b)),comts(b,ts(c,d)),ists1(ts(c,d),ts(d,c),b,comts(c,d)),assts1(d,c,b)):is(ts(b,ts(c,d)),ts(d,ts(c,b)))
1907 anders:=tr3is(nat,ts(ts(a,b),ts(c,d)),ts(a,ts(b,ts(c,d))),ts(a,ts(d,ts(c,b))),ts(ts(a,d),ts(c,b)),assts1(a,b,ts(c,d)),ists2(ts(b,ts(c,d)),ts(d,ts(c,b)),a,t2),assts2(a,d,ts(c,b))):is(ts(ts(a,b),ts(c,d)),ts(ts(a,d),ts(c,b)))
1908 -ii1
1909 y@[z:frac]
1910 1z:=num(z):nat
1911 2z:=den(z):nat
1912 [e:eq(x,y)][f:eq(y,z)]
1913 +139
1914 t1:=ists12(ts(1x,2y),ts(1y,2x),ts(1y,2z),ts(1z,2y),e,f):is(ts(ts(1x,2y),ts(1y,2z)),ts(ts(1y,2x),ts(1z,2y)))
1915 t2:=stets(1x,2y,1y,2z):is(ts(ts(1x,2y),ts(1y,2z)),ts(ts(1x,2z),ts(1y,2y)))
1916 t3:=tris(nat,ts(ts(1y,2x),ts(1z,2y)),ts(ts(1y,2y),ts(1z,2x)),ts(ts(1z,2x),ts(1y,2y)),stets(1y,2x,1z,2y),comts(ts(1y,2y),ts(1z,2x))):is(ts(ts(1y,2x),ts(1z,2y)),ts(ts(1z,2x),ts(1y,2y)))
1917 t4:=tr3is(nat,ts(ts(1x,2z),ts(1y,2y)),ts(ts(1x,2y),ts(1y,2z)),ts(ts(1y,2x),ts(1z,2y)),ts(ts(1z,2x),ts(1y,2y)),symis(nat,ts(ts(1x,2y),ts(1y,2z)),ts(ts(1x,2z),ts(1y,2y)),t2),t1,t3):is(ts(ts(1x,2z),ts(1y,2y)),ts(ts(1z,2x),ts(1y,2y)))
1918 -139
1919 satz39:=satz33b(ts(1x,2z),ts(1z,2x),ts(1y,2y),t4".139"):eq(x,z)
1920 +*139
1921 f@anders:=tr4is(nat,ts(ts(1x,2z),ts(1y,2y)),ts(ts(1x,2y),ts(1y,2z)),ts(ts(1y,2x),ts(1z,2y)),ts(ts(1y,2y),ts(1z,2x)),ts(ts(1z,2x),ts(1y,2y)),stets(1x,2z,1y,2y),ists12(ts(1x,2y),ts(1y,2x),ts(1y,2z),ts(1z,2y),e,f),stets(1y,2x,1z,2y),comts(ts(1y,2y),ts(1z,2x))):is(ts(ts(1x,2z),ts(1y,2y)),ts(ts(1z,2x),ts(1y,2y)))
1922 -139
1923 f@treq:=satz39:eq(x,z)
1924 z@[e:eq(z,x)][f:eq(z,y)]
1925 treq1:=treq(x,z,y,symeq(z,x,e),f):eq(x,y)
1926 z@[e:eq(x,z)][f:eq(y,z)]
1927 treq2:=treq(x,z,y,e,symeq(y,z,f)):eq(x,y)
1928 z@[u:frac][e:eq(x,y)][f:eq(y,z)][g:eq(z,u)]
1929 tr3eq:=treq(x,y,u,e,treq(y,z,u,f,g)):eq(x,u)
1930 u@[v:frac][e:eq(x,y)][f:eq(y,z)][g:eq(z,u)][h:eq(u,v)]
1931 tr4eq:=tr3eq(x,y,z,v,e,f,treq(z,u,v,g,h)):eq(x,v)
1932 x@[n:nat]
1933 satz40:=eqi1(ts(1x,n),ts(2x,n),tris(nat,ts(1x,ts(2x,n)),ts(1x,ts(n,2x)),ts(ts(1x,n),2x),ists2(ts(2x,n),ts(n,2x),1x,comts(2x,n)),assts2(1x,n,2x))):eq(x,fr(ts(1x,n),ts(2x,n)))
1934 satz40a:=symeq(x,fr(ts(1x,n),ts(2x,n)),satz40):eq(fr(ts(1x,n),ts(2x,n)),x)
1935 x2@[n:nat]
1936 satz40b:=eqi12(ts(x1,n),ts(x2,n),tris(nat,ts(x1,ts(x2,n)),ts(x1,ts(n,x2)),ts(ts(x1,n),x2),ists2(ts(x2,n),ts(n,x2),x1,comts(x2,n)),assts2(x1,n,x2))):eq(fr(x1,x2),fr(ts(x1,n),ts(x2,n)))
1937 satz40c:=symeq(fr(x1,x2),fr(ts(x1,n),ts(x2,n)),satz40b):eq(fr(ts(x1,n),ts(x2,n)),fr(x1,x2))
1938 y@moref:=more(ts(1x,2y),ts(1y,2x)):'prop'
1939 lessf:=less(ts(1x,2y),ts(1y,2x)):'prop'
1940 y2@[m:more(ts(x1,y2),ts(y1,x2))]
1941 morefi12:=ismore12(ts(x1,y2),ts(num(fr(x1,x2)),den(fr(y1,y2))),ts(y1,x2),ts(num(fr(y1,y2)),den(fr(x1,x2))),12isnd(x1,x2,y1,y2),12isnd(y1,y2,x1,x2),m):moref(fr(x1,x2),fr(y1,y2))
1942 y2@[l:less(ts(x1,y2),ts(y1,x2))]
1943 lessfi12:=isless12(ts(x1,y2),ts(num(fr(x1,x2)),den(fr(y1,y2))),ts(y1,x2),ts(num(fr(y1,y2)),den(fr(x1,x2))),12isnd(x1,x2,y1,y2),12isnd(y1,y2,x1,x2),l):lessf(fr(x1,x2),fr(y1,y2))
1944 n2@[m:more(ts(1x,n2),ts(n1,2x))]
1945 morefi1:=ismore12(ts(1x,n2),ts(1x,den(fr(n1,n2))),ts(n1,2x),ts(num(fr(n1,n2)),2x),n2isnd(x,n1,n2),1disnd(x,n1,n2),m):moref(x,fr(n1,n2))
1946 n2@[m:more(ts(n1,2x),ts(1x,n2))]
1947 morefi2:=ismore12(ts(n1,2x),ts(num(fr(n1,n2)),2x),ts(1x,n2),ts(1x,den(fr(n1,n2))),1disnd(x,n1,n2),n2isnd(x,n1,n2),m):moref(fr(n1,n2),x)
1948 n2@[l:less(ts(1x,n2),ts(n1,2x))]
1949 lessfi1:=isless12(ts(1x,n2),ts(1x,den(fr(n1,n2))),ts(n1,2x),ts(num(fr(n1,n2)),2x),n2isnd(x,n1,n2),1disnd(x,n1,n2),l):lessf(x,fr(n1,n2))
1950 n2@[l:less(ts(n1,2x),ts(1x,n2))]
1951 lessfi2:=isless12(ts(n1,2x),ts(num(fr(n1,n2)),2x),ts(1x,n2),ts(1x,den(fr(n1,n2))),1disnd(x,n1,n2),n2isnd(x,n1,n2),l):lessf(fr(n1,n2),x)
1952 y@satz41:=satz10(ts(1x,2y),ts(1y,2x)):orec3(eq(x,y),moref(x,y),lessf(x,y))
1953 satz41a:=satz10a(ts(1x,2y),ts(1y,2x)):or3(eq(x,y),moref(x,y),lessf(x,y))
1954 satz41b:=satz10b(ts(1x,2y),ts(1y,2x)):ec3(eq(x,y),moref(x,y),lessf(x,y))
1955 [m:moref(x,y)]
1956 satz42:=satz11(ts(1x,2y),ts(1y,2x),m):lessf(y,x)
1957 y@[l:lessf(x,y)]
1958 satz43:=satz12(ts(1x,2y),ts(1y,2x),l):moref(y,x)
1959 u@1u:=num(u):nat
1960 2u:=den(u):nat
1961 [m:moref(x,y)][e:eq(x,z)][f:eq(y,u)]
1962 +244
1963 t1:=ists12(ts(1y,2u),ts(1u,2y),ts(1z,2x),ts(1x,2z),f,symeq(x,z,e)):is(ts(ts(1y,2u),ts(1z,2x)),ts(ts(1u,2y),ts(1x,2z)))
1964 t2:=tr3is(nat,ts(ts(1y,2x),ts(1z,2u)),ts(ts(1y,2u),ts(1z,2x)),ts(ts(1u,2y),ts(1x,2z)),ts(ts(1u,2z),ts(1x,2y)),stets(1y,2x,1z,2u),t1,stets(1u,2y,1x,2z)):is(ts(ts(1y,2x),ts(1z,2u)),ts(ts(1u,2z),ts(1x,2y)))
1965 t3:=ismore1(ts(ts(1u,2z),ts(1x,2y)),ts(ts(1y,2x),ts(1z,2u)),ts(ts(1u,2z),ts(1y,2x)),symis(nat,ts(ts(1y,2x),ts(1z,2u)),ts(ts(1u,2z),ts(1x,2y)),t2),satz32d(ts(1x,2y),ts(1y,2x),ts(1u,2z),m)):more(ts(ts(1y,2x),ts(1z,2u)),ts(ts(1u,2z),ts(1y,2x)))
1966 -244
1967 satz44:=satz33a(ts(1z,2u),ts(1u,2z),ts(1y,2x),ismore1(ts(ts(1y,2x),ts(1z,2u)),ts(ts(1z,2u),ts(1y,2x)),ts(ts(1u,2z),ts(1y,2x)),comts(ts(1y,2x),ts(1z,2u)),t3".244")):moref(z,u)
1968 u@[e:eq(x,y)][f:eq(z,u)][m:moref(x,z)]
1969 eqmoref12:=satz44(x,z,y,u,m,e,f):moref(y,u)
1970 z@[e:eq(x,y)][m:moref(x,z)]
1971 eqmoref1:=satz44(x,z,y,z,m,e,refeq(z)):moref(y,z)
1972 e@[m:moref(z,x)]
1973 eqmoref2:=satz44(z,x,z,y,m,refeq(z),e):moref(z,y)
1974 u@[l:lessf(x,y)][e:eq(x,z)][f:eq(y,u)]
1975 satz45:=satz42(u,z,satz44(y,x,u,z,satz43(x,y,l),f,e)):lessf(z,u)
1976 u@[e:eq(x,y)][f:eq(z,u)][l:lessf(x,z)]
1977 eqlessf12:=satz45(x,z,y,u,l,e,f):lessf(y,u)
1978 z@[e:eq(x,y)][l:lessf(x,z)]
1979 eqlessf1:=satz45(x,z,y,z,l,e,refeq(z)):lessf(y,z)
1980 e@[l:lessf(z,x)]
1981 eqlessf2:=satz45(z,x,z,y,l,refeq(z),e):lessf(z,y)
1982 y@moreq:=or(moref(x,y),eq(x,y)):'prop'
1983 lesseq:=or(lessf(x,y),eq(x,y)):'prop'
1984 [e:eq(x,y)]
1985 moreqi2:=ori2(moref(x,y),eq(x,y),e):moreq(x,y)
1986 lesseqi2:=ori2(lessf(x,y),eq(x,y),e):lesseq(x,y)
1987 y@[m:moref(x,y)]
1988 moreqi1:=ori1(moref(x,y),eq(x,y),m):moreq(x,y)
1989 y@[l:lessf(x,y)]
1990 lesseqi1:=ori1(lessf(x,y),eq(x,y),l):lesseq(x,y)
1991 y@[m:moreq(x,y)]
1992 satz41c:=th7"l.ec3"(eq(x,y),moref(x,y),lessf(x,y),satz41b,comor(moref(x,y),eq(x,y),m)):not(lessf(x,y))
1993 y@[l:lesseq(x,y)]
1994 satz41d:=th9"l.ec3"(eq(x,y),moref(x,y),lessf(x,y),satz41b,l):not(moref(x,y))
1995 y@[n:not(moref(x,y))]
1996 satz41e:=th2"l.or3"(eq(x,y),moref(x,y),lessf(x,y),satz41a,n):lesseq(x,y)
1997 y@[n:not(lessf(x,y))]
1998 satz41f:=comor(eq(x,y),moref(x,y),th3"l.or3"(eq(x,y),moref(x,y),lessf(x,y),satz41a,n)):moreq(x,y)
1999 y@[m:moref(x,y)]
2000 satz41g:=th3"l.or"(lessf(x,y),eq(x,y),ec3e23(eq(x,y),moref(x,y),lessf(x,y),satz41b,m),ec3e21(eq(x,y),moref(x,y),lessf(x,y),satz41b,m)):not(lesseq(x,y))
2001 y@[l:lessf(x,y)]
2002 satz41h:=th3"l.or"(moref(x,y),eq(x,y),ec3e32(eq(x,y),moref(x,y),lessf(x,y),satz41b,l),ec3e31(eq(x,y),moref(x,y),lessf(x,y),satz41b,l)):not(moreq(x,y))
2003 y@[n:not(moreq(x,y))]
2004 satz41j:=or3e3(eq(x,y),moref(x,y),lessf(x,y),satz41a,th5"l.or"(moref(x,y),eq(x,y),n),th4"l.or"(moref(x,y),eq(x,y),n)):lessf(x,y)
2005 y@[n:not(lesseq(x,y))]
2006 satz41k:=or3e2(eq(x,y),moref(x,y),lessf(x,y),satz41a,th4"l.or"(lessf(x,y),eq(x,y),n),th5"l.or"(lessf(x,y),eq(x,y),n)):moref(x,y)
2007 u@[m:moreq(x,y)][e:eq(x,z)][f:eq(y,u)]
2008 +246
2009 [n:moref(x,y)]
2010 t1:=ori1(moref(z,u),eq(z,u),satz44(n,e,f)):moreq(z,u)
2011 f@[g:eq(x,y)]
2012 t2:=ori2(moref(z,u),eq(z,u),tr3eq(z,x,y,u,symeq(x,z,e),g,f)):moreq(z,u)
2013 -246
2014 satz46:=orapp(moref(x,y),eq(x,y),moreq(z,u),m,[t:moref(x,y)]t1".246"(t),[t:eq(x,y)]t2".246"(t)):moreq(z,u)
2015 u@[e:eq(x,y)][f:eq(z,u)][m:moreq(x,z)]
2016 eqmoreq12:=satz46(x,z,y,u,m,e,f):moreq(y,u)
2017 z@[e:eq(x,y)][m:moreq(x,z)]
2018 eqmoreq1:=satz46(x,z,y,z,m,e,refeq(z)):moreq(y,z)
2019 e@[m:moreq(z,x)]
2020 eqmoreq2:=satz46(z,x,z,y,m,refeq(z),e):moreq(z,y)
2021 u@[l:lesseq(x,y)][e:eq(x,z)][f:eq(y,u)]
2022 +247
2023 [k:lessf(x,y)]
2024 t1:=ori1(lessf(z,u),eq(z,u),satz45(k,e,f)):lesseq(z,u)
2025 f@[g:eq(x,y)]
2026 t2:=ori2(lessf(z,u),eq(z,u),tr3eq(z,x,y,u,symeq(x,z,e),g,f)):lesseq(z,u)
2027 -247
2028 satz47:=orapp(lessf(x,y),eq(x,y),lesseq(z,u),l,[t:lessf(x,y)]t1".247"(t),[t:eq(x,y)]t2".247"(t)):lesseq(z,u)
2029 u@[e:eq(x,y)][f:eq(z,u)][l:lesseq(x,z)]
2030 eqlesseq12:=satz47(x,z,y,u,l,e,f):lesseq(y,u)
2031 z@[e:eq(x,y)][l:lesseq(x,z)]
2032 eqlesseq1:=satz47(x,z,y,z,l,e,refeq(z)):lesseq(y,z)
2033 e@[l:lesseq(z,x)]
2034 eqlesseq2:=satz47(z,x,z,y,l,refeq(z),e):lesseq(z,y)
2035 y@[m:moreq(x,y)]
2036 satz48:=th9"l.or"(moref(x,y),eq(x,y),lessf(y,x),eq(y,x),m,[t:moref(x,y)]satz42(x,y,t),[t:eq(x,y)]satz38(x,y,t)):lesseq(y,x)
2037 y@[l:lesseq(x,y)]
2038 satz49:=th9"l.or"(lessf(x,y),eq(x,y),moref(y,x),eq(y,x),l,[t:lessf(x,y)]satz43(x,y,t),[t:eq(x,y)]satz38(x,y,t)):moreq(y,x)
2039 z@[l:lessf(x,y)][k:lessf(y,z)]
2040 +250
2041 t1:=satz34a(ts(1x,2y),ts(1y,2x),ts(1y,2z),ts(1z,2y),l,k):less(ts(ts(1x,2y),ts(1y,2z)),ts(ts(1y,2x),ts(1z,2y)))
2042 t2:=isless12(ts(ts(1x,2y),ts(1y,2z)),ts(ts(1x,2z),ts(1y,2y)),ts(ts(1y,2x),ts(1z,2y)),ts(ts(1z,2x),ts(1y,2y)),stets(1x,2y,1y,2z),tris(nat,ts(ts(1y,2x),ts(1z,2y)),ts(ts(1y,2y),ts(1z,2x)),ts(ts(1z,2x),ts(1y,2y)),stets(1y,2x,1z,2y),comts(ts(1y,2y),ts(1z,2x))),t1):less(ts(ts(1x,2z),ts(1y,2y)),ts(ts(1z,2x),ts(1y,2y)))
2043 -250
2044 satz50:=satz33c(ts(1x,2z),ts(1z,2x),ts(1y,2y),t2".250"):lessf(x,z)
2045 trlessf:=satz50:lessf(x,z)
2046 z@[m:moref(x,y)][n:moref(y,z)]
2047 trmoref:=satz43(z,x,satz50(z,y,x,satz42(y,z,n),satz42(x,y,m))):moref(x,z)
2048 z@[l:lesseq(x,y)][k:lessf(y,z)]
2049 satz51a:=orapp(lessf(x,y),eq(x,y),lessf(x,z),l,[t:lessf(x,y)]satz50(t,k),[t:eq(x,y)]eqlessf1(y,x,z,symeq(x,y,t),k)):lessf(x,z)
2050 z@[l:lessf(x,y)][k:lesseq(y,z)]
2051 satz51b:=orapp(lessf(y,z),eq(y,z),lessf(x,z),k,[t:lessf(y,z)]satz50(l,t),[t:eq(y,z)]eqlessf2(y,z,x,t,l)):lessf(x,z)
2052 z@[m:moreq(x,y)][n:moref(y,z)]
2053 satz51c:=satz43(z,x,satz51b(z,y,x,satz42(y,z,n),satz48(x,y,m))):moref(x,z)
2054 z@[m:moref(x,y)][n:moreq(y,z)]
2055 satz51d:=satz43(z,x,satz51a(z,y,x,satz48(y,z,n),satz42(x,y,m))):moref(x,z)
2056 z@[l:lesseq(x,y)][k:lesseq(y,z)]
2057 +252
2058 [e:eq(x,y)][f:eq(y,z)]
2059 t1:=ori2(lessf(x,z),eq(x,z),treq(x,y,z,e,f)):lesseq(x,z)
2060 e@[j:lessf(y,z)]
2061 t2:=ori1(lessf(x,z),eq(x,z),satz51a(l,j)):lesseq(x,z)
2062 e@t3:=orapp(lessf(y,z),eq(y,z),lesseq(x,z),k,[t:lessf(y,z)]t2(t),[t:eq(y,z)]t1(t)):lesseq(x,z)
2063 k@[j:lessf(x,y)]
2064 t4:=ori1(lessf(x,z),eq(x,z),satz51b(j,k)):lesseq(x,z)
2065 -252
2066 satz52:=orapp(lessf(x,y),eq(x,y),lesseq(x,z),l,[t:lessf(x,y)]t4".252"(t),[t:eq(x,y)]t3".252"(t)):lesseq(x,z)
2067 trlesseq:=satz52:lesseq(x,z)
2068 +*252
2069 k@[j:lessf(x,y)]
2070 t5:=ori1(lessf(x,z),eq(x,z),satz51b(j,k)):lesseq(x,z)
2071 k@[e:eq(x,y)]
2072 t6:=eqlesseq1(y,x,z,symeq(x,y,e),k):lesseq(x,z)
2073 k@anders:=orapp(lessf(x,y),eq(x,y),lesseq(x,z),l,[t:lessf(x,y)]t5(t),[t:eq(x,y)]t6(t)):lesseq(x,z)
2074 -252
2075 z@[m:moreq(x,y)][n:moreq(y,z)]
2076 trmoreq:=satz49(z,x,satz52(z,y,x,satz48(y,z,n),satz48(x,y,m))):moreq(x,z)
2077 +253
2078 x@t1:=ismore1(pl(ts(1x,2x),ts(1x,2x)),ts(pl(1x,1x),2x),ts(1x,2x),distpt1(1x,1x,2x),satz18(ts(1x,2x),ts(1x,2x))):more(ts(pl(1x,1x),2x),ts(1x,2x))
2079 t2:=morefi2(pl(1x,1x),2x,t1):moref(fr(pl(1x,1x),2x),x)
2080 -253
2081 x@satz53:=somei(frac,[t:frac]moref(t,x),fr(pl(1x,1x),2x),t2".253"):some"l"(frac,[t:frac]moref(t,x))
2082 +254
2083 t1:=isless2(pl(ts(1x,2x),ts(1x,2x)),ts(1x,pl(2x,2x)),ts(1x,2x),distpt2(1x,2x,2x),satz18a(ts(1x,2x),ts(1x,2x))):less(ts(1x,2x),ts(1x,pl(2x,2x)))
2084 t2:=lessfi2(1x,pl(2x,2x),t1):lessf(fr(1x,pl(2x,2x)),x)
2085 -254
2086 satz54:=somei(frac,[t:frac]lessf(t,x),fr(1x,pl(2x,2x)),t2".254"):some"l"(frac,[t:frac]lessf(t,x))
2087 y@[l:lessf(x,y)]
2088 +255
2089 t1:=satz19f(ts(1x,2y),ts(1y,2x),ts(1x,2x),l):less(pl(ts(1x,2x),ts(1x,2y)),pl(ts(1x,2x),ts(1y,2x)))
2090 t2:=satz19c(ts(1x,2y),ts(1y,2x),ts(1y,2y),l):less(pl(ts(1x,2y),ts(1y,2y)),pl(ts(1y,2x),ts(1y,2y)))
2091 t3:=isless12(pl(ts(1x,2x),ts(1x,2y)),ts(1x,pl(2x,2y)),pl(ts(1x,2x),ts(1y,2x)),ts(pl(1x,1y),2x),distpt2(1x,2x,2y),distpt1(1x,1y,2x),t1):less(ts(1x,pl(2x,2y)),ts(pl(1x,1y),2x))
2092 t4:=lessfi1(pl(1x,1y),pl(2x,2y),t3):lessf(x,fr(pl(1x,1y),pl(2x,2y)))
2093 t5:=isless12(pl(ts(1x,2y),ts(1y,2y)),ts(pl(1x,1y),2y),pl(ts(1y,2x),ts(1y,2y)),ts(1y,pl(2x,2y)),distpt1(1x,1y,2y),distpt2(1y,2x,2y),t2):less(ts(pl(1x,1y),2y),ts(1y,pl(2x,2y)))
2094 t6:=lessfi2(y,pl(1x,1y),pl(2x,2y),t5):lessf(fr(pl(1x,1y),pl(2x,2y)),y)
2095 t7:=andi(lessf(x,fr(pl(1x,1y),pl(2x,2y))),lessf(fr(pl(1x,1y),pl(2x,2y)),y),t4,t6):and(lessf(x,fr(pl(1x,1y),pl(2x,2y))),lessf(fr(pl(1x,1y),pl(2x,2y)),y))
2096 -255
2097 satz55:=somei(frac,[t:frac]and(lessf(x,t),lessf(t,y)),fr(pl(1x,1y),pl(2x,2y)),t7".255"):some"l"(frac,[t:frac]and(lessf(x,t),lessf(t,y)))
2098 y@pf:=fr(pl(ts(1x,2y),ts(1y,2x)),ts(2x,2y)):frac
2099 +ii3
2100 y2@t1:=ispl12(ts(num(fr(x1,x2)),den(fr(y1,y2))),ts(x1,y2),ts(num(fr(y1,y2)),den(fr(x1,x2))),ts(y1,x2),ndis12(x1,x2,y1,y2),ndis12(y1,y2,x1,x2)):is(pl(ts(num(fr(x1,x2)),den(fr(y1,y2))),ts(num(fr(y1,y2)),den(fr(x1,x2)))),pl(ts(x1,y2),ts(y1,x2)))
2101 t2:=ists12(den(fr(x1,x2)),x2,den(fr(y1,y2)),y2,denis(x1,x2),denis(y1,y2)):is(ts(den(fr(x1,x2)),den(fr(y1,y2))),ts(x2,y2))
2102 -ii3
2103 y2@pf12:=isnd(pl(ts(num(fr(x1,x2)),den(fr(y1,y2))),ts(num(fr(y1,y2)),den(fr(x1,x2)))),ts(den(fr(x1,x2)),den(fr(y1,y2))),pl(ts(x1,y2),ts(y1,x2)),ts(x2,y2),t1".ii3",t2".ii3"):is"e"(frac,pf(fr(x1,x2),fr(y1,y2)),fr(pl(ts(x1,y2),ts(y1,x2)),ts(x2,y2)))
2104 +*ii3
2105 n2@t3:=ispl12(ts(1x,den(fr(n1,n2))),ts(1x,n2),ts(num(fr(n1,n2)),2x),ts(n1,2x),ndisn2(x,n1,n2),ndis1d(x,n1,n2)):is(pl(ts(1x,den(fr(n1,n2))),ts(num(fr(n1,n2)),2x)),pl(ts(1x,n2),ts(n1,2x)))
2106 t4:=ists2(den(fr(n1,n2)),n2,2x,denis(n1,n2)):is(ts(2x,den(fr(n1,n2))),ts(2x,n2))
2107 -ii3
2108 n2@pf1:=isnd(pl(ts(1x,den(fr(n1,n2))),ts(num(fr(n1,n2)),2x)),ts(2x,den(fr(n1,n2))),pl(ts(1x,n2),ts(n1,2x)),ts(2x,n2),t3".ii3",t4".ii3"):is"e"(frac,pf(x,fr(n1,n2)),fr(pl(ts(1x,n2),ts(n1,2x)),ts(2x,n2)))
2109 +*ii3
2110 n2@t5:=ispl12(ts(num(fr(n1,n2)),2x),ts(n1,2x),ts(1x,den(fr(n1,n2))),ts(1x,n2),ndis1d(x,n1,n2),ndisn2(x,n1,n2)):is(pl(ts(num(fr(n1,n2)),2x),ts(1x,den(fr(n1,n2)))),pl(ts(n1,2x),ts(1x,n2)))
2111 t6:=ists1(den(fr(n1,n2)),n2,2x,denis(n1,n2)):is(ts(den(fr(n1,n2)),2x),ts(n2,2x))
2112 -ii3
2113 n2@pf2:=isnd(pl(ts(num(fr(n1,n2)),2x),ts(1x,den(fr(n1,n2)))),ts(den(fr(n1,n2)),2x),pl(ts(n1,2x),ts(1x,n2)),ts(n2,2x),t5".ii3",t6".ii3"):is"e"(frac,pf(fr(n1,n2),x),fr(pl(ts(n1,2x),ts(1x,n2)),ts(n2,2x)))
2114 y2@pfeq12a:=refeq1(pf(fr(x1,x2),fr(y1,y2)),fr(pl(ts(x1,y2),ts(y1,x2)),ts(x2,y2)),pf12):eq(pf(fr(x1,x2),fr(y1,y2)),fr(pl(ts(x1,y2),ts(y1,x2)),ts(x2,y2)))
2115 pfeq12b:=refeq2(pf(fr(x1,x2),fr(y1,y2)),fr(pl(ts(x1,y2),ts(y1,x2)),ts(x2,y2)),pf12):eq(fr(pl(ts(x1,y2),ts(y1,x2)),ts(x2,y2)),pf(fr(x1,x2),fr(y1,y2)))
2116 n2@pfeq1a:=refeq1(pf(x,fr(n1,n2)),fr(pl(ts(1x,n2),ts(n1,2x)),ts(2x,n2)),pf1):eq(pf(x,fr(n1,n2)),fr(pl(ts(1x,n2),ts(n1,2x)),ts(2x,n2)))
2117 pfeq1b:=refeq2(pf(x,fr(n1,n2)),fr(pl(ts(1x,n2),ts(n1,2x)),ts(2x,n2)),pf1):eq(fr(pl(ts(1x,n2),ts(n1,2x)),ts(2x,n2)),pf(x,fr(n1,n2)))
2118 pfeq2a:=refeq1(pf(fr(n1,n2),x),fr(pl(ts(n1,2x),ts(1x,n2)),ts(n2,2x)),pf2):eq(pf(fr(n1,n2),x),fr(pl(ts(n1,2x),ts(1x,n2)),ts(n2,2x)))
2119 pfeq2b:=refeq2(pf(fr(n1,n2),x),fr(pl(ts(n1,2x),ts(1x,n2)),ts(n2,2x)),pf2):eq(fr(pl(ts(n1,2x),ts(1x,n2)),ts(n2,2x)),pf(fr(n1,n2),x))
2120 u@[e:eq(x,y)][f:eq(z,u)]
2121 +356
2122 t1:=ists1(ts(1x,2y),ts(1y,2x),ts(2z,2u),e):is(ts(ts(1x,2y),ts(2z,2u)),ts(ts(1y,2x),ts(2z,2u)))
2123 t2:=t1(z,u,x,y,f,e):is(ts(ts(1z,2u),ts(2x,2y)),ts(ts(1u,2z),ts(2x,2y)))
2124 t3:=tr3is(nat,ts(ts(1x,2z),ts(2y,2u)),ts(ts(1x,2z),ts(2u,2y)),ts(ts(1x,2y),ts(2u,2z)),ts(ts(1x,2y),ts(2z,2u)),ists2(ts(2y,2u),ts(2u,2y),ts(1x,2z),comts(2y,2u)),stets(1x,2z,2u,2y),ists2(ts(2u,2z),ts(2z,2u),ts(1x,2y),comts(2u,2z))):is(ts(ts(1x,2z),ts(2y,2u)),ts(ts(1x,2y),ts(2z,2u)))
2125 t4:=tr4is(nat,ts(ts(1x,2z),ts(2y,2u)),ts(ts(1x,2y),ts(2z,2u)),ts(ts(1y,2x),ts(2z,2u)),ts(ts(1y,2u),ts(2z,2x)),ts(ts(1y,2u),ts(2x,2z)),t3,t1,stets(1y,2x,2z,2u),ists2(ts(2z,2x),ts(2x,2z),ts(1y,2u),comts(2z,2x))):is(ts(ts(1x,2z),ts(2y,2u)),ts(ts(1y,2u),ts(2x,2z)))
2126 t5:=tr4is(nat,ts(ts(1z,2x),ts(2y,2u)),ts(ts(1z,2u),ts(2y,2x)),ts(ts(1z,2u),ts(2x,2y)),ts(ts(1u,2z),ts(2x,2y)),ts(ts(1u,2y),ts(2x,2z)),stets(1z,2x,2y,2u),ists2(ts(2y,2x),ts(2x,2y),ts(1z,2u),comts(2y,2x)),t2,stets(1u,2z,2x,2y)):is(ts(ts(1z,2x),ts(2y,2u)),ts(ts(1u,2y),ts(2x,2z)))
2127 t6:=ispl12(ts(ts(1x,2z),ts(2y,2u)),ts(ts(1y,2u),ts(2x,2z)),ts(ts(1z,2x),ts(2y,2u)),ts(ts(1u,2y),ts(2x,2z)),t4,t5):is(pl(ts(ts(1x,2z),ts(2y,2u)),ts(ts(1z,2x),ts(2y,2u))),pl(ts(ts(1y,2u),ts(2x,2z)),ts(ts(1u,2y),ts(2x,2z))))
2128 t7:=tr3is(nat,ts(pl(ts(1x,2z),ts(1z,2x)),ts(2y,2u)),pl(ts(ts(1x,2z),ts(2y,2u)),ts(ts(1z,2x),ts(2y,2u))),pl(ts(ts(1y,2u),ts(2x,2z)),ts(ts(1u,2y),ts(2x,2z))),ts(pl(ts(1y,2u),ts(1u,2y)),ts(2x,2z)),disttp1(ts(1x,2z),ts(1z,2x),ts(2y,2u)),t6,distpt1(ts(1y,2u),ts(1u,2y),ts(2x,2z))):is(ts(pl(ts(1x,2z),ts(1z,2x)),ts(2y,2u)),ts(pl(ts(1y,2u),ts(1u,2y)),ts(2x,2z)))
2129 -356
2130 satz56:=eqi12(pl(ts(1x,2z),ts(1z,2x)),ts(2x,2z),pl(ts(1y,2u),ts(1u,2y)),ts(2y,2u),t7".356"):eq(pf(x,z),pf(y,u))
2131 eqpf12:=satz56:eq(pf(x,z),pf(y,u))
2132 z@[e:eq(x,y)]
2133 eqpf1:=eqpf12(x,y,z,z,e,refeq(z)):eq(pf(x,z),pf(y,z))
2134 eqpf2:=eqpf12(z,z,x,y,refeq(z),e):eq(pf(z,x),pf(z,y))
2135 x2@[n:nat]
2136 satz57:=tr3eq(pf(fr(x1,n),fr(x2,n)),fr(pl(ts(x1,n),ts(x2,n)),ts(n,n)),fr(ts(pl(x1,x2),n),ts(n,n)),fr(pl(x1,x2),n),pfeq12a(x1,n,x2,n),eqn(pl(ts(x1,n),ts(x2,n)),ts(n,n),ts(pl(x1,x2),n),distpt1(x1,x2,n)),satz40c(pl(x1,x2),n,n)):eq(pf(fr(x1,n),fr(x2,n)),fr(pl(x1,x2),n))
2137 satz57a:=symeq(pf(fr(x1,n),fr(x2,n)),fr(pl(x1,x2),n),satz57):eq(fr(pl(x1,x2),n),pf(fr(x1,n),fr(x2,n)))
2138 y@satz58:=eqnd(pl(ts(1x,2y),ts(1y,2x)),ts(2x,2y),pl(ts(1y,2x),ts(1x,2y)),ts(2y,2x),compl(ts(1x,2y),ts(1y,2x)),comts(2x,2y)):eq(pf(x,y),pf(y,x))
2139 compf:=satz58:eq(pf(x,y),pf(y,x))
2140 +359
2141 z@t1:=tr3is(nat,ts(ts(1y,2x),2z),ts(ts(2x,1y),2z),ts(2x,ts(1y,2z)),ts(ts(1y,2z),2x),ists1(ts(1y,2x),ts(2x,1y),2z,comts(1y,2x)),assts1(2x,1y,2z),comts(2x,ts(1y,2z))):is(ts(ts(1y,2x),2z),ts(ts(1y,2z),2x))
2142 t2:=ispl12(ts(ts(1x,2y),2z),ts(1x,ts(2y,2z)),ts(ts(1y,2x),2z),ts(ts(1y,2z),2x),assts1(1x,2y,2z),t1):is(pl(ts(ts(1x,2y),2z),ts(ts(1y,2x),2z)),pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)))
2143 t3:=tris(nat,ts(pl(ts(1x,2y),ts(1y,2x)),2z),pl(ts(ts(1x,2y),2z),ts(ts(1y,2x),2z)),pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)),disttp1(ts(1x,2y),ts(1y,2x),2z),t2):is(ts(pl(ts(1x,2y),ts(1y,2x)),2z),pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)))
2144 t4:=tris(nat,ts(1z,ts(2x,2y)),ts(1z,ts(2y,2x)),ts(ts(1z,2y),2x),ists2(ts(2x,2y),ts(2y,2x),1z,comts(2x,2y)),assts2(1z,2y,2x)):is(ts(1z,ts(2x,2y)),ts(ts(1z,2y),2x))
2145 t5:=ispl12(ts(pl(ts(1x,2y),ts(1y,2x)),2z),pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)),ts(1z,ts(2x,2y)),ts(ts(1z,2y),2x),t3,t4):is(pl(ts(pl(ts(1x,2y),ts(1y,2x)),2z),ts(1z,ts(2x,2y))),pl(pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)),ts(ts(1z,2y),2x)))
2146 t6:=ispl2(pl(ts(ts(1y,2z),2x),ts(ts(1z,2y),2x)),ts(pl(ts(1y,2z),ts(1z,2y)),2x),ts(1x,ts(2y,2z)),distpt1(ts(1y,2z),ts(1z,2y),2x)):is(pl(ts(1x,ts(2y,2z)),pl(ts(ts(1y,2z),2x),ts(ts(1z,2y),2x))),pl(ts(1x,ts(2y,2z)),ts(pl(ts(1y,2z),ts(1z,2y)),2x)))
2147 t7:=tr3is(nat,pl(ts(pl(ts(1x,2y),ts(1y,2x)),2z),ts(1z,ts(2x,2y))),pl(pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)),ts(ts(1z,2y),2x)),pl(ts(1x,ts(2y,2z)),pl(ts(ts(1y,2z),2x),ts(ts(1z,2y),2x))),pl(ts(1x,ts(2y,2z)),ts(pl(ts(1y,2z),ts(1z,2y)),2x)),t5,asspl1(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x),ts(ts(1z,2y),2x)),t6):is(pl(ts(pl(ts(1x,2y),ts(1y,2x)),2z),ts(1z,ts(2x,2y))),pl(ts(1x,ts(2y,2z)),ts(pl(ts(1y,2z),ts(1z,2y)),2x)))
2148 -359
2149 z@satz59:=tr3eq(pf(pf(x,y),z),fr(pl(ts(pl(ts(1x,2y),ts(1y,2x)),2z),ts(1z,ts(2x,2y))),ts(ts(2x,2y),2z)),fr(pl(ts(1x,ts(2y,2z)),ts(pl(ts(1y,2z),ts(1z,2y)),2x)),ts(2x,ts(2y,2z))),pf(x,pf(y,z)),pfeq2a(z,pl(ts(1x,2y),ts(1y,2x)),ts(2x,2y)),eqnd(pl(ts(pl(ts(1x,2y),ts(1y,2x)),2z),ts(1z,ts(2x,2y))),ts(ts(2x,2y),2z),pl(ts(1x,ts(2y,2z)),ts(pl(ts(1y,2z),ts(1z,2y)),2x)),ts(2x,ts(2y,2z)),t7".359",assts1(2x,2y,2z)),pfeq1b(x,pl(ts(1y,2z),ts(1z,2y)),ts(2y,2z))):eq(pf(pf(x,y),z),pf(x,pf(y,z)))
2150 asspf1:=satz59:eq(pf(pf(x,y),z),pf(x,pf(y,z)))
2151 asspf2:=symeq(pf(pf(x,y),z),pf(x,pf(y,z)),asspf1):eq(pf(x,pf(y,z)),pf(pf(x,y),z))
2152 c@stets1:=tr3is(nat,ts(ts(a,b),c),ts(a,ts(b,c)),ts(a,ts(c,b)),ts(ts(a,c),b),assts1(a,b,c),ists2(ts(b,c),ts(c,b),a,comts(b,c)),assts2(a,c,b)):is(ts(ts(a,b),c),ts(ts(a,c),b))
2153 +*359
2154 z@t8:=tris(nat,ts(pl(ts(1x,2y),ts(1y,2x)),2z),pl(ts(ts(1x,2y),2z),ts(ts(1y,2x),2z)),pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)),disttp1(ts(1x,2y),ts(1y,2x),2z),ispl12(ts(ts(1x,2y),2z),ts(1x,ts(2y,2z)),ts(ts(1y,2x),2z),ts(ts(1y,2z),2x),assts1(1x,2y,2z),stets1(1y,2x,2z))):is(ts(pl(ts(1x,2y),ts(1y,2x)),2z),pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)))
2155 t9:=tris(nat,ts(1z,ts(2x,2y)),ts(ts(1z,2x),2y),ts(ts(1z,2y),2x),assts2(1z,2x,2y),stets1(1z,2x,2y)):is(ts(1z,ts(2x,2y)),ts(ts(1z,2y),2x))
2156 anderst7:=tr3is(nat,pl(ts(pl(ts(1x,2y),ts(1y,2x)),2z),ts(1z,ts(2x,2y))),pl(pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)),ts(ts(1z,2y),2x)),pl(ts(1x,ts(2y,2z)),pl(ts(ts(1y,2z),2x),ts(ts(1z,2y),2x))),pl(ts(1x,ts(2y,2z)),ts(pl(ts(1y,2z),ts(1z,2y)),2x)),ispl12(ts(pl(ts(1x,2y),ts(1y,2x)),2z),pl(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x)),ts(1z,ts(2x,2y)),ts(ts(1z,2y),2x),t8,t9),asspl1(ts(1x,ts(2y,2z)),ts(ts(1y,2z),2x),ts(ts(1z,2y),2x)),ispl2(pl(ts(ts(1y,2z),2x),ts(ts(1z,2y),2x)),ts(pl(ts(1y,2z),ts(1z,2y)),2x),ts(1x,ts(2y,2z)),distpt1(ts(1y,2z),ts(1z,2y),2x))):is(pl(ts(pl(ts(1x,2y),ts(1y,2x)),2z),ts(1z,ts(2x,2y))),pl(ts(1x,ts(2y,2z)),ts(pl(ts(1y,2z),ts(1z,2y)),2x)))
2157 -359
2158 +360
2159 y@t1:=satz18(ts(1x,2y),ts(1y,2x)):more(pl(ts(1x,2y),ts(1y,2x)),ts(1x,2y))
2160 t2:=satz32a(pl(ts(1x,2y),ts(1y,2x)),ts(1x,2y),2x,t1):more(ts(pl(ts(1x,2y),ts(1y,2x)),2x),ts(ts(1x,2y),2x))
2161 t3:=tris(nat,ts(ts(1x,2y),2x),ts(1x,ts(2y,2x)),ts(1x,ts(2x,2y)),assts1(1x,2y,2x),ists2(ts(2y,2x),ts(2x,2y),1x,comts(2y,2x))):is(ts(ts(1x,2y),2x),ts(1x,ts(2x,2y)))
2162 t4:=ismore2(ts(ts(1x,2y),2x),ts(1x,ts(2x,2y)),ts(pl(ts(1x,2y),ts(1y,2x)),2x),t3,t2):more(ts(pl(ts(1x,2y),ts(1y,2x)),2x),ts(1x,ts(2x,2y)))
2163 -360
2164 y@satz60:=morefi2(pl(ts(1x,2y),ts(1y,2x)),ts(2x,2y),t4".360"):moref(pf(x,y),x)
2165 satz60a:=satz42(pf(x,y),x,satz60):lessf(x,pf(x,y))
2166 z@[m:moref(x,y)]
2167 +361
2168 t1:=satz32a(ts(1x,2y),ts(1y,2x),2z,m):more(ts(ts(1x,2y),2z),ts(ts(1y,2x),2z))
2169 t2:=ismore12(ts(ts(1x,2y),2z),ts(ts(1x,2z),2y),ts(ts(1y,2x),2z),ts(ts(1y,2z),2x),stets1(1x,2y,2z),stets1(1y,2x,2z),t1):more(ts(ts(1x,2z),2y),ts(ts(1y,2z),2x))
2170 t3:=stets1(1z,2x,2y):is(ts(ts(1z,2x),2y),ts(ts(1z,2y),2x))
2171 t4:=satz19h(ts(ts(1z,2x),2y),ts(ts(1z,2y),2x),ts(ts(1x,2z),2y),ts(ts(1y,2z),2x),t3,t2):more(pl(ts(ts(1x,2z),2y),ts(ts(1z,2x),2y)),pl(ts(ts(1y,2z),2x),ts(ts(1z,2y),2x)))
2172 t5:=ismore12(pl(ts(ts(1x,2z),2y),ts(ts(1z,2x),2y)),ts(pl(ts(1x,2z),ts(1z,2x)),2y),pl(ts(ts(1y,2z),2x),ts(ts(1z,2y),2x)),ts(pl(ts(1y,2z),ts(1z,2y)),2x),distpt1(ts(1x,2z),ts(1z,2x),2y),distpt1(ts(1y,2z),ts(1z,2y),2x),t4):more(ts(pl(ts(1x,2z),ts(1z,2x)),2y),ts(pl(ts(1y,2z),ts(1z,2y)),2x))
2173 t6:=satz32a(ts(pl(ts(1x,2z),ts(1z,2x)),2y),ts(pl(ts(1y,2z),ts(1z,2y)),2x),2z,t5):more(ts(ts(pl(ts(1x,2z),ts(1z,2x)),2y),2z),ts(ts(pl(ts(1y,2z),ts(1z,2y)),2x),2z))
2174 t7:=ismore12(ts(ts(pl(ts(1x,2z),ts(1z,2x)),2y),2z),ts(pl(ts(1x,2z),ts(1z,2x)),ts(2y,2z)),ts(ts(pl(ts(1y,2z),ts(1z,2y)),2x),2z),ts(pl(ts(1y,2z),ts(1z,2y)),ts(2x,2z)),assts1(pl(ts(1x,2z),ts(1z,2x)),2y,2z),assts1(pl(ts(1y,2z),ts(1z,2y)),2x,2z),t6):more(ts(pl(ts(1x,2z),ts(1z,2x)),ts(2y,2z)),ts(pl(ts(1y,2z),ts(1z,2y)),ts(2x,2z)))
2175 -361
2176 satz61:=morefi12(pl(ts(1x,2z),ts(1z,2x)),ts(2x,2z),pl(ts(1y,2z),ts(1z,2y)),ts(2y,2z),t7".361"):moref(pf(x,z),pf(y,z))
2177 z@[m:moref(x,y)]
2178 satz62a:=satz61(m):moref(pf(x,z),pf(y,z))
2179 z@[e:eq(x,y)]
2180 satz62b:=eqpf1(x,y,z,e):eq(pf(x,z),pf(y,z))
2181 z@[l:lessf(x,y)]
2182 satz62c:=satz42(pf(y,z),pf(x,z),satz61(y,x,z,satz43(l))):lessf(pf(x,z),pf(y,z))
2183 m@satz62d:=eqmoref12(pf(x,z),pf(z,x),pf(y,z),pf(z,y),compf(x,z),compf(y,z),satz62a):moref(pf(z,x),pf(z,y))
2184 e@satz62e:=eqpf2(x,y,z,e):eq(pf(z,x),pf(z,y))
2185 l@satz62f:=eqlessf12(pf(x,z),pf(z,x),pf(y,z),pf(z,y),compf(x,z),compf(y,z),satz62c):lessf(pf(z,x),pf(z,y))
2186 u@[e:eq(x,y)][m:moref(z,u)]
2187 satz62g:=eqmoref2(pf(x,u),pf(y,u),pf(x,z),eqpf1(x,y,u,e),satz62d(z,u,x,m)):moref(pf(x,z),pf(y,u))
2188 satz62h:=eqmoref12(pf(x,z),pf(z,x),pf(y,u),pf(u,y),compf(x,z),compf(y,u),satz62g):moref(pf(z,x),pf(u,y))
2189 e@[l:lessf(z,u)]
2190 satz62j:=eqlessf2(pf(x,u),pf(y,u),pf(x,z),eqpf1(x,y,u,e),satz62f(z,u,x,l)):lessf(pf(x,z),pf(y,u))
2191 satz62k:=eqlessf12(pf(x,z),pf(z,x),pf(y,u),pf(u,y),compf(x,z),compf(y,u),satz62j):lessf(pf(z,x),pf(u,y))
2192 +363
2193 z@t1:=satz41a(x,y):or3(eq(x,y),moref(x,y),lessf(x,y))
2194 t2:=satz41b(pf(x,z),pf(y,z)):ec3(eq(pf(x,z),pf(y,z)),moref(pf(x,z),pf(y,z)),lessf(pf(x,z),pf(y,z)))
2195 -363
2196 z@[m:moref(pf(x,z),pf(y,z))]
2197 satz63a:=th11"l.ec3"(eq(x,y),moref(x,y),lessf(x,y),eq(pf(x,z),pf(y,z)),moref(pf(x,z),pf(y,z)),lessf(pf(x,z),pf(y,z)),t1".363",t2".363",[u:eq(x,y)]satz62b(x,y,z,u),[u:moref(x,y)]satz62a(x,y,z,u),[u:lessf(x,y)]satz62c(x,y,z,u),m):moref(x,y)
2198 z@[e:eq(pf(x,z),pf(y,z))]
2199 satz63b:=th10"l.ec3"(eq(x,y),moref(x,y),lessf(x,y),eq(pf(x,z),pf(y,z)),moref(pf(x,z),pf(y,z)),lessf(pf(x,z),pf(y,z)),t1".363",t2".363",[u:eq(x,y)]satz62b(x,y,z,u),[u:moref(x,y)]satz62a(x,y,z,u),[u:lessf(x,y)]satz62c(x,y,z,u),e):eq(x,y)
2200 z@[l:lessf(pf(x,z),pf(y,z))]
2201 satz63c:=th12"l.ec3"(eq(x,y),moref(x,y),lessf(x,y),eq(pf(x,z),pf(y,z)),moref(pf(x,z),pf(y,z)),lessf(pf(x,z),pf(y,z)),t1".363",t2".363",[u:eq(x,y)]satz62b(x,y,z,u),[u:moref(x,y)]satz62a(x,y,z,u),[u:lessf(x,y)]satz62c(x,y,z,u),l):lessf(x,y)
2202 z@[m:moref(pf(z,x),pf(z,y))]
2203 satz63d:=satz63a(eqmoref12(pf(z,x),pf(x,z),pf(z,y),pf(y,z),compf(z,x),compf(z,y),m)):moref(x,y)
2204 z@[e:eq(pf(z,x),pf(z,y))]
2205 satz63e:=satz63b(tr3eq(pf(x,z),pf(z,x),pf(z,y),pf(y,z),compf(x,z),e,compf(z,y))):eq(x,y)
2206 z@[f:lessf(pf(z,x),pf(z,y))]
2207 satz63f:=satz63c(eqlessf12(pf(z,x),pf(x,z),pf(z,y),pf(y,z),compf(z,x),compf(z,y),f)):lessf(x,y)
2208 u@[m:moref(x,y)][n:moref(z,u)]
2209 +364
2210 t1:=satz61(x,y,z,m):moref(pf(x,z),pf(y,z))
2211 t2:=eqmoref12(pf(z,y),pf(y,z),pf(u,y),pf(y,u),compf(z,y),compf(u,y),satz61(z,u,y,n)):moref(pf(y,z),pf(y,u))
2212 -364
2213 satz64:=trmoref(pf(x,z),pf(y,z),pf(y,u),t1".364",t2".364"):moref(pf(x,z),pf(y,u))
2214 u@[l:lessf(x,y)][k:lessf(z,u)]
2215 satz64a:=satz42(pf(y,u),pf(x,z),satz64(y,x,u,z,satz43(x,y,l),satz43(z,u,k))):lessf(pf(x,z),pf(y,u))
2216 u@[m:moreq(x,y)][n:moref(z,u)]
2217 satz65a:=orapp(moref(x,y),eq(x,y),moref(pf(x,z),pf(y,u)),m,[v:moref(x,y)]satz64(v,n),[v:eq(x,y)]satz62g(v,n)):moref(pf(x,z),pf(y,u))
2218 u@[m:moref(x,y)][n:moreq(z,u)]
2219 satz65b:=orapp(moref(z,u),eq(z,u),moref(pf(x,z),pf(y,u)),n,[v:moref(z,u)]satz64(m,v),[v:eq(z,u)]eqmoref2(pf(y,z),pf(y,u),pf(x,z),eqpf2(z,u,y,v),satz61(x,y,z,m))):moref(pf(x,z),pf(y,u))
2220 u@[l:lesseq(x,y)][k:lessf(z,u)]
2221 satz65c:=satz42(pf(y,u),pf(x,z),satz65a(y,x,u,z,satz49(x,y,l),satz43(z,u,k))):lessf(pf(x,z),pf(y,u))
2222 u@[l:lessf(x,y)][k:lesseq(z,u)]
2223 satz65d:=satz42(pf(y,u),pf(x,z),satz65b(y,x,u,z,satz43(x,y,l),satz49(z,u,k))):lessf(pf(x,z),pf(y,u))
2224 u@[m:moreq(x,y)][n:moreq(z,u)]
2225 +366
2226 [e:eq(x,y)][f:eq(z,u)]
2227 t1:=moreqi2(pf(x,z),pf(y,u),satz56(e,f)):moreq(pf(x,z),pf(y,u))
2228 e@[o:moref(z,u)]
2229 t2:=moreqi1(pf(x,z),pf(y,u),satz65a(m,o)):moreq(pf(x,z),pf(y,u))
2230 e@t3:=orapp(moref(z,u),eq(z,u),moreq(pf(x,z),pf(y,u)),n,[v:moref(z,u)]t2(v),[v:eq(z,u)]t1(v)):moreq(pf(x,z),pf(y,u))
2231 n@[o:moref(x,y)]
2232 t4:=moreqi1(pf(x,z),pf(y,u),satz65b(o,n)):moreq(pf(x,z),pf(y,u))
2233 -366
2234 satz66:=orapp(moref(x,y),eq(x,y),moreq(pf(x,z),pf(y,u)),m,[v:moref(x,y)]t4".366"(v),[v:eq(x,y)]t3".366"(v)):moreq(pf(x,z),pf(y,u))
2235 u@[l:lesseq(x,y)][k:lesseq(z,u)]
2236 satz66a:=satz48(pf(y,u),pf(x,z),satz66(y,x,u,z,satz49(x,y,l),satz49(z,u,k))):lesseq(pf(x,z),pf(y,u))
2237 y@[l:lesseq(x,y)]
2238 +367
2239 [v:frac][e:eq(pf(y,v),x)]
2240 t1:=eqmoref1(pf(y,v),x,y,e,satz60(y,v)):moref(x,y)
2241 v@t2:=th3"l.imp"(eq(pf(y,v),x),moref(x,y),satz41d(x,y,l),[t:eq(pf(y,v),x)]t1(t)):not(eq(pf(y,v),x))
2242 -367
2243 vorbemerkung67:=th5"l.some"(frac,[v:frac]eq(pf(y,v),x),[v:frac]t2".367"(v)):not(some"l"(frac,[t:frac]eq(pf(y,t),x)))
2244 y@[v:frac][w:frac][e:eq(pf(y,v),x)][f:eq(pf(y,w),x)]
2245 satz67b:=satz63e(v,w,y,treq2(pf(y,v),pf(y,w),x,e,f)):eq(v,w)
2246 y@[m:moref(x,y)]
2247 +*367
2248 m@t3:=onei(nat,[t:nat]diffprop(ts(1x,2y),ts(1y,2x),t),satz8b(ts(1x,2y),ts(1y,2x)),m):one([t:nat]diffprop(ts(1x,2y),ts(1y,2x),t))
2249 vo:=ind(nat,[t:nat]diffprop(ts(1x,2y),ts(1y,2x),t),t3):nat
2250 t4:=oneax(nat,[t:nat]diffprop(ts(1x,2y),ts(1y,2x),t),t3):is(ts(1x,2y),pl(ts(1y,2x),vo))
2251 w:=fr(vo,ts(2x,2y)):frac
2252 t5:=treq(y,fr(ts(1y,2x),ts(2y,2x)),fr(ts(1y,2x),ts(2x,2y)),satz40(y,2x),eqd(ts(1y,2x),ts(2y,2x),ts(2x,2y),comts(2y,2x))):eq(y,fr(ts(1y,2x),ts(2x,2y)))
2253 t6:=tr4eq(pf(y,w),pf(fr(ts(1y,2x),ts(2x,2y)),fr(vo,ts(2x,2y))),fr(pl(ts(1y,2x),vo),ts(2x,2y)),fr(ts(1x,2y),ts(2x,2y)),x,eqpf1(y,fr(ts(1y,2x),ts(2x,2y)),w,t5),satz57(ts(1y,2x),vo,ts(2x,2y)),eqn(pl(ts(1y,2x),vo),ts(2x,2y),ts(1x,2y),symis(nat,ts(1x,2y),pl(ts(1y,2x),vo),t4)),satz40a(x,2y)):eq(pf(y,w),x)
2254 -367
2255 m@satz67a:=somei(frac,[t:frac]eq(pf(y,t),x),w".367",t6".367"):some"l"(frac,[t:frac]eq(pf(y,t),x))
2256 mf:=w".367":frac
2257 satz67c:=t6".367":eq(pf(y,mf(x,y,m)),x)
2258 satz67d:=symeq(pf(y,mf(x,y,m)),x,satz67c):eq(x,pf(y,mf(x,y,m)))
2259 y@[v:frac][m:moref(x,y)][e:eq(pf(y,v),x)]
2260 satz67e:=satz67b(v,mf(x,y,m),e,satz67c(m)):eq(v,mf(x,y,m))
2261 y@tf:=fr(ts(1x,1y),ts(2x,2y)):frac
2262 +ii4
2263 y2@t1:=ists12(num(fr(x1,x2)),x1,num(fr(y1,y2)),y1,numis(x1,x2),numis(y1,y2)):is(ts(num(fr(x1,x2)),num(fr(y1,y2))),ts(x1,y1))
2264 t2:=ists12(den(fr(x1,x2)),x2,den(fr(y1,y2)),y2,denis(x1,x2),denis(y1,y2)):is(ts(den(fr(x1,x2)),den(fr(y1,y2))),ts(x2,y2))
2265 -ii4
2266 y2@tf12:=isnd(ts(num(fr(x1,x2)),num(fr(y1,y2))),ts(den(fr(x1,x2)),den(fr(y1,y2))),ts(x1,y1),ts(x2,y2),t1".ii4",t2".ii4"):is"e"(frac,tf(fr(x1,x2),fr(y1,y2)),fr(ts(x1,y1),ts(x2,y2)))
2267 +*ii4
2268 n2@t3:=ists2(num(fr(n1,n2)),n1,1x,numis(n1,n2)):is(ts(1x,num(fr(n1,n2))),ts(1x,n1))
2269 t4:=ists2(den(fr(n1,n2)),n2,2x,denis(n1,n2)):is(ts(2x,den(fr(n1,n2))),ts(2x,n2))
2270 -ii4
2271 n2@tf1:=isnd(ts(1x,num(fr(n1,n2))),ts(2x,den(fr(n1,n2))),ts(1x,n1),ts(2x,n2),t3".ii4",t4".ii4"):is"e"(frac,tf(x,fr(n1,n2)),fr(ts(1x,n1),ts(2x,n2)))
2272 +*ii4
2273 n2@t5:=ists1(num(fr(n1,n2)),n1,1x,numis(n1,n2)):is(ts(num(fr(n1,n2)),1x),ts(n1,1x))
2274 t6:=ists1(den(fr(n1,n2)),n2,2x,denis(n1,n2)):is(ts(den(fr(n1,n2)),2x),ts(n2,2x))
2275 -ii4
2276 n2@tf2:=isnd(ts(num(fr(n1,n2)),1x),ts(den(fr(n1,n2)),2x),ts(n1,1x),ts(n2,2x),t5".ii4",t6".ii4"):is"e"(frac,tf(fr(n1,n2),x),fr(ts(n1,1x),ts(n2,2x)))
2277 y2@tfeq12a:=refeq1(tf(fr(x1,x2),fr(y1,y2)),fr(ts(x1,y1),ts(x2,y2)),tf12):eq(tf(fr(x1,x2),fr(y1,y2)),fr(ts(x1,y1),ts(x2,y2)))
2278 tfeq12b:=refeq2(tf(fr(x1,x2),fr(y1,y2)),fr(ts(x1,y1),ts(x2,y2)),tf12):eq(fr(ts(x1,y1),ts(x2,y2)),tf(fr(x1,x2),fr(y1,y2)))
2279 n2@tfeq1a:=refeq1(tf(x,fr(n1,n2)),fr(ts(1x,n1),ts(2x,n2)),tf1):eq(tf(x,fr(n1,n2)),fr(ts(1x,n1),ts(2x,n2)))
2280 tfeq1b:=refeq2(tf(x,fr(n1,n2)),fr(ts(1x,n1),ts(2x,n2)),tf1):eq(fr(ts(1x,n1),ts(2x,n2)),tf(x,fr(n1,n2)))
2281 tfeq2a:=refeq1(tf(fr(n1,n2),x),fr(ts(n1,1x),ts(n2,2x)),tf2):eq(tf(fr(n1,n2),x),fr(ts(n1,1x),ts(n2,2x)))
2282 tfeq2b:=refeq2(tf(fr(n1,n2),x),fr(ts(n1,1x),ts(n2,2x)),tf2):eq(fr(ts(n1,1x),ts(n2,2x)),tf(fr(n1,n2),x))
2283 u@[e:eq(x,y)][f:eq(z,u)]
2284 +468
2285 t1:=ists12(ts(1x,2y),ts(1y,2x),ts(1z,2u),ts(1u,2z),e,f):is(ts(ts(1x,2y),ts(1z,2u)),ts(ts(1y,2x),ts(1u,2z)))
2286 -468
2287 d@stets2:=tr3is(nat,ts(ts(a,b),ts(c,d)),ts(ts(a,b),ts(d,c)),ts(ts(a,c),ts(d,b)),ts(ts(a,c),ts(b,d)),ists2(ts(c,d),ts(d,c),ts(a,b),comts(c,d)),stets(a,b,d,c),ists2(ts(d,b),ts(b,d),ts(a,c),comts(d,b))):is(ts(ts(a,b),ts(c,d)),ts(ts(a,c),ts(b,d)))
2288 +*468
2289 f@t2:=tr3is(nat,ts(ts(1x,1z),ts(2y,2u)),ts(ts(1x,2y),ts(1z,2u)),ts(ts(1y,2x),ts(1u,2z)),ts(ts(1y,1u),ts(2x,2z)),stets2(1x,1z,2y,2u),t1,stets2(1y,2x,1u,2z)):is(ts(ts(1x,1z),ts(2y,2u)),ts(ts(1y,1u),ts(2x,2z)))
2290 -468
2291 f@satz68:=eqi12(ts(1x,1z),ts(2x,2z),ts(1y,1u),ts(2y,2u),t2".468"):eq(tf(x,z),tf(y,u))
2292 eqtf12:=satz68:eq(tf(x,z),tf(y,u))
2293 z@[e:eq(x,y)]
2294 eqtf1:=eqtf12(x,y,z,z,e,refeq(z)):eq(tf(x,z),tf(y,z))
2295 eqtf2:=eqtf12(z,z,x,y,refeq(z),e):eq(tf(z,x),tf(z,y))
2296 y@satz69:=eqnd(ts(1x,1y),ts(2x,2y),ts(1y,1x),ts(2y,2x),comts(1x,1y),comts(2x,2y)):eq(tf(x,y),tf(y,x))
2297 comtf:=satz69:eq(tf(x,y),tf(y,x))
2298 z@satz70:=tr3eq(tf(tf(x,y),z),fr(ts(ts(1x,1y),1z),ts(ts(2x,2y),2z)),fr(ts(1x,ts(1y,1z)),ts(2x,ts(2y,2z))),tf(x,tf(y,z)),tfeq2a(z,ts(1x,1y),ts(2x,2y)),eqnd(ts(ts(1x,1y),1z),ts(ts(2x,2y),2z),ts(1x,ts(1y,1z)),ts(2x,ts(2y,2z)),assts1(1x,1y,1z),assts1(2x,2y,2z)),tfeq1b(x,ts(1y,1z),ts(2y,2z))):eq(tf(tf(x,y),z),tf(x,tf(y,z)))
2299 asstf1:=satz70:eq(tf(tf(x,y),z),tf(x,tf(y,z)))
2300 asstf2:=symeq(tf(tf(x,y),z),tf(x,tf(y,z)),asstf1):eq(tf(x,tf(y,z)),tf(tf(x,y),z))
2301 +471
2302 t1:=tr3eq(tf(x,pf(y,z)),fr(ts(1x,pl(ts(1y,2z),ts(1z,2y))),ts(2x,ts(2y,2z))),fr(pl(ts(1x,ts(1y,2z)),ts(1x,ts(1z,2y))),ts(2x,ts(2y,2z))),pf(fr(ts(1x,ts(1y,2z)),ts(2x,ts(2y,2z))),fr(ts(1x,ts(1z,2y)),ts(2x,ts(2y,2z)))),tfeq1a(x,pl(ts(1y,2z),ts(1z,2y)),ts(2y,2z)),eqn(ts(1x,pl(ts(1y,2z),ts(1z,2y))),ts(2x,ts(2y,2z)),pl(ts(1x,ts(1y,2z)),ts(1x,ts(1z,2y))),disttp2(1x,ts(1y,2z),ts(1z,2y))),satz57a(ts(1x,ts(1y,2z)),ts(1x,ts(1z,2y)),ts(2x,ts(2y,2z)))):eq(tf(x,pf(y,z)),pf(fr(ts(1x,ts(1y,2z)),ts(2x,ts(2y,2z))),fr(ts(1x,ts(1z,2y)),ts(2x,ts(2y,2z)))))
2303 t2:=treq(fr(ts(1x,ts(1y,2z)),ts(2x,ts(2y,2z))),fr(ts(ts(1x,1y),2z),ts(ts(2x,2y),2z)),tf(x,y),eqnd(ts(1x,ts(1y,2z)),ts(2x,ts(2y,2z)),ts(ts(1x,1y),2z),ts(ts(2x,2y),2z),assts2(1x,1y,2z),assts2(2x,2y,2z)),satz40c(ts(1x,1y),ts(2x,2y),2z)):eq(fr(ts(1x,ts(1y,2z)),ts(2x,ts(2y,2z))),tf(x,y))
2304 t3:=treq(fr(ts(1x,ts(1z,2y)),ts(2x,ts(2y,2z))),fr(ts(1x,ts(1z,2y)),ts(2x,ts(2z,2y))),tf(x,z),eqd(ts(1x,ts(1z,2y)),ts(2x,ts(2y,2z)),ts(2x,ts(2z,2y)),ists2(ts(2y,2z),ts(2z,2y),2x,comts(2y,2z))),t2(x,z,y)):eq(fr(ts(1x,ts(1z,2y)),ts(2x,ts(2y,2z))),tf(x,z))
2305 -471
2306 satz71:=treq(tf(x,pf(y,z)),pf(fr(ts(1x,ts(1y,2z)),ts(2x,ts(2y,2z))),fr(ts(1x,ts(1z,2y)),ts(2x,ts(2y,2z)))),pf(tf(x,y),tf(x,z)),t1".471",eqpf12(fr(ts(1x,ts(1y,2z)),ts(2x,ts(2y,2z))),tf(x,y),fr(ts(1x,ts(1z,2y)),ts(2x,ts(2y,2z))),tf(x,z),t2".471",t3".471")):eq(tf(x,pf(y,z)),pf(tf(x,y),tf(x,z)))
2307 disttpf1:=tr3eq(tf(pf(x,y),z),tf(z,pf(x,y)),pf(tf(z,x),tf(z,y)),pf(tf(x,z),tf(y,z)),comtf(pf(x,y),z),satz71(z,x,y),eqpf12(tf(z,x),tf(x,z),tf(z,y),tf(y,z),comtf(z,x),comtf(z,y))):eq(tf(pf(x,y),z),pf(tf(x,z),tf(y,z)))
2308 disttpf2:=satz71:eq(tf(x,pf(y,z)),pf(tf(x,y),tf(x,z)))
2309 distptf1:=symeq(tf(pf(x,y),z),pf(tf(x,z),tf(y,z)),disttpf1):eq(pf(tf(x,z),tf(y,z)),tf(pf(x,y),z))
2310 distptf2:=symeq(tf(x,pf(y,z)),pf(tf(x,y),tf(x,z)),disttpf2):eq(pf(tf(x,y),tf(x,z)),tf(x,pf(y,z)))
2311 [m:moref(x,y)]
2312 +472
2313 t1:=satz32a(ts(1x,2y),ts(1y,2x),ts(1z,2z),m):more(ts(ts(1x,2y),ts(1z,2z)),ts(ts(1y,2x),ts(1z,2z)))
2314 t2:=ismore12(ts(ts(1x,2y),ts(1z,2z)),ts(ts(1x,1z),ts(2y,2z)),ts(ts(1y,2x),ts(1z,2z)),ts(ts(1y,1z),ts(2x,2z)),stets2(1x,2y,1z,2z),stets2(1y,2x,1z,2z),t1):more(ts(ts(1x,1z),ts(2y,2z)),ts(ts(1y,1z),ts(2x,2z)))
2315 -472
2316 satz72a:=morefi12(ts(1x,1z),ts(2x,2z),ts(1y,1z),ts(2y,2z),t2".472"):moref(tf(x,z),tf(y,z))
2317 z@[e:eq(x,y)]
2318 satz72b:=satz68(x,y,z,z,e,refeq(z)):eq(tf(x,z),tf(y,z))
2319 z@[l:lessf(x,y)]
2320 satz72c:=satz42(tf(y,z),tf(x,z),satz72a(y,x,z,satz43(x,y,l))):lessf(tf(x,z),tf(y,z))
2321 m@satz72d:=eqmoref12(tf(x,z),tf(z,x),tf(y,z),tf(z,y),comtf(x,z),comtf(y,z),satz72a):moref(tf(z,x),tf(z,y))
2322 e@satz72e:=eqtf2(x,y,z,e):eq(tf(z,x),tf(z,y))
2323 l@satz72f:=eqlessf12(tf(x,z),tf(z,x),tf(y,z),tf(z,y),comtf(x,z),comtf(y,z),satz72c):lessf(tf(z,x),tf(z,y))
2324 u@[e:eq(x,y)][m:moref(z,u)]
2325 satz72g:=eqmoref2(tf(x,u),tf(y,u),tf(x,z),eqtf1(x,y,u,e),satz72d(z,u,x,m)):moref(tf(x,z),tf(y,u))
2326 satz72h:=eqmoref12(tf(x,z),tf(z,x),tf(y,u),tf(u,y),comtf(x,z),comtf(y,u),satz72g):moref(tf(z,x),tf(u,y))
2327 e@[l:lessf(z,u)]
2328 satz72j:=eqlessf2(tf(x,u),tf(y,u),tf(x,z),eqtf1(x,y,u,e),satz72f(z,u,x,l)):lessf(tf(x,z),tf(y,u))
2329 satz72k:=eqlessf12(tf(x,z),tf(z,x),tf(y,u),tf(u,y),comtf(x,z),comtf(y,u),satz72j):lessf(tf(z,x),tf(u,y))
2330 +473
2331 z@t1:=satz41a(x,y):or3(eq(x,y),moref(x,y),lessf(x,y))
2332 t2:=satz41b(tf(x,z),tf(y,z)):ec3(eq(tf(x,z),tf(y,z)),moref(tf(x,z),tf(y,z)),lessf(tf(x,z),tf(y,z)))
2333 -473
2334 z@[m:moref(tf(x,z),tf(y,z))]
2335 satz73a:=th11"l.ec3"(eq(x,y),moref(x,y),lessf(x,y),eq(tf(x,z),tf(y,z)),moref(tf(x,z),tf(y,z)),lessf(tf(x,z),tf(y,z)),t1".473",t2".473",[u:eq(x,y)]satz72b(x,y,z,u),[u:moref(x,y)]satz72a(x,y,z,u),[u:lessf(x,y)]satz72c(x,y,z,u),m):moref(x,y)
2336 z@[e:eq(tf(x,z),tf(y,z))]
2337 satz73b:=th10"l.ec3"(eq(x,y),moref(x,y),lessf(x,y),eq(tf(x,z),tf(y,z)),moref(tf(x,z),tf(y,z)),lessf(tf(x,z),tf(y,z)),t1".473",t2".473",[u:eq(x,y)]satz72b(x,y,z,u),[u:moref(x,y)]satz72a(x,y,z,u),[u:lessf(x,y)]satz72c(x,y,z,u),e):eq(x,y)
2338 z@[l:lessf(tf(x,z),tf(y,z))]
2339 satz73c:=th12"l.ec3"(eq(x,y),moref(x,y),lessf(x,y),eq(tf(x,z),tf(y,z)),moref(tf(x,z),tf(y,z)),lessf(tf(x,z),tf(y,z)),t1".473",t2".473",[u:eq(x,y)]satz72b(x,y,z,u),[u:moref(x,y)]satz72a(x,y,z,u),[u:lessf(x,y)]satz72c(x,y,z,u),l):lessf(x,y)
2340 z@[m:moref(tf(z,x),tf(z,y))]
2341 satz73d:=satz73a(eqmoref12(tf(z,x),tf(x,z),tf(z,y),tf(y,z),comtf(z,x),comtf(z,y),m)):moref(x,y)
2342 z@[e:eq(tf(z,x),tf(z,y))]
2343 satz73e:=satz73b(tr3eq(tf(x,z),tf(z,x),tf(z,y),tf(y,z),comtf(x,z),e,comtf(z,y))):eq(x,y)
2344 z@[l:lessf(tf(z,x),tf(z,y))]
2345 satz73f:=satz73c(eqlessf12(tf(z,x),tf(x,z),tf(z,y),tf(y,z),comtf(z,x),comtf(z,y),l)):lessf(x,y)
2346 u@[m:moref(x,y)][n:moref(z,u)]
2347 +474
2348 t1:=satz72a(x,y,z,m):moref(tf(x,z),tf(y,z))
2349 t2:=eqmoref12(tf(z,y),tf(y,z),tf(u,y),tf(y,u),comtf(z,y),comtf(u,y),satz72a(z,u,y,n)):moref(tf(y,z),tf(y,u))
2350 -474
2351 satz74:=trmoref(tf(x,z),tf(y,z),tf(y,u),t1".474",t2".474"):moref(tf(x,z),tf(y,u))
2352 u@[l:lessf(x,y)][k:lessf(z,u)]
2353 satz74a:=satz42(tf(y,u),tf(x,z),satz74(y,x,u,z,satz43(x,y,l),satz43(z,u,k))):lessf(tf(x,z),tf(y,u))
2354 u@[m:moreq(x,y)][n:moref(z,u)]
2355 satz75a:=orapp(moref(x,y),eq(x,y),moref(tf(x,z),tf(y,u)),m,[v:moref(x,y)]satz74(v,n),[v:eq(x,y)]satz72g(v,n)):moref(tf(x,z),tf(y,u))
2356 u@[m:moref(x,y)][n:moreq(z,u)]
2357 satz75b:=orapp(moref(z,u),eq(z,u),moref(tf(x,z),tf(y,u)),n,[v:moref(z,u)]satz74(m,v),[v:eq(z,u)]eqmoref2(tf(y,z),tf(y,u),tf(x,z),eqtf2(z,u,y,v),satz72a(m))):moref(tf(x,z),tf(y,u))
2358 u@[l:lesseq(x,y)][k:lessf(z,u)]
2359 satz75c:=satz42(tf(y,u),tf(x,z),satz75a(y,x,u,z,satz49(x,y,l),satz43(z,u,k))):lessf(tf(x,z),tf(y,u))
2360 u@[l:lessf(x,y)][k:lesseq(z,u)]
2361 satz75d:=satz42(tf(y,u),tf(x,z),satz75b(y,x,u,z,satz43(x,y,l),satz49(z,u,k))):lessf(tf(x,z),tf(y,u))
2362 u@[m:moreq(x,y)][n:moreq(z,u)]
2363 +476
2364 [e:eq(x,y)][f:eq(z,u)]
2365 t1:=moreqi2(tf(x,z),tf(y,u),satz68(e,f)):moreq(tf(x,z),tf(y,u))
2366 e@[o:moref(z,u)]
2367 t2:=moreqi1(tf(x,z),tf(y,u),satz75a(m,o)):moreq(tf(x,z),tf(y,u))
2368 e@t3:=orapp(moref(z,u),eq(z,u),moreq(tf(x,z),tf(y,u)),n,[v:moref(z,u)]t2(v),[v:eq(z,u)]t1(v)):moreq(tf(x,z),tf(y,u))
2369 n@[o:moref(x,y)]
2370 t4:=moreqi1(tf(x,z),tf(y,u),satz75b(o,n)):moreq(tf(x,z),tf(y,u))
2371 -476
2372 satz76:=orapp(moref(x,y),eq(x,y),moreq(tf(x,z),tf(y,u)),m,[v:moref(x,y)]t4".476"(v),[v:eq(x,y)]t3".476"(v)):moreq(tf(x,z),tf(y,u))
2373 u@[l:lesseq(x,y)][k:lesseq(z,u)]
2374 satz76a:=satz48(tf(y,u),tf(x,z),satz76(y,x,u,z,satz49(x,y,l),satz49(z,u,k))):lesseq(tf(x,z),tf(y,u))
2375 y@[v:frac][w:frac][e:eq(tf(y,v),x)][f:eq(tf(y,w),x)]
2376 satz77b:=satz73e(v,w,y,treq2(tf(y,v),tf(y,w),x,e,f)):eq(v,w)
2377 +477
2378 y@v:=fr(ts(1x,2y),ts(2x,1y)):frac
2379 t1:=tr4eq(tf(y,v),tf(v,y),fr(ts(ts(1x,2y),1y),ts(ts(2x,1y),2y)),fr(ts(1x,ts(1y,2y)),ts(2x,ts(1y,2y))),x,comtf(y,v),tfeq2a(y,ts(1x,2y),ts(2x,1y)),eqnd(ts(ts(1x,2y),1y),ts(ts(2x,1y),2y),ts(1x,ts(1y,2y)),ts(2x,ts(1y,2y)),tris(nat,ts(ts(1x,2y),1y),ts(1x,ts(2y,1y)),ts(1x,ts(1y,2y)),assts1(1x,2y,1y),ists2(ts(2y,1y),ts(1y,2y),1x,comts(2y,1y))),assts1(2x,1y,2y)),satz40a(x,ts(1y,2y))):eq(tf(y,v),x)
2380 -477
2381 y@satz77a:=somei(frac,[t:frac]eq(tf(y,t),x),v".477",t1".477"):some"l"(frac,[t:frac]eq(tf(y,t),x))
2382 +rt
2383 @eq:=[x:frac][y:frac]eq"n"(x,y):[x:frac][y:frac]'prop'
2384 refeq:=[x:frac]refeq"n"(x):[x:frac]<x><x>eq
2385 symeq:=[x:frac][y:frac][t:<y><x>eq]symeq"n"(x,y,t):[x:frac][y:frac][t:<y><x>eq]<x><y>eq
2386 treq:=[x:frac][y:frac][z:frac][t:<y><x>eq][u:<z><y>eq]treq"n"(x,y,z,t,u):[x:frac][y:frac][z:frac][t:<y><x>eq][u:<z><y>eq]<z><x>eq
2387 [x:frac][s:set(frac)]
2388 inf:=esti(frac,x,s):'prop'
2389 @rat:=ect"eq"(frac,eq,refeq,symeq,treq):'type'
2390 [x0:rat][y0:rat]
2391 is:=is"e"(rat,x0,y0):'prop'
2392 nis:=not(is(x0,y0)):'prop'
2393 @[p:[x:rat]'prop']
2394 some:=some"l"(rat,p):'prop'
2395 all:=all"l"(rat,p):'prop'
2396 one:=one"e"(rat,p):'prop'
2397 x0@[s:set(rat)]
2398 in:=esti(rat,x0,s):'prop'
2399 x@ratof:=ectelt"eq"(frac,eq,refeq,symeq,treq,x):rat
2400 x0@class:=ecect"eq"(frac,eq,refeq,symeq,treq,x0):set(frac)
2401 x@inclass:=th5"eq.4"(frac,eq,refeq,symeq,treq,x):inf(x,class(ratof(x)))
2402 x0@[x:frac][y:frac][xix0:inf(x,class(x0))][e:eq"n"(x,y)]
2403 lemmaeq1:=th8"eq.4"(frac,eq,refeq,symeq,treq,x0,x,xix0,y,e):inf(y,class(x0))
2404 x0@[a:'prop'][a1:[x:frac][xi:inf(x,class(x0))]a]
2405 ratapp1:=th3"eq.4"(frac,eq,refeq,symeq,treq,x0,a,a1):a
2406 y0@[a:'prop'][a1:[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]a][x:frac][xix0:inf(x,class(x0))]
2407 +ii5
2408 t1:=ratapp1(y0,a,[y:frac][yi:inf(y,class(y0))]<yi><xix0><y><x>a1):a
2409 -ii5
2410 a1@ratapp2:=ratapp1(x0,a,[x:frac][xi:inf(x,class(x0))]t1".ii5"(x,xi)):a
2411 y0@[z0:rat][a:'prop'][a1:[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]a][x:frac][xix0:inf(x,class(x0))]
2412 +*ii5
2413 xix0@t2:=ratapp2(y0,z0,a,[y:frac][z:frac][yi:inf(y,class(y0))][zi:inf(z,class(z0))]<zi><yi><xix0><z><y><x>a1):a
2414 -ii5
2415 a1@ratapp3:=ratapp1(x0,a,[x:frac][xi:inf(x,class(x0))]t2".ii5"(x,xi)):a
2416 z0@[u0:rat][a:'prop'][a1:[x:frac][y:frac][z:frac][u:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]a][x:frac][xix0:inf(x,class(x0))]
2417 +*ii5
2418 xix0@t3:=ratapp3(y0,z0,u0,a,[y:frac][z:frac][u:frac][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]<ui><zi><yi><xix0><u><z><y><x>a1):a
2419 -ii5
2420 a1@ratapp4:=ratapp1(x0,a,[x:frac][xi:inf(x,class(x0))]t3".ii5"(x,xi)):a
2421 y0@[x1:frac][y1:frac][x1ix0:inf(x1,class(x0))][y1iy0:inf(y1,class(y0))][e:eq"n"(x1,y1)]
2422 isi:=th3"eq.5"(frac,eq,refeq,symeq,treq,x0,y0,x1,x1ix0,y1,y1iy0,e):is(x0,y0)
2423 y1iy0@[i:is(x0,y0)]
2424 ise:=th5"eq.5"(frac,eq,refeq,symeq,treq,x0,y0,x1,x1ix0,y1,y1iy0,i):eq"n"(x1,y1)
2425 y1iy0@[n:not(eq"n"(x1,y1))]
2426 nisi:=th3"l.imp"(is(x0,y0),eq"n"(x1,y1),n,[t:is(x0,y0)]ise(t)):nis(x0,y0)
2427 y1iy0@[n:nis(x0,y0)]
2428 nise:=th3"l.imp"(eq"n"(x1,y1),is(x0,y0),n,[t:eq"n"(x1,y1)]isi(t)):not(eq"n"(x1,y1))
2429 @[alpha:'type'][f:[x:frac][y:frac]alpha]
2430 fixf:=fixfu2"eq"(frac,eq,refeq,symeq,treq,alpha,f):'prop'
2431 y0@[alpha:'type'][f:[x:frac][y:frac]alpha][ff:fixf(alpha,f)]
2432 indrat:=indeq2"eq"(frac,eq,refeq,symeq,treq,alpha,f,ff,x0,y0):alpha
2433 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))]
2434 isindrat:=th1"eq.11"(frac,eq,refeq,symeq,treq,alpha,f,ff,x0,y0,x,xix0,y,yiy0):is"e"(alpha,<y><x>f,indrat)
2435 x0@satz78:=refis(rat,x0):is(x0,x0)
2436 y0@[i:is(x0,y0)]
2437 satz79:=symis(rat,x0,y0,i):is(y0,x0)
2438 z0@[i:is(x0,y0)][j:is(y0,z0)]
2439 satz80:=tris(rat,x0,y0,z0,i,j):is(x0,z0)
2440 y0@more:=some"l"(frac,[x:frac]some"l"(frac,[y:frac]and3(inf(x,class(x0)),inf(y,class(y0)),moref(x,y)))):'prop'
2441 +*ii5
2442 y1@propm:=and3(inf(x1,class(x0)),inf(y1,class(y0)),moref(x1,y1)):'prop'
2443 -ii5
2444 y0@[m:more(x0,y0)][x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][t:frac][s:some"l"(frac,[u:frac]propm".ii5"(t,u))][u:frac][p:propm".ii5"(t,u)]
2445 +*ii5
2446 p@t4:=and3e1(inf(t,class(x0)),inf(u,class(y0)),moref(t,u),p):inf(t,class(x0))
2447 t5:=and3e2(inf(t,class(x0)),inf(u,class(y0)),moref(t,u),p):inf(u,class(y0))
2448 t6:=and3e3(inf(t,class(x0)),inf(u,class(y0)),moref(t,u),p):moref(t,u)
2449 t7:=satz44(t,u,x,y,t6,ise(x0,x0,t,x,t4,xix0,refis(rat,x0)),ise(y0,y0,u,y,t5,yiy0,refis(rat,y0))):moref(x,y)
2450 s@t8:=someapp(frac,[u:frac]propm(t,u),s,moref(x,y),[u:frac][v:propm(t,u)]t7(u,v)):moref(x,y)
2451 -ii5
2452 yiy0@also18:=someapp(frac,[t:frac]some"l"(frac,[u:frac]propm".ii5"(t,u)),m,moref(x,y),[t:frac][v:some"l"(frac,[u:frac]propm".ii5"(t,u))]t8".ii5"(t,v)):moref(x,y)
2453 y1iy0@[m:moref(x1,y1)]
2454 +*ii5
2455 m@t9:=and3i(inf(x1,class(x0)),inf(y1,class(y0)),moref(x1,y1),x1ix0,y1iy0,m):propm(x1,y1)
2456 t10:=somei(frac,[t:frac]propm(x1,t),y1,t9):some"l"(frac,[t:frac]propm(x1,t))
2457 -ii5
2458 m@morei:=somei(frac,[u:frac]some"l"(frac,[t:frac]propm".ii5"(u,t)),x1,t10".ii5"):more(x0,y0)
2459 y1iy0@[m:more(x0,y0)]
2460 moree:=also18(m,x1,y1,x1ix0,y1iy0):moref(x1,y1)
2461 z0@[i:is(x0,y0)][m:more(x0,z0)]
2462 ismore1:=isp(rat,[t:rat]more(t,z0),x0,y0,m,i):more(y0,z0)
2463 i@[m:more(z0,x0)]
2464 ismore2:=isp(rat,[t:rat]more(z0,t),x0,y0,m,i):more(z0,y0)
2465 u0@[i:is(x0,y0)][j:is(z0,u0)][m:more(x0,z0)]
2466 ismore12:=ismore2(z0,u0,y0,j,ismore1(x0,y0,z0,i,m)):more(y0,u0)
2467 y0@less:=some"l"(frac,[x:frac]some"l"(frac,[y:frac]and3(inf(x,class(x0)),inf(y,class(y0)),lessf(x,y)))):'prop'
2468 +*ii5
2469 y1@propl:=and3(inf(x1,class(x0)),inf(y1,class(y0)),lessf(x1,y1)):'prop'
2470 -ii5
2471 y0@[l:less(x0,y0)][x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][t:frac][s:some"l"(frac,[u:frac]propl".ii5"(t,u))][u:frac][p:propl".ii5"(t,u)]
2472 +*ii5
2473 p@t11:=and3e1(inf(t,class(x0)),inf(u,class(y0)),lessf(t,u),p):inf(t,class(x0))
2474 t12:=and3e2(inf(t,class(x0)),inf(u,class(y0)),lessf(t,u),p):inf(u,class(y0))
2475 t13:=and3e3(inf(t,class(x0)),inf(u,class(y0)),lessf(t,u),p):lessf(t,u)
2476 t14:=satz45(t,u,x,y,t13,ise(x0,x0,t,x,t11,xix0,refis(rat,x0)),ise(y0,y0,u,y,t12,yiy0,refis(rat,y0))):lessf(x,y)
2477 s@t15:=someapp(frac,[u:frac]propl(t,u),s,lessf(x,y),[u:frac][v:propl(t,u)]t14(u,v)):lessf(x,y)
2478 -ii5
2479 yiy0@also19:=someapp(frac,[t:frac]some"l"(frac,[u:frac]propl".ii5"(t,u)),l,lessf(x,y),[t:frac][v:some"l"(frac,[u:frac]propl".ii5"(t,u))]t15".ii5"(t,v)):lessf(x,y)
2480 y1iy0@[l:lessf(x1,y1)]
2481 +*ii5
2482 l@t16:=and3i(inf(x1,class(x0)),inf(y1,class(y0)),lessf(x1,y1),x1ix0,y1iy0,l):propl(x1,y1)
2483 t17:=somei(frac,[t:frac]propl(x1,t),y1,t16):some"l"(frac,[t:frac]propl(x1,t))
2484 -ii5
2485 l@lessi:=somei(frac,[u:frac]some"l"(frac,[t:frac]propl".ii5"(u,t)),x1,t17".ii5"):less(x0,y0)
2486 y1iy0@[l:less(x0,y0)]
2487 lesse:=also19(l,x1,y1,x1ix0,y1iy0):lessf(x1,y1)
2488 z0@[i:is(x0,y0)][l:less(x0,z0)]
2489 isless1:=isp(rat,[t:rat]less(t,z0),x0,y0,l,i):less(y0,z0)
2490 i@[l:less(z0,x0)]
2491 isless2:=isp(rat,[t:rat]less(z0,t),x0,y0,l,i):less(z0,y0)
2492 u0@[i:is(x0,y0)][j:is(z0,u0)][l:less(x0,z0)]
2493 isless12:=isless2(z0,u0,y0,j,isless1(x0,y0,z0,i,l)):less(y0,u0)
2494 +581
2495 y1iy0@t1:=satz41a(x1,y1):or3(eq"n"(x1,y1),moref(x1,y1),lessf(x1,y1))
2496 [e:eq"n"(x1,y1)]
2497 t2:=or3i1(is(x0,y0),more(x0,y0),less(x0,y0),isi(e)):or3(is(x0,y0),more(x0,y0),less(x0,y0))
2498 y1iy0@[m:moref(x1,y1)]
2499 t3:=or3i2(is(x0,y0),more(x0,y0),less(x0,y0),morei(m)):or3(is(x0,y0),more(x0,y0),less(x0,y0))
2500 y1iy0@[l:lessf(x1,y1)]
2501 t4:=or3i3(is(x0,y0),more(x0,y0),less(x0,y0),lessi(l)):or3(is(x0,y0),more(x0,y0),less(x0,y0))
2502 y1iy0@t5:=or3app(eq"n"(x1,y1),moref(x1,y1),lessf(x1,y1),or3(is(x0,y0),more(x0,y0),less(x0,y0)),t1,[t:eq"n"(x1,y1)]t2(t),[t:moref(x1,y1)]t3(t),[t:lessf(x1,y1)]t4(t)):or3(is(x0,y0),more(x0,y0),less(x0,y0))
2503 t6:=satz41b(x1,y1):ec3(eq"n"(x1,y1),moref(x1,y1),lessf(x1,y1))
2504 [i:is(x0,y0)]
2505 t7:=th3"l.imp"(more(x0,y0),moref(x1,y1),ec3e12(eq"n"(x1,y1),moref(x1,y1),lessf(x1,y1),t6,ise(i)),[t:more(x0,y0)]moree(t)):not(more(x0,y0))
2506 y1iy0@[m:more(x0,y0)]
2507 t8:=th3"l.imp"(less(x0,y0),lessf(x1,y1),ec3e23(eq"n"(x1,y1),moref(x1,y1),lessf(x1,y1),t6,moree(m)),[t:less(x0,y0)]lesse(t)):not(less(x0,y0))
2508 y1iy0@[l:less(x0,y0)]
2509 t9:=th3"l.imp"(is(x0,y0),eq"n"(x1,y1),ec3e31(eq"n"(x1,y1),moref(x1,y1),lessf(x1,y1),t6,lesse(l)),[t:is(x0,y0)]ise(t)):nis(x0,y0)
2510 y1iy0@t10:=th6"l.ec3"(is(x0,y0),more(x0,y0),less(x0,y0),th1"l.ec"(is(x0,y0),more(x0,y0),[t:is(x0,y0)]t7(t)),th1"l.ec"(more(x0,y0),less(x0,y0),[t:more(x0,y0)]t8(t)),th1"l.ec"(less(x0,y0),is(x0,y0),[t:less(x0,y0)]t9(t))):ec3(is(x0,y0),more(x0,y0),less(x0,y0))
2511 t11:=orec3i(is(x0,y0),more(x0,y0),less(x0,y0),t5,t10):orec3(is(x0,y0),more(x0,y0),less(x0,y0))
2512 -581
2513 y0@satz81:=ratapp2(orec3(is(x0,y0),more(x0,y0),less(x0,y0)),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t11".581"(x,y,xi,yi)):orec3(is(x0,y0),more(x0,y0),less(x0,y0))
2514 satz81a:=orec3e1(is(x0,y0),more(x0,y0),less(x0,y0),satz81):or3(is(x0,y0),more(x0,y0),less(x0,y0))
2515 satz81b:=orec3e2(is(x0,y0),more(x0,y0),less(x0,y0),satz81):ec3(is(x0,y0),more(x0,y0),less(x0,y0))
2516 [m:more(x0,y0)]
2517 +582
2518 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))]
2519 t1:=lessi(y0,x0,y,x,yiy0,xix0,satz42(x,y,moree(x0,y0,x,y,xix0,yiy0,m))):less(y0,x0)
2520 -582
2521 satz82:=ratapp2(less(y0,x0),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t1".582"(x,y,xi,yi)):less(y0,x0)
2522 y0@[l:less(x0,y0)]
2523 +583
2524 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))]
2525 t1:=morei(y0,x0,y,x,yiy0,xix0,satz43(x,y,lesse(x0,y0,x,y,xix0,yiy0,l))):more(y0,x0)
2526 -583
2527 satz83:=ratapp2(more(y0,x0),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t1".583"(x,y,xi,yi)):more(y0,x0)
2528 y0@moreis:=or(more(x0,y0),is(x0,y0)):'prop'
2529 [m:more(x0,y0)]
2530 moreisi1:=ori1(more,is,m):moreis(x0,y0)
2531 y0@[i:is(x0,y0)]
2532 moreisi2:=ori2(more,is,i):moreis(x0,y0)
2533 y1iy0@[m:moreq(x1,y1)]
2534 moreisi:=orapp(moref(x1,y1),eq"n"(x1,y1),moreis,m,[t:moref(x1,y1)]moreisi1(morei(t)),[t:eq"n"(x1,y1)]moreisi2(isi(t))):moreis(x0,y0)
2535 y1iy0@[m:moreis(x0,y0)]
2536 moreise:=orapp(more,is,moreq(x1,y1),m,[t:more]moreqi1(x1,y1,moree(t)),[t:is]moreqi2(x1,y1,ise(t))):moreq(x1,y1)
2537 z0@[i:is(x0,y0)][m:moreis(x0,z0)]
2538 ismoreis1:=isp(rat,[t:rat]moreis(t,z0),x0,y0,m,i):moreis(y0,z0)
2539 i@[m:moreis(z0,x0)]
2540 ismoreis2:=isp(rat,[t:rat]moreis(z0,t),x0,y0,m,i):moreis(z0,y0)
2541 u0@[i:is(x0,y0)][j:is(z0,u0)][m:moreis(x0,z0)]
2542 ismoreis12:=ismoreis2(z0,u0,y0,j,ismoreis1(x0,y0,z0,i,m)):moreis(y0,u0)
2543 y0@lessis:=or(less(x0,y0),is(x0,y0)):'prop'
2544 [l:less(x0,y0)]
2545 lessisi1:=ori1(less,is,l):lessis(x0,y0)
2546 y0@[i:is(x0,y0)]
2547 lessisi2:=ori2(less,is,i):lessis(x0,y0)
2548 y1iy0@[l:lesseq(x1,y1)]
2549 lessisi:=orapp(lessf(x1,y1),eq"n"(x1,y1),lessis,l,[t:lessf(x1,y1)]lessisi1(lessi(t)),[t:eq"n"(x1,y1)]lessisi2(isi(t))):lessis(x0,y0)
2550 y1iy0@[l:lessis(x0,y0)]
2551 lessise:=orapp(less,is,lesseq(x1,y1),l,[t:less]lesseqi1(x1,y1,lesse(t)),[t:is]lesseqi2(x1,y1,ise(t))):lesseq(x1,y1)
2552 z0@[i:is(x0,y0)][l:lessis(x0,z0)]
2553 islessis1:=isp(rat,[t:rat]lessis(t,z0),x0,y0,l,i):lessis(y0,z0)
2554 i@[l:lessis(z0,x0)]
2555 islessis2:=isp(rat,[t:rat]lessis(z0,t),x0,y0,l,i):lessis(z0,y0)
2556 u0@[i:is(x0,y0)][j:is(z0,u0)][l:lessis(x0,z0)]
2557 islessis12:=islessis2(z0,u0,y0,j,islessis1(x0,y0,z0,i,l)):lessis(y0,u0)
2558 y0@[m:moreis(x0,y0)]
2559 satz81c:=th7"l.ec3"(is(x0,y0),more(x0,y0),less(x0,y0),satz81b,comor(more(x0,y0),is(x0,y0),m)):not(less(x0,y0))
2560 y0@[l:lessis(x0,y0)]
2561 satz81d:=th9"l.ec3"(is(x0,y0),more(x0,y0),less(x0,y0),satz81b,l):not(more(x0,y0))
2562 y0@[n:not(more(x0,y0))]
2563 satz81e:=th2"l.or3"(is(x0,y0),more(x0,y0),less(x0,y0),satz81a,n):lessis(x0,y0)
2564 y0@[n:not(less(x0,y0))]
2565 satz81f:=comor(is(x0,y0),more(x0,y0),th3"l.or3"(is(x0,y0),more(x0,y0),less(x0,y0),satz81a,n)):moreis(x0,y0)
2566 y0@[m:more(x0,y0)]
2567 satz81g:=th3"l.or"(less(x0,y0),is(x0,y0),ec3e23(is(x0,y0),more(x0,y0),less(x0,y0),satz81b,m),ec3e21(is(x0,y0),more(x0,y0),less(x0,y0),satz81b,m)):not(lessis(x0,y0))
2568 y0@[l:less(x0,y0)]
2569 satz81h:=th3"l.or"(more(x0,y0),is(x0,y0),ec3e32(is(x0,y0),more(x0,y0),less(x0,y0),satz81b,l),ec3e31(is(x0,y0),more(x0,y0),less(x0,y0),satz81b,l)):not(moreis(x0,y0))
2570 y0@[n:not(moreis(x0,y0))]
2571 satz81j:=or3e3(is(x0,y0),more(x0,y0),less(x0,y0),satz81a,th5"l.or"(more(x0,y0),is(x0,y0),n),th4"l.or"(more(x0,y0),is(x0,y0),n)):less(x0,y0)
2572 y0@[n:not(lessis(x0,y0))]
2573 satz81k:=or3e2(is(x0,y0),more(x0,y0),less(x0,y0),satz81a,th4"l.or"(less(x0,y0),is(x0,y0),n),th5"l.or"(less(x0,y0),is(x0,y0),n)):more(x0,y0)
2574 y0@[m:moreis(x0,y0)]
2575 +584
2576 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))]
2577 t1:=lessisi(y0,x0,y,x,yiy0,xix0,satz48(x,y,moreise(x0,y0,x,y,xix0,yiy0,m))):lessis(y0,x0)
2578 -584
2579 satz84:=ratapp2(lessis(y0,x0),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t1".584"(x,y,xi,yi)):lessis(y0,x0)
2580 y0@[l:lessis(x0,y0)]
2581 +585
2582 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))]
2583 t1:=moreisi(y0,x0,y,x,yiy0,xix0,satz49(x,y,lessise(x0,y0,x,y,xix0,yiy0,l))):moreis(y0,x0)
2584 -585
2585 satz85:=ratapp2(moreis(y0,x0),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t1".585"(x,y,xi,yi)):moreis(y0,x0)
2586 z0@[l:less(x0,y0)][k:less(y0,z0)]
2587 +586
2588 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2589 t1:=lessi(x0,z0,x,z,xix0,ziz0,satz50(x,y,z,lesse(x0,y0,x,y,xix0,yiy0,l),lesse(y0,z0,y,z,yiy0,ziz0,k))):less(x0,z0)
2590 -586
2591 satz86:=ratapp3(less(x0,z0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t1".586"(x,y,z,xi,yi,zi)):less(x0,z0)
2592 trless:=satz86:less(x0,z0)
2593 z0@[m:more(x0,y0)][n:more(y0,z0)]
2594 trmore:=satz83(z0,x0,satz86(z0,y0,x0,satz82(y0,z0,n),satz82(m))):more(x0,z0)
2595 z0@[l:lessis(x0,y0)][k:less(y0,z0)]
2596 +587
2597 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2598 t1:=lessi(x0,z0,x,z,xix0,ziz0,satz51a(x,y,z,lessise(x0,y0,x,y,xix0,yiy0,l),lesse(y0,z0,y,z,yiy0,ziz0,k))):less(x0,z0)
2599 -587
2600 satz87a:=ratapp3(less(x0,z0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t1".587"(x,y,z,xi,yi,zi)):less(x0,z0)
2601 z0@[l:less(x0,y0)][k:lessis(y0,z0)]
2602 +*587
2603 k@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2604 t2:=lessi(x0,z0,x,z,xix0,ziz0,satz51b(x,y,z,lesse(x0,y0,x,y,xix0,yiy0,l),lessise(y0,z0,y,z,yiy0,ziz0,k))):less(x0,z0)
2605 -587
2606 k@satz87b:=ratapp3(less(x0,z0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t2".587"(x,y,z,xi,yi,zi)):less(x0,z0)
2607 z0@[m:moreis(x0,y0)][n:more(y0,z0)]
2608 satz87c:=satz83(z0,x0,satz87b(z0,y0,x0,satz82(y0,z0,n),satz84(m))):more(x0,z0)
2609 z0@[m:more(x0,y0)][n:moreis(y0,z0)]
2610 satz87d:=satz83(z0,x0,satz87a(z0,y0,x0,satz84(y0,z0,n),satz82(m))):more(x0,z0)
2611 z0@[l:lessis(x0,y0)][k:lessis(y0,z0)]
2612 +588
2613 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2614 t1:=lessisi(x0,z0,x,z,xix0,ziz0,satz52(x,y,z,lessise(x0,y0,x,y,xix0,yiy0,l),lessise(y0,z0,y,z,yiy0,ziz0,k))):lessis(x0,z0)
2615 -588
2616 satz88:=ratapp3(lessis(x0,z0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t1".588"(x,y,z,xi,yi,zi)):lessis(x0,z0)
2617 trlessis:=satz88:lessis(x0,z0)
2618 z0@[m:moreis(x0,y0)][n:moreis(y0,z0)]
2619 trmoreis:=satz85(z0,x0,satz88(z0,y0,x0,satz84(y0,z0,n),satz84(m))):moreis(x0,z0)
2620 +589
2621 x0@[x:frac][xix0:inf(x,class(x0))][z:frac][m:moref(z,x)]
2622 t1:=somei(rat,[t:rat]more(t,x0),ratof(z),morei(ratof(z),x0,z,x,inclass(z),xix0,m)):some([t:rat]more(t,x0))
2623 xix0@t2:=someapp(frac,[t:frac]moref(t,x),satz53(x),some([t:rat]more(t,x0)),[t:frac][u:moref(t,x)]t1(t,u)):some([t:rat]more(t,x0))
2624 -589
2625 x0@satz89:=ratapp1(some([t:rat]more(t,x0)),[x:frac][xi:inf(x,class(x0))]t2".589"(x,xi)):some([t:rat]more(t,x0))
2626 +590
2627 z"rt.589"@[l:lessf(z,x)]
2628 t1:=somei(rat,[t:rat]less(t,x0),ratof(z),lessi(ratof(z),x0,z,x,inclass(z),xix0,l)):some([t:rat]less(t,x0))
2629 xix0"rt.589"@t2:=someapp(frac,[t:frac]lessf(t,x),satz54(x),some([t:rat]less(t,x0)),[t:frac][u:lessf(t,x)]t1(t,u)):some([t:rat]less(t,x0))
2630 -590
2631 satz90:=ratapp1(some([t:rat]less(t,x0)),[x:frac][xi:inf(x,class(x0))]t2".590"(x,xi)):some([t:rat]less(t,x0))
2632 y0@[l:less(x0,y0)]
2633 +591
2634 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][z:frac][a:and(lessf(x,z),lessf(z,y))]
2635 t1:=lessi(x0,ratof(z),x,z,xix0,inclass(z),ande1(lessf(x,z),lessf(z,y),a)):less(x0,ratof(z))
2636 t2:=lessi(ratof(z),y0,z,y,inclass(z),yiy0,ande2(lessf(x,z),lessf(z,y),a)):less(ratof(z),y0)
2637 t3:=andi(less(x0,ratof(z)),less(ratof(z),y0),t1,t2):and(less(x0,ratof(z)),less(ratof(z),y0))
2638 t4:=somei(rat,[t:rat]and(less(x0,t),less(t,y0)),ratof(z),t3):some([t:rat]and(less(x0,t),less(t,y0)))
2639 yiy0@t5:=someapp(frac,[t:frac]and(lessf(x,t),lessf(t,y)),satz55(x,y,lesse(x,y,xix0,yiy0,l)),some([t:rat]and(less(x0,t),less(t,y0))),[t:frac][u:and(lessf(x,t),lessf(t,y))]t4(t,u)):some([t:rat]and(less(x0,t),less(t,y0)))
2640 -591
2641 satz91:=ratapp2(some([t:rat]and(less(x0,t),less(t,y0))),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t5".591"(x,y,xi,yi)):some([t:rat]and(less(x0,t),less(t,y0)))
2642 @plusfrt:=[x:frac][y:frac]ratof(pf(x,y)):[x:frac][y:frac]rat
2643 [x:frac][y:frac][z:frac][u:frac][e:eq"n"(x,y)][f:eq"n"(z,u)]
2644 +*ii5
2645 f@t18:=isi(ratof(pf(x,z)),ratof(pf(y,u)),pf(x,z),pf(y,u),inclass(pf(x,z)),inclass(pf(y,u)),satz56(x,y,z,u,e,f)):is(<z><x>plusfrt,<u><y>plusfrt)
2646 -ii5
2647 @fplusfrt:=[x:frac][y:frac][z:frac][u:frac][v:<y><x>eq][w:<u><z>eq]t18".ii5"(x,y,z,u,v,w):fixf(rat,plusfrt)
2648 y0@pl:=indrat(rat,plusfrt,fplusfrt):rat
2649 +*ii5
2650 y1iy0@t19:=isindrat(rat,plusfrt,fplusfrt,x1,y1,x1ix0,y1iy0):is(ratof(pf(x1,y1)),pl(x0,y0))
2651 -ii5
2652 y1iy0@picp:=isp(rat,[t:rat]inf(pf(x1,y1),class(t)),ratof(pf(x1,y1)),pl(x0,y0),inclass(pf(x1,y1)),t19".ii5"):inf(pf(x1,y1),class(pl(x0,y0)))
2653 z0@[i:is(x0,y0)]
2654 ispl1:=isf(rat,rat,[t:rat]pl(t,z0),x0,y0,i):is(pl(x0,z0),pl(y0,z0))
2655 ispl2:=isf(rat,rat,[t:rat]pl(z0,t),x0,y0,i):is(pl(z0,x0),pl(z0,y0))
2656 u0@[i:is(x0,y0)][j:is(z0,u0)]
2657 ispl12:=tris(rat,pl(x0,z0),pl(y0,z0),pl(y0,u0),ispl1(i),ispl2(z0,u0,y0,j)):is(pl(x0,z0),pl(y0,u0))
2658 +592
2659 y1iy0@t1:=isi(pl(x0,y0),pl(y0,x0),pf(x1,y1),pf(y1,x1),picp,picp(y0,x0,y1,x1,y1iy0,x1ix0),satz58(x1,y1)):is(pl(x0,y0),pl(y0,x0))
2660 -592
2661 y0@satz92:=ratapp2(is(pl(x0,y0),pl(y0,x0)),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t1".592"(x,y,xi,yi)):is(pl(x0,y0),pl(y0,x0))
2662 compl:=satz92:is(pl(x0,y0),pl(y0,x0))
2663 +593
2664 z0@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2665 t1:=picp(pl(x0,y0),z0,pf(x,y),z,picp(x0,y0,x,y,xix0,yiy0),ziz0):inf(pf(pf(x,y),z),class(pl(pl(x0,y0),z0)))
2666 t2:=picp(x0,pl(y0,z0),x,pf(y,z),xix0,picp(y0,z0,y,z,yiy0,ziz0)):inf(pf(x,pf(y,z)),class(pl(x0,pl(y0,z0))))
2667 t3:=isi(pl(pl(x0,y0),z0),pl(x0,pl(y0,z0)),pf(pf(x,y),z),pf(x,pf(y,z)),t1,t2,satz59(x,y,z)):is(pl(pl(x0,y0),z0),pl(x0,pl(y0,z0)))
2668 -593
2669 z0@satz93:=ratapp3(is(pl(pl(x0,y0),z0),pl(x0,pl(y0,z0))),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t3".593"(x,y,z,xi,yi,zi)):is(pl(pl(x0,y0),z0),pl(x0,pl(y0,z0)))
2670 asspl1:=satz93:is(pl(pl(x0,y0),z0),pl(x0,pl(y0,z0)))
2671 asspl2:=symis(rat,pl(pl(x0,y0),z0),pl(x0,pl(y0,z0)),satz93):is(pl(x0,pl(y0,z0)),pl(pl(x0,y0),z0))
2672 +594
2673 y1iy0@t1:=morei(pl(x0,y0),x0,pf(x1,y1),x1,picp,x1ix0,satz60(x1,y1)):more(pl(x0,y0),x0)
2674 -594
2675 y0@satz94:=ratapp2(more(pl(x0,y0),x0),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t1".594"(x,y,xi,yi)):more(pl(x0,y0),x0)
2676 satz94a:=satz82(pl(x0,y0),x0,satz94):less(x0,pl(x0,y0))
2677 z0@[m:more(x0,y0)]
2678 +595
2679 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2680 t1:=morei(pl(x0,z0),pl(y0,z0),pf(x,z),pf(y,z),picp(x0,z0,x,z,xix0,ziz0),picp(y0,z0,y,z,yiy0,ziz0),satz61(x,y,z,moree(x0,y0,x,y,xix0,yiy0,m))):more(pl(x0,z0),pl(y0,z0))
2681 -595
2682 satz95:=ratapp3(more(pl(x0,z0),pl(y0,z0)),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t1".595"(x,y,z,xi,yi,zi)):more(pl(x0,z0),pl(y0,z0))
2683 z0@[m:more(x0,y0)]
2684 +596
2685 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2686 t1:=morei(pl(x0,z0),pl(y0,z0),pf(x,z),pf(y,z),picp(x0,z0,x,z,xix0,ziz0),picp(y0,z0,y,z,yiy0,ziz0),satz62a(x,y,z,moree(x0,y0,x,y,xix0,yiy0,m))):more(pl(x0,z0),pl(y0,z0))
2687 -596
2688 satz96a:=ratapp3(more(pl(x0,z0),pl(y0,z0)),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t1".596"(x,y,z,xi,yi,zi)):more(pl(x0,z0),pl(y0,z0))
2689 z0@[i:is(x0,y0)]
2690 +*596
2691 i@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2692 t2:=isi(pl(x0,z0),pl(y0,z0),pf(x,z),pf(y,z),picp(x0,z0,x,z,xix0,ziz0),picp(y0,z0,y,z,yiy0,ziz0),satz62b(x,y,z,ise(x0,y0,x,y,xix0,yiy0,i))):is(pl(x0,z0),pl(y0,z0))
2693 -596
2694 i@satz96b:=ratapp3(is(pl(x0,z0),pl(y0,z0)),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t2".596"(x,y,z,xi,yi,zi)):is(pl(x0,z0),pl(y0,z0))
2695 z0@[l:less(x0,y0)]
2696 +*596
2697 l@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2698 t3:=lessi(pl(x0,z0),pl(y0,z0),pf(x,z),pf(y,z),picp(x0,z0,x,z,xix0,ziz0),picp(y0,z0,y,z,yiy0,ziz0),satz62c(x,y,z,lesse(x0,y0,x,y,xix0,yiy0,l))):less(pl(x0,z0),pl(y0,z0))
2699 -596
2700 l@satz96c:=ratapp3(less(pl(x0,z0),pl(y0,z0)),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t3".596"(x,y,z,xi,yi,zi)):less(pl(x0,z0),pl(y0,z0))
2701 +*596
2702 m@andersa:=satz95(m):more(pl(x0,z0),pl(y0,z0))
2703 i@andersb:=ispl1(x0,y0,z0,i):is(pl(x0,z0),pl(y0,z0))
2704 l@andersc:=satz82(pl(y0,z0),pl(x0,z0),satz95(y0,x0,z0,satz83(l))):less(pl(x0,z0),pl(y0,z0))
2705 -596
2706 m@satz96d:=ismore12(pl(x0,z0),pl(z0,x0),pl(y0,z0),pl(z0,y0),compl(x0,z0),compl(y0,z0),satz96a):more(pl(z0,x0),pl(z0,y0))
2707 i@satz96e:=ispl2(x0,y0,z0,i):is(pl(z0,x0),pl(z0,y0))
2708 l@satz96f:=isless12(pl(x0,z0),pl(z0,x0),pl(y0,z0),pl(z0,y0),compl(x0,z0),compl(y0,z0),satz96c):less(pl(z0,x0),pl(z0,y0))
2709 z0@[m:more(pl(x0,z0),pl(y0,z0))]
2710 +597
2711 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2712 t1:=morei(x0,y0,x,y,xix0,yiy0,satz63a(x,y,z,moree(pl(x0,z0),pl(y0,z0),pf(x,z),pf(y,z),picp(x0,z0,x,z,xix0,ziz0),picp(y0,z0,y,z,yiy0,ziz0),m))):more(x0,y0)
2713 -597
2714 satz97a:=ratapp3(more(x0,y0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t1".597"(x,y,z,xi,yi,zi)):more(x0,y0)
2715 z0@[i:is(pl(x0,z0),pl(y0,z0))]
2716 +*597
2717 i@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2718 t2:=isi(x0,y0,x,y,xix0,yiy0,satz63b(x,y,z,ise(pl(x0,z0),pl(y0,z0),pf(x,z),pf(y,z),picp(x0,z0,x,z,xix0,ziz0),picp(y0,z0,y,z,yiy0,ziz0),i))):is(x0,y0)
2719 -597
2720 i@satz97b:=ratapp3(is(x0,y0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t2".597"(x,y,z,xi,yi,zi)):is(x0,y0)
2721 z0@[l:less(pl(x0,z0),pl(y0,z0))]
2722 +*597
2723 l@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2724 t3:=lessi(x0,y0,x,y,xix0,yiy0,satz63c(x,y,z,lesse(pl(x0,z0),pl(y0,z0),pf(x,z),pf(y,z),picp(x0,z0,x,z,xix0,ziz0),picp(y0,z0,y,z,yiy0,ziz0),l))):less(x0,y0)
2725 -597
2726 l@satz97c:=ratapp3(less(x0,y0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t3".597"(x,y,z,xi,yi,zi)):less(x0,y0)
2727 +*597
2728 l@anders:=satz82(y0,x0,satz97a(y0,x0,z0,satz83(pl(x0,z0),pl(y0,z0),l))):less(x0,y0)
2729 -597
2730 u0@[m:more(x0,y0)][n:more(z0,u0)]
2731 +598
2732 [x:frac][y:frac][z:frac][u:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))][uiu0:inf(u,class(u0))]
2733 t1:=morei(pl(x0,z0),pl(y0,u0),pf(x,z),pf(y,u),picp(x0,z0,x,z,xix0,ziz0),picp(y0,u0,y,u,yiy0,uiu0),satz64(x,y,z,u,moree(x0,y0,x,y,xix0,yiy0,m),moree(z0,u0,z,u,ziz0,uiu0,n))):more(pl(x0,z0),pl(y0,u0))
2734 -598
2735 satz98:=ratapp4(more(pl(x0,z0),pl(y0,u0)),[x:frac][y:frac][z:frac][u:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]t1".598"(x,y,z,u,xi,yi,zi,ui)):more(pl(x0,z0),pl(y0,u0))
2736 u0@[l:less(x0,y0)][k:less(z0,u0)]
2737 satz98a:=satz82(pl(y0,u0),pl(x0,z0),satz98(y0,x0,u0,z0,satz83(l),satz83(z0,u0,k))):less(pl(x0,z0),pl(y0,u0))
2738 u0@[m:moreis(x0,y0)][n:more(z0,u0)]
2739 +599
2740 [x:frac][y:frac][z:frac][u:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))][uiu0:inf(u,class(u0))]
2741 t1:=morei(pl(x0,z0),pl(y0,u0),pf(x,z),pf(y,u),picp(x0,z0,x,z,xix0,ziz0),picp(y0,u0,y,u,yiy0,uiu0),satz65a(x,y,z,u,moreise(x0,y0,x,y,xix0,yiy0,m),moree(z0,u0,z,u,ziz0,uiu0,n))):more(pl(x0,z0),pl(y0,u0))
2742 -599
2743 satz99a:=ratapp4(more(pl(x0,z0),pl(y0,u0)),[x:frac][y:frac][z:frac][u:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]t1".599"(x,y,z,u,xi,yi,zi,ui)):more(pl(x0,z0),pl(y0,u0))
2744 u0@[m:more(x0,y0)][n:moreis(z0,u0)]
2745 +*599
2746 n@[x:frac][y:frac][z:frac][u:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))][uiu0:inf(u,class(u0))]
2747 t2:=morei(pl(x0,z0),pl(y0,u0),pf(x,z),pf(y,u),picp(x0,z0,x,z,xix0,ziz0),picp(y0,u0,y,u,yiy0,uiu0),satz65b(x,y,z,u,moree(x0,y0,x,y,xix0,yiy0,m),moreise(z0,u0,z,u,ziz0,uiu0,n))):more(pl(x0,z0),pl(y0,u0))
2748 -599
2749 n@satz99b:=ratapp4(more(pl(x0,z0),pl(y0,u0)),[x:frac][y:frac][z:frac][u:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]t2".599"(x,y,z,u,xi,yi,zi,ui)):more(pl(x0,z0),pl(y0,u0))
2750 u0@[l:lessis(x0,y0)][k:less(z0,u0)]
2751 satz99c:=satz82(pl(y0,u0),pl(x0,z0),satz99a(y0,x0,u0,z0,satz85(l),satz83(z0,u0,k))):less(pl(x0,z0),pl(y0,u0))
2752 u0@[l:less(x0,y0)][k:lessis(z0,u0)]
2753 satz99d:=satz82(pl(y0,u0),pl(x0,z0),satz99b(y0,x0,u0,z0,satz83(l),satz85(z0,u0,k))):less(pl(x0,z0),pl(y0,u0))
2754 u0@[m:moreis(x0,y0)][n:moreis(z0,u0)]
2755 +5100
2756 [x:frac][y:frac][z:frac][u:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))][uiu0:inf(u,class(u0))]
2757 t1:=moreisi(pl(x0,z0),pl(y0,u0),pf(x,z),pf(y,u),picp(x0,z0,x,z,xix0,ziz0),picp(y0,u0,y,u,yiy0,uiu0),satz66(x,y,z,u,moreise(x0,y0,x,y,xix0,yiy0,m),moreise(z0,u0,z,u,ziz0,uiu0,n))):moreis(pl(x0,z0),pl(y0,u0))
2758 -5100
2759 satz100:=ratapp4(moreis(pl(x0,z0),pl(y0,u0)),[x:frac][y:frac][z:frac][u:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]t1".5100"(x,y,z,u,xi,yi,zi,ui)):moreis(pl(x0,z0),pl(y0,u0))
2760 u0@[l:lessis(x0,y0)][k:lessis(z0,u0)]
2761 satz100a:=satz84(pl(y0,u0),pl(x0,z0),satz100(y0,x0,u0,z0,satz85(l),satz85(z0,u0,k))):lessis(pl(x0,z0),pl(y0,u0))
2762 y0@[l:lessis(x0,y0)]
2763 +5101
2764 [v0:rat][i:is(pl(y0,v0),x0)]
2765 t1:=ismore1(pl(y0,v0),x0,y0,i,satz94(y0,v0)):more(x0,y0)
2766 v0@t2:=th3"l.imp"(is(pl(y0,v0),x0),more(x0,y0),satz81d(x0,y0,l),[t:is(pl(y0,v0),x0)]t1(t)):nis(pl(y0,v0),x0)
2767 -5101
2768 vorbemerkung101:=th5"l.some"(rat,[v:rat]is(pl(y0,v),x0),[v:rat]t2".5101"(v)):not(some([t:rat]is(pl(y0,t),x0)))
2769 y0@[m:more(x0,y0)]
2770 +*5101
2771 m@[x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][v:frac][e:eq"n"(pf(y,v),x)]
2772 t3:=isi(pl(y0,ratof(v)),x0,pf(y,v),x,picp(y0,ratof(v),y,v,yiy0,inclass(v)),xix0,e):is(pl(y0,ratof(v)),x0)
2773 t4:=somei(rat,[t:rat]is(pl(y0,t),x0),ratof(v),t3):some([t:rat]is(pl(y0,t),x0))
2774 yiy0@t5:=someapp(frac,[t:frac]eq"n"(pf(y,t),x),satz67a(x,y,moree(x0,y0,x,y,xix0,yiy0,m)),some([t:rat]is(pl(y0,t),x0)),[t:frac][u:eq"n"(pf(y,t),x)]t4(t,u)):some([t:rat]is(pl(y0,t),x0))
2775 -5101
2776 m@satz101a:=ratapp2(some([t:rat]is(pl(y0,t),x0)),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t5".5101"(x,y,xi,yi)):some([t:rat]is(pl(y0,t),x0))
2777 y0@[v0:rat][w0:rat][i:is(pl(y0,v0),x0)][j:is(pl(y0,w0),x0)]
2778 +*5101
2779 j@[x:frac][y:frac][v:frac][w:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][viv0:inf(v,class(v0))][wiw0:inf(w,class(w0))]
2780 t6:=isi(v0,w0,v,w,viv0,wiw0,satz67b(x,y,v,w,ise(pl(y0,v0),x0,pf(y,v),x,picp(y0,v0,y,v,yiy0,viv0),xix0,i),ise(pl(y0,w0),x0,pf(y,w),x,picp(y0,w0,y,w,yiy0,wiw0),xix0,j))):is(v0,w0)
2781 -5101
2782 j@satz101b:=ratapp4(v0,w0,is(v0,w0),[x:frac][y:frac][v:frac][w:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][vi:inf(v,class(v0))][wi:inf(w,class(w0))]t6".5101"(x,y,v,w,xi,yi,vi,wi)):is(v0,w0)
2783 +*5101
2784 y0@t7:=[t:rat][u:rat][v:is(pl(y0,t),x0)][w:is(pl(y0,u),x0)]satz101b(t,u,v,w):amone(rat,[t:rat]is(pl(y0,t),x0))
2785 -5101
2786 m@satz101:=onei(rat,[t:rat]is(pl(y0,t),x0),t7".5101",satz101a):one([t:rat]is(pl(y0,t),x0))
2787 mn:=ind(rat,[t:rat]is(pl(y0,t),x0),satz101):rat
2788 satz101c:=oneax(rat,[t:rat]is(pl(y0,t),x0),satz101):is(pl(y0,mn(x0,y0,m)),x0)
2789 satz101d:=symis(rat,pl(y0,mn(x0,y0,m)),x0,satz101c):is(x0,pl(y0,mn(x0,y0,m)))
2790 satz101e:=tris(rat,pl(mn(x0,y0,m),y0),pl(y0,mn(x0,y0,m)),x0,compl(mn(x0,y0,m),y0),satz101c):is(pl(mn(x0,y0,m),y0),x0)
2791 satz101f:=symis(rat,pl(mn(x0,y0,m),y0),x0,satz101e):is(x0,pl(mn(x0,y0,m),y0))
2792 y0@[v0:rat][m:more(x0,y0)][i:is(pl(y0,v0),x0)]
2793 satz101g:=satz101b(v0,mn(x0,y0,m),i,satz101c(m)):is(v0,mn(x0,y0,m))
2794 @timesfrt:=[x:frac][y:frac]ratof(tf(x,y)):[x:frac][y:frac]rat
2795 +*ii5
2796 f@t20:=isi(ratof(tf(x,z)),ratof(tf(y,u)),tf(x,z),tf(y,u),inclass(tf(x,z)),inclass(tf(y,u)),satz68(x,y,z,u,e,f)):is(<z><x>timesfrt,<u><y>timesfrt)
2797 -ii5
2798 @ftimesfrt:=[x:frac][y:frac][z:frac][u:frac][v:<y><x>eq][w:<u><z>eq]t20".ii5"(x,y,z,u,v,w):fixf(rat,timesfrt)
2799 y0@ts:=indrat(rat,timesfrt,ftimesfrt):rat
2800 +*ii5
2801 y1iy0@t21:=isindrat(rat,timesfrt,ftimesfrt,x1,y1,x1ix0,y1iy0):is(ratof(tf(x1,y1)),ts(x0,y0))
2802 -ii5
2803 y1iy0@tict:=isp(rat,[t:rat]inf(tf(x1,y1),class(t)),ratof(tf(x1,y1)),ts(x0,y0),inclass(tf(x1,y1)),t21".ii5"):inf(tf(x1,y1),class(ts(x0,y0)))
2804 z0@[i:is(x0,y0)]
2805 ists1:=isf(rat,rat,[t:rat]ts(t,z0),x0,y0,i):is(ts(x0,z0),ts(y0,z0))
2806 ists2:=isf(rat,rat,[t:rat]ts(z0,t),x0,y0,i):is(ts(z0,x0),ts(z0,y0))
2807 u0@[i:is(x0,y0)][j:is(z0,u0)]
2808 ists12:=tris(rat,ts(x0,z0),ts(y0,z0),ts(y0,u0),ists1(i),ists2(z0,u0,y0,j)):is(ts(x0,z0),ts(y0,u0))
2809 +5102
2810 y1iy0@t1:=isi(ts(x0,y0),ts(y0,x0),tf(x1,y1),tf(y1,x1),tict,tict(y0,x0,y1,x1,y1iy0,x1ix0),satz69(x1,y1)):is(ts(x0,y0),ts(y0,x0))
2811 -5102
2812 y0@satz102:=ratapp2(is(ts(x0,y0),ts(y0,x0)),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t1".5102"(x,y,xi,yi)):is(ts(x0,y0),ts(y0,x0))
2813 comts:=satz102:is(ts(x0,y0),ts(y0,x0))
2814 +5103
2815 ziz0"rt.593"@t1:=tict(ts(x0,y0),z0,tf(x,y),z,tict(x0,y0,x,y,xix0,yiy0),ziz0):inf(tf(tf(x,y),z),class(ts(ts(x0,y0),z0)))
2816 t2:=tict(x0,ts(y0,z0),x,tf(y,z),xix0,tict(y0,z0,y,z,yiy0,ziz0)):inf(tf(x,tf(y,z)),class(ts(x0,ts(y0,z0))))
2817 t3:=isi(ts(ts(x0,y0),z0),ts(x0,ts(y0,z0)),tf(tf(x,y),z),tf(x,tf(y,z)),t1,t2,satz70(x,y,z)):is(ts(ts(x0,y0),z0),ts(x0,ts(y0,z0)))
2818 -5103
2819 z0@satz103:=ratapp3(is(ts(ts(x0,y0),z0),ts(x0,ts(y0,z0))),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t3".5103"(x,y,z,xi,yi,zi)):is(ts(ts(x0,y0),z0),ts(x0,ts(y0,z0)))
2820 assts1:=satz103:is(ts(ts(x0,y0),z0),ts(x0,ts(y0,z0)))
2821 assts2:=symis(rat,ts(ts(x0,y0),z0),ts(x0,ts(y0,z0)),satz103):is(ts(x0,ts(y0,z0)),ts(ts(x0,y0),z0))
2822 +5104
2823 ziz0"rt.593"@t1:=tict(x0,pl(y0,z0),x,pf(y,z),xix0,picp(y0,z0,y,z,yiy0,ziz0)):inf(tf(x,pf(y,z)),class(ts(x0,pl(y0,z0))))
2824 t2:=picp(ts(x0,y0),ts(x0,z0),tf(x,y),tf(x,z),tict(x0,y0,x,y,xix0,yiy0),tict(x0,z0,x,z,xix0,ziz0)):inf(pf(tf(x,y),tf(x,z)),class(pl(ts(x0,y0),ts(x0,z0))))
2825 t3:=isi(ts(x0,pl(y0,z0)),pl(ts(x0,y0),ts(x0,z0)),tf(x,pf(y,z)),pf(tf(x,y),tf(x,z)),t1,t2,satz71(x,y,z)):is(ts(x0,pl(y0,z0)),pl(ts(x0,y0),ts(x0,z0)))
2826 -5104
2827 satz104:=ratapp3(is(ts(x0,pl(y0,z0)),pl(ts(x0,y0),ts(x0,z0))),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t3".5104"(x,y,z,xi,yi,zi)):is(ts(x0,pl(y0,z0)),pl(ts(x0,y0),ts(x0,z0)))
2828 disttp1:=tr3is(rat,ts(pl(x0,y0),z0),ts(z0,pl(x0,y0)),pl(ts(z0,x0),ts(z0,y0)),pl(ts(x0,z0),ts(y0,z0)),comts(pl(x0,y0),z0),satz104(z0,x0,y0),ispl12(ts(z0,x0),ts(x0,z0),ts(z0,y0),ts(y0,z0),comts(z0,x0),comts(z0,y0))):is(ts(pl(x0,y0),z0),pl(ts(x0,z0),ts(y0,z0)))
2829 disttp2:=satz104:is(ts(x0,pl(y0,z0)),pl(ts(x0,y0),ts(x0,z0)))
2830 distpt1:=symis(rat,ts(pl(x0,y0),z0),pl(ts(x0,z0),ts(y0,z0)),disttp1):is(pl(ts(x0,z0),ts(y0,z0)),ts(pl(x0,y0),z0))
2831 distpt2:=symis(rat,ts(x0,pl(y0,z0)),pl(ts(x0,y0),ts(x0,z0)),disttp2):is(pl(ts(x0,y0),ts(x0,z0)),ts(x0,pl(y0,z0)))
2832 [m:more(x0,y0)]
2833 +5105
2834 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2835 t1:=morei(ts(x0,z0),ts(y0,z0),tf(x,z),tf(y,z),tict(x0,z0,x,z,xix0,ziz0),tict(y0,z0,y,z,yiy0,ziz0),satz72a(x,y,z,moree(x0,y0,x,y,xix0,yiy0,m))):more(ts(x0,z0),ts(y0,z0))
2836 -5105
2837 satz105a:=ratapp3(more(ts(x0,z0),ts(y0,z0)),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t1".5105"(x,y,z,xi,yi,zi)):more(ts(x0,z0),ts(y0,z0))
2838 z0@[i:is(x0,y0)]
2839 +*5105
2840 i@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2841 t2:=isi(ts(x0,z0),ts(y0,z0),tf(x,z),tf(y,z),tict(x0,z0,x,z,xix0,ziz0),tict(y0,z0,y,z,yiy0,ziz0),satz72b(x,y,z,ise(x0,y0,x,y,xix0,yiy0,i))):is(ts(x0,z0),ts(y0,z0))
2842 -5105
2843 i@satz105b:=ratapp3(is(ts(x0,z0),ts(y0,z0)),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t2".5105"(x,y,z,xi,yi,zi)):is(ts(x0,z0),ts(y0,z0))
2844 z0@[l:less(x0,y0)]
2845 +*5105
2846 l@[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]
2847 t3:=lessi(ts(x0,z0),ts(y0,z0),tf(x,z),tf(y,z),tict(x0,z0,x,z,xi,zi),tict(y0,z0,y,z,yi,zi),satz72c(x,y,z,lesse(x0,y0,x,y,xi,yi,l))):less(ts(x0,z0),ts(y0,z0))
2848 -5105
2849 l@satz105c:=ratapp3(less(ts(x0,z0),ts(y0,z0)),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t3".5105"(x,y,z,xi,yi,zi)):less(ts(x0,z0),ts(y0,z0))
2850 +*5105
2851 i@andersb:=ists1(x0,y0,z0,i):is(ts(x0,z0),ts(y0,z0))
2852 l@andersc:=satz82(ts(y0,z0),ts(x0,z0),satz105a(y0,x0,z0,satz83(l))):less(ts(x0,z0),ts(y0,z0))
2853 -5105
2854 m@satz105d:=ismore12(ts(x0,z0),ts(z0,x0),ts(y0,z0),ts(z0,y0),comts(x0,z0),comts(y0,z0),satz105a):more(ts(z0,x0),ts(z0,y0))
2855 i@satz105e:=ists2(x0,y0,z0,i):is(ts(z0,x0),ts(z0,y0))
2856 l@satz105f:=isless12(ts(x0,z0),ts(z0,x0),ts(y0,z0),ts(z0,y0),comts(x0,z0),comts(y0,z0),satz105c):less(ts(z0,x0),ts(z0,y0))
2857 z0@[m:more(ts(x0,z0),ts(y0,z0))]
2858 +5106
2859 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2860 t1:=morei(x0,y0,x,y,xix0,yiy0,satz73a(x,y,z,moree(ts(x0,z0),ts(y0,z0),tf(x,z),tf(y,z),tict(x0,z0,x,z,xix0,ziz0),tict(y0,z0,y,z,yiy0,ziz0),m))):more(x0,y0)
2861 -5106
2862 satz106a:=ratapp3(more(x0,y0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t1".5106"(x,y,z,xi,yi,zi)):more(x0,y0)
2863 z0@[i:is(ts(x0,z0),ts(y0,z0))]
2864 +*5106
2865 i@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2866 t2:=isi(x0,y0,x,y,xix0,yiy0,satz73b(x,y,z,ise(ts(x0,z0),ts(y0,z0),tf(x,z),tf(y,z),tict(x0,z0,x,z,xix0,ziz0),tict(y0,z0,y,z,yiy0,ziz0),i))):is(x0,y0)
2867 -5106
2868 i@satz106b:=ratapp3(is(x0,y0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t2".5106"(x,y,z,xi,yi,zi)):is(x0,y0)
2869 z0@[l:less(ts(x0,z0),ts(y0,z0))]
2870 +*5106
2871 l@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))]
2872 t3:=lessi(x0,y0,x,y,xix0,yiy0,satz73c(x,y,z,lesse(ts(x0,z0),ts(y0,z0),tf(x,z),tf(y,z),tict(x0,z0,x,z,xix0,ziz0),tict(y0,z0,y,z,yiy0,ziz0),l))):less(x0,y0)
2873 -5106
2874 l@satz106c:=ratapp3(less(x0,y0),[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))]t3".5106"(x,y,z,xi,yi,zi)):less(x0,y0)
2875 +*5106
2876 l@anders:=satz82(y0,x0,satz106a(y0,x0,z0,satz83(ts(x0,z0),ts(y0,z0),l))):less(x0,y0)
2877 -5106
2878 u0@[m:more(x0,y0)][n:more(z0,u0)]
2879 +5107
2880 [x:frac][y:frac][z:frac][u:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))][uiu0:inf(u,class(u0))]
2881 t1:=morei(ts(x0,z0),ts(y0,u0),tf(x,z),tf(y,u),tict(x0,z0,x,z,xix0,ziz0),tict(y0,u0,y,u,yiy0,uiu0),satz74(x,y,z,u,moree(x0,y0,x,y,xix0,yiy0,m),moree(z0,u0,z,u,ziz0,uiu0,n))):more(ts(x0,z0),ts(y0,u0))
2882 -5107
2883 satz107:=ratapp4(more(ts(x0,z0),ts(y0,u0)),[x:frac][y:frac][z:frac][u:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]t1".5107"(x,y,z,u,xi,yi,zi,ui)):more(ts(x0,z0),ts(y0,u0))
2884 u0@[l:less(x0,y0)][k:less(z0,u0)]
2885 satz107a:=satz82(ts(y0,u0),ts(x0,z0),satz107(y0,x0,u0,z0,satz83(l),satz83(z0,u0,k))):less(ts(x0,z0),ts(y0,u0))
2886 u0@[m:moreis(x0,y0)][n:more(z0,u0)]
2887 +5108
2888 [x:frac][y:frac][z:frac][u:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))][uiu0:inf(u,class(u0))]
2889 t1:=morei(ts(x0,z0),ts(y0,u0),tf(x,z),tf(y,u),tict(x0,z0,x,z,xix0,ziz0),tict(y0,u0,y,u,yiy0,uiu0),satz75a(x,y,z,u,moreise(x0,y0,x,y,xix0,yiy0,m),moree(z0,u0,z,u,ziz0,uiu0,n))):more(ts(x0,z0),ts(y0,u0))
2890 -5108
2891 satz108a:=ratapp4(more(ts(x0,z0),ts(y0,u0)),[x:frac][y:frac][z:frac][u:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]t1".5108"(x,y,z,u,xi,yi,zi,ui)):more(ts(x0,z0),ts(y0,u0))
2892 u0@[m:more(x0,y0)][n:moreis(z0,u0)]
2893 +*5108
2894 n@[x:frac][y:frac][z:frac][u:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))][uiu0:inf(u,class(u0))]
2895 t2:=morei(ts(x0,z0),ts(y0,u0),tf(x,z),tf(y,u),tict(x0,z0,x,z,xix0,ziz0),tict(y0,u0,y,u,yiy0,uiu0),satz75b(x,y,z,u,moree(x0,y0,x,y,xix0,yiy0,m),moreise(z0,u0,z,u,ziz0,uiu0,n))):more(ts(x0,z0),ts(y0,u0))
2896 -5108
2897 n@satz108b:=ratapp4(more(ts(x0,z0),ts(y0,u0)),[x:frac][y:frac][z:frac][u:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]t2".5108"(x,y,z,u,xi,yi,zi,ui)):more(ts(x0,z0),ts(y0,u0))
2898 u0@[l:lessis(x0,y0)][k:less(z0,u0)]
2899 satz108c:=satz82(ts(y0,u0),ts(x0,z0),satz108a(y0,x0,u0,z0,satz85(l),satz83(z0,u0,k))):less(ts(x0,z0),ts(y0,u0))
2900 u0@[l:less(x0,y0)][k:lessis(z0,u0)]
2901 satz108d:=satz82(ts(y0,u0),ts(x0,z0),satz108b(y0,x0,u0,z0,satz83(l),satz85(z0,u0,k))):less(ts(x0,z0),ts(y0,u0))
2902 u0@[m:moreis(x0,y0)][n:moreis(z0,u0)]
2903 +5109
2904 [x:frac][y:frac][z:frac][u:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))][uiu0:inf(u,class(u0))]
2905 t1:=moreisi(ts(x0,z0),ts(y0,u0),tf(x,z),tf(y,u),tict(x0,z0,x,z,xix0,ziz0),tict(y0,u0,y,u,yiy0,uiu0),satz76(x,y,z,u,moreise(x0,y0,x,y,xix0,yiy0,m),moreise(z0,u0,z,u,ziz0,uiu0,n))):moreis(ts(x0,z0),ts(y0,u0))
2906 -5109
2907 satz109:=ratapp4(moreis(ts(x0,z0),ts(y0,u0)),[x:frac][y:frac][z:frac][u:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))][ui:inf(u,class(u0))]t1".5109"(x,y,z,u,xi,yi,zi,ui)):moreis(ts(x0,z0),ts(y0,u0))
2908 u0@[l:lessis(x0,y0)][k:lessis(z0,u0)]
2909 satz109a:=satz84(ts(y0,u0),ts(x0,z0),satz109(y0,x0,u0,z0,satz85(l),satz85(z0,u0,k))):lessis(ts(x0,z0),ts(y0,u0))
2910 +5110
2911 y1iy0@[v:frac][e:eq"n"(tf(y1,v),x1)]
2912 t1:=isi(ts(y0,ratof(v)),x0,tf(y1,v),x1,tict(y0,ratof(v),y1,v,y1iy0,inclass(v)),x1ix0,e):is(ts(y0,ratof(v)),x0)
2913 t2:=somei(rat,[t:rat]is(ts(y0,t),x0),ratof(v),t1):some([t:rat]is(ts(y0,t),x0))
2914 y1iy0@t3:=someapp(frac,[t:frac]eq"n"(tf(y1,t),x1),satz77a(x1,y1),some([t:rat]is(ts(y0,t),x0)),[t:frac][u:eq"n"(tf(y1,t),x1)]t2(t,u)):some([t:rat]is(ts(y0,t),x0))
2915 -5110
2916 y0@satz110a:=ratapp2(some([t:rat]is(ts(y0,t),x0)),[x:frac][y:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))]t3".5110"(x,y,xi,yi)):some([t:rat]is(ts(y0,t),x0))
2917 [v0:rat][w0:rat][i:is(ts(y0,v0),x0)][j:is(ts(y0,w0),x0)]
2918 +*5110
2919 j@[x:frac][y:frac][v:frac][w:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][viv0:inf(v,class(v0))][wiw0:inf(w,class(w0))]
2920 t4:=isi(v0,w0,v,w,viv0,wiw0,satz77b(x,y,v,w,ise(ts(y0,v0),x0,tf(y,v),x,tict(y0,v0,y,v,yiy0,viv0),xix0,i),ise(ts(y0,w0),x0,tf(y,w),x,tict(y0,w0,y,w,yiy0,wiw0),xix0,j))):is(v0,w0)
2921 -5110
2922 j@satz110b:=ratapp4(v0,w0,is(v0,w0),[x:frac][y:frac][v:frac][w:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][vi:inf(v,class(v0))][wi:inf(w,class(w0))]t4".5110"(x,y,v,w,xi,yi,vi,wi)):is(v0,w0)
2923 +*5110
2924 y0@t5:=[t:rat][u:rat][v:is(ts(y0,t),x0)][w:is(ts(y0,u),x0)]satz110b(t,u,v,w):amone(rat,[t:rat]is(ts(y0,t),x0))
2925 -5110
2926 y0@satz110:=onei(rat,[t:rat]is(ts(y0,t),x0),t5".5110",satz110a):one([t:rat]is(ts(y0,t),x0))
2927 -rt
2928 @[x:nat][y:nat]
2929 +5111
2930 t1:=tris(nat,ts(num(fr(x,1)),den(fr(y,1))),ts(x,1),x,ndis12(x,1,y,1),satz28a(x)):is(ts(num(fr(x,1)),den(fr(y,1))),x)
2931 t2:=symis(nat,ts(num(fr(x,1)),den(fr(y,1))),x,t1):is(x,ts(num(fr(x,1)),den(fr(y,1))))
2932 -5111
2933 [m:moref(fr(x,1),fr(y,1))]
2934 satz111a:=ismore12(ts(num(fr(x,1)),den(fr(y,1))),x,ts(num(fr(y,1)),den(fr(x,1))),y,t1".5111",t1".5111"(y,x),m):more(x,y)
2935 y@[e:eq(fr(x,1),fr(y,1))]
2936 satz111b:=tr3is(nat,x,ts(num(fr(x,1)),den(fr(y,1))),ts(num(fr(y,1)),den(fr(x,1))),y,t2".5111",e,t1".5111"(y,x)):is(x,y)
2937 y@[l:lessf(fr(x,1),fr(y,1))]
2938 satz111c:=isless12(ts(num(fr(x,1)),den(fr(y,1))),x,ts(num(fr(y,1)),den(fr(x,1))),y,t1".5111",t1".5111"(y,x),l):less(x,y)
2939 y@[m:more(x,y)]
2940 satz111d:=ismore12(x,ts(num(fr(x,1)),den(fr(y,1))),y,ts(num(fr(y,1)),den(fr(x,1))),t2".5111",t2".5111"(y,x),m):moref(fr(x,1),fr(y,1))
2941 y@[i:is(x,y)]
2942 satz111e:=tr3is(nat,ts(num(fr(x,1)),den(fr(y,1))),x,y,ts(num(fr(y,1)),den(fr(x,1))),t1".5111",i,t2".5111"(y,x)):eq(fr(x,1),fr(y,1))
2943 y@[l:less(x,y)]
2944 satz111f:=isless12(x,ts(num(fr(x,1)),den(fr(y,1))),y,ts(num(fr(y,1)),den(fr(x,1))),t2".5111",t2".5111"(y,x),l):lessf(fr(x,1),fr(y,1))
2945 +*rt
2946 @[x0:rat][x:nat]
2947 natprop:=inf(fr(x,1),class(x0)):'prop'
2948 x0@natrt:=some"n"([t:nat]natprop(t)):'prop'
2949 +*ii5
2950 y0@[x:nat][y:nat][npx:natprop(x0,x)][npy:natprop(y0,y)][i:is(x0,y0)]
2951 t22:=satz111b(x,y,ise(x0,y0,fr(x,1),fr(y,1),npx,npy,i)):is"n"(x,y)
2952 x0@t23:=[t:nat][u:nat][v:natprop(x0,t)][w:natprop(x0,u)]t22(x0,x0,t,u,v,w,refis(rat,x0)):amone(nat,[t:nat]natprop(x0,t))
2953 -ii5
2954 x0@[nx0:natrt(x0)]
2955 satz111g:=onei(nat,[t:nat]natprop(t),t23".ii5",nx0):one"n"([t:nat]natprop(x0,t))
2956 nofrt:=ind(nat,[t:nat]natprop(t),satz111g):nat
2957 inclassn:=oneax(nat,[t:nat]natprop(t),satz111g):inf(fr(nofrt,1),class(x0))
2958 [y0:rat][ny0:natrt(y0)][i:is(x0,y0)]
2959 isrten:=t22".ii5"(x0,y0,nofrt(x0,nx0),nofrt(y0,ny0),inclassn(x0,nx0),inclassn(y0,ny0),i):is"n"(nofrt(x0,nx0),nofrt(y0,ny0))
2960 ny0@[i:is"n"(nofrt(x0,nx0),nofrt(y0,ny0))]
2961 isrtin:=isi(x0,y0,fr(nofrt(x0,nx0),1),fr(nofrt(y0,ny0),1),inclassn(x0,nx0),inclassn(y0,ny0),eqn"n"(nofrt(x0,nx0),1,nofrt(y0,ny0),i)):is(x0,y0)
2962 @[x:nat]
2963 rtofn:=ratof(fr(x,1)):rat
2964 natrti:=somei(nat,[t:nat]natprop(rtofn,t),x,inclass(fr(x,1))):natrt(rtofn(x))
2965 [y:nat][i:is"n"(x,y)]
2966 isnert:=isf(nat,rat,[t:nat]rtofn(t),x,y,i):is(rtofn(x),rtofn(y))
2967 y@[i:is(rtofn(x),rtofn(y))]
2968 isnirt:=t22".ii5"(rtofn(x),rtofn(y),x,y,inclass(fr(x,1)),inclass(fr(y,1)),i):is"n"(x,y)
2969 nx0@isrtn1:=isi(x0,rtofn(nofrt(x0,nx0)),fr(nofrt(x0,nx0),1),fr(nofrt(x0,nx0),1),inclassn(x0,nx0),inclass(fr(nofrt(x0,nx0),1)),refeq"n"(fr(nofrt(x0,nx0),1))):is(x0,rtofn(nofrt(x0,nx0)))
2970 x@isnrt1:=t22".ii5"(rtofn(x),rtofn(x),x,nofrt(rtofn(x),natrti(x)),inclass(fr(x,1)),inclassn(rtofn(x),natrti(x)),refis(rat,rtofn(x))):is"n"(x,nofrt(rtofn(x),natrti(x)))
2971 -rt
2972 @[x:nat][y:nat]
2973 satz112a:=satz57(x,y,1):eq(pf(fr(x,1),fr(y,1)),fr(pl(x,y),1))
2974 satz112b:=treq(tf(fr(x,1),fr(y,1)),fr(ts(x,y),ts(1,1)),fr(ts(x,y),1),tfeq12a(x,1,y,1),eqd(ts(x,y),ts(1,1),1,satz28a(1))):eq(tf(fr(x,1),fr(y,1)),fr(ts(x,y),1))
2975 +*rt
2976 ny0@satz112c:=lemmaeq1(pl(x0,y0),pf(fr(nofrt(x0,nx0),1),fr(nofrt(y0,ny0),1)),fr(pl"n"(nofrt(x0,nx0),nofrt(y0,ny0)),1),picp(x0,y0,fr(nofrt(x0,nx0),1),fr(nofrt(y0,ny0),1),inclassn(x0,nx0),inclassn(y0,ny0)),satz112a(nofrt(x0,nx0),nofrt(y0,ny0))):inf(fr(pl"n"(nofrt(x0,nx0),nofrt(y0,ny0)),1),class(pl(x0,y0)))
2977 satz112d:=somei(nat,[t:nat]natprop(pl(x0,y0),t),pl"n"(nofrt(x0,nx0),nofrt(y0,ny0)),satz112c):natrt(pl(x0,y0))
2978 satz112e:=lemmaeq1(ts(x0,y0),tf(fr(nofrt(x0,nx0),1),fr(nofrt(y0,ny0),1)),fr(ts"n"(nofrt(x0,nx0),nofrt(y0,ny0)),1),tict(x0,y0,fr(nofrt(x0,nx0),1),fr(nofrt(y0,ny0),1),inclassn(x0,nx0),inclassn(y0,ny0)),satz112b(nofrt(x0,nx0),nofrt(y0,ny0))):inf(fr(ts"n"(nofrt(x0,nx0),nofrt(y0,ny0)),1),class(ts(x0,y0)))
2979 satz112f:=somei(nat,[t:nat]natprop(ts(x0,y0),t),ts"n"(nofrt(x0,nx0),nofrt(y0,ny0)),satz112e):natrt(ts(x0,y0))
2980 [m:more(x0,y0)]
2981 +5112
2982 t1:=satz111a(nofrt(x0,nx0),nofrt(y0,ny0),moree(x0,y0,fr(nofrt(x0,nx0),1),fr(nofrt(y0,ny0),1),inclassn(x0,nx0),inclassn(y0,ny0),m)):more"n"(nofrt(x0,nx0),nofrt(y0,ny0))
2983 [z:nat][d:diffprop(nofrt(x0,nx0),nofrt(y0,ny0),z)]
2984 t2:=tris(nat,nofrt(x0,nx0),pl"n"(nofrt(y0,ny0),z),pl"n"(nofrt(y0,ny0),nofrt(rtofn(z),natrti(z))),d,ispl2"n"(z,nofrt(rtofn(z),natrti(z)),nofrt(y0,ny0),isnrt1(z))):is"n"(nofrt(x0,nx0),pl"n"(nofrt(y0,ny0),nofrt(rtofn(z),natrti(z))))
2985 t3:=isi(x0,pl(y0,rtofn(z)),fr(nofrt(x0,nx0),1),fr(pl"n"(nofrt(y0,ny0),nofrt(rtofn(z),natrti(z))),1),inclassn(x0,nx0),satz112c(y0,ny0,rtofn(z),natrti(z)),eqn(nofrt(x0,nx0),1,pl"n"(nofrt(y0,ny0),nofrt(rtofn(z),natrti(z))),t2)):is(x0,pl(y0,rtofn(z)))
2986 t4:=satz101g(x0,y0,rtofn(z),m,symis(rat,x0,pl(y0,rtofn(z)),t3)):is(rtofn(z),mn(x0,y0,m))
2987 t5:=isp(rat,[t:rat]natrt(t),rtofn(z),mn(x0,y0,m),natrti(z),t4):natrt(mn(x0,y0,m))
2988 -5112
2989 satz112g:=someapp(nat,[t:nat]diffprop(nofrt(x0,nx0),nofrt(y0,ny0),t),t1".5112",natrt(mn(x0,y0,m)),[t:nat][u:diffprop(nofrt(x0,nx0),nofrt(y0,ny0),t)]t5".5112"(t,u)):natrt(mn(x0,y0,m))
2990 @[x:nat][y:nat]
2991 satz112h:=isi(pl(rtofn(x),rtofn(y)),rtofn(pl"n"(x,y)),pf(fr(x,1),fr(y,1)),fr(pl"n"(x,y),1),picp(rtofn(x),rtofn(y),fr(x,1),fr(y,1),inclass(fr(x,1)),inclass(fr(y,1))),inclass(fr(pl"n"(x,y),1)),satz112a(x,y)):is(pl(rtofn(x),rtofn(y)),rtofn(pl"n"(x,y)))
2992 satz112j:=isi(ts(rtofn(x),rtofn(y)),rtofn(ts"n"(x,y)),tf(fr(x,1),fr(y,1)),fr(ts"n"(x,y),1),tict(rtofn(x),rtofn(y),fr(x,1),fr(y,1),inclass(fr(x,1)),inclass(fr(y,1))),inclass(fr(ts"n"(x,y),1)),satz112b(x,y)):is(ts(rtofn(x),rtofn(y)),rtofn(ts"n"(x,y)))
2993 +nt
2994 @natt:=ot(rat,[t:rat]natrt(t)):'type'
2995 nx0@ntofrt:=out(rat,[t:rat]natrt(t),x0,nx0):natt
2996 @[xt:natt][yt:natt]
2997 is:=is"e"(natt,xt,yt):'prop'
2998 nis:=not(is(xt,yt)):'prop'
2999 @[p:[x:natt]'prop']
3000 all:=all"l"(natt,p):'prop'
3001 some:=some"l"(natt,p):'prop'
3002 one:=one"e"(natt,p):'prop'
3003 xt@[st:set(natt)]
3004 in:=esti(natt,xt,st):'prop'
3005 xt@rtofnt:=in"e"(rat,[t:rat]natrt(t),xt):rat
3006 natrti:=inp(rat,[t:rat]natrt(t),xt):natrt(rtofnt(xt))
3007 ny0@[i:is"rt"(x0,y0)]
3008 isrtent:=isouti(rat,[t:rat]natrt(t),x0,nx0,y0,ny0,i):is(ntofrt(x0,nx0),ntofrt(y0,ny0))
3009 ny0@[i:is(ntofrt(x0,nx0),ntofrt(y0,ny0))]
3010 isrtint:=isoute(rat,[t:rat]natrt(t),x0,nx0,y0,ny0,i):is"rt"(x0,y0)
3011 yt@[i:is(xt,yt)]
3012 isntert:=isini(rat,[t:rat]natrt(t),xt,yt,i):is"rt"(rtofnt(xt),rtofnt(yt))
3013 yt@[i:is"rt"(rtofnt(xt),rtofnt(yt))]
3014 isntirt:=isine(rat,[t:rat]natrt(t),xt,yt,i):is(xt,yt)
3015 nx0@isrtnt1:=isinout(rat,[t:rat]natrt(t),x0,nx0):is"rt"(x0,rtofnt(ntofrt(x0,nx0)))
3016 xt@isntrt1:=isoutin(rat,[t:rat]natrt(t),xt):is(xt,ntofrt(rtofnt(xt),natrti(xt)))
3017 x@ntofn:=ntofrt(rtofn(x),natrti"rt"(x)):natt
3018 y@[i:is"n"(x,y)]
3019 isnent:=isrtent(rtofn(x),natrti"rt"(x),rtofn(y),natrti"rt"(y),isnert(x,y,i)):is(ntofn(x),ntofn(y))
3020 y@[i:is(ntofn(x),ntofn(y))]
3021 isnint:=isnirt(x,y,isrtint(rtofn"rt"(x),natrti"rt"(x),rtofn(y),natrti"rt"(y),i)):is"n"(x,y)
3022 xt@nofnt:=nofrt(rtofnt(xt),natrti(xt)):nat
3023 yt@[i:is(xt,yt)]
3024 isnten:=isrten(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt),isntert(xt,yt,i)):is"n"(nofnt(xt),nofnt(yt))
3025 yt@[i:is"n"(nofnt(xt),nofnt(yt))]
3026 isntin:=isntirt(xt,yt,isrtin(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt),i)):is(xt,yt)
3027 +ii5
3028 x@t24:=isrtnt1(rtofn(x),natrti"rt"(x)):is"rt"(rtofn(x),rtofnt(ntofn(x)))
3029 t25:=isrten(rtofn(x),natrti"rt"(x),rtofnt(ntofn(x)),natrti(ntofn(x)),t24):is"n"(nofrt(rtofn(x),natrti"rt"(x)),nofnt(ntofn(x)))
3030 -ii5
3031 x@isnnt1:=tris(nat,x,nofrt(rtofn(x),natrti"rt"(x)),nofnt(ntofn(x)),isnrt1(x),t25".ii5"):is"n"(x,nofnt(ntofn(x)))
3032 +*ii5
3033 xt@t26:=isrtn1(rtofnt(xt),natrti(xt)):is"rt"(rtofnt(xt),rtofn(nofnt(xt)))
3034 t27:=isrtent(rtofnt(xt),natrti(xt),rtofn(nofnt(xt)),natrti"rt"(nofnt(xt)),t26):is(ntofrt(rtofnt(xt),natrti(xt)),ntofn(nofnt(xt)))
3035 -ii5
3036 xt@isntn1:=tris(natt,xt,ntofrt(rtofnt(xt),natrti(xt)),ntofn(nofnt(xt)),isntrt1(xt),t27".ii5"):is(xt,ntofn(nofnt(xt)))
3037 x@isnnt2:=symis(nat,x,nofnt(ntofn(x)),isnnt1):is"n"(nofnt(ntofn(x)),x)
3038 xt@isntn2:=symis(natt,xt,ntofn(nofnt(xt)),isntn1):is(ntofn(nofnt(xt)),xt)
3039 @1t:=ntofn(1):natt
3040 suct:=[x:natt]ntofn(<nofnt(x)>suc):[x:natt]natt
3041 +5113
3042 xt@[i:is(<xt>suct,1t)]
3043 t1:=isnint(<nofnt(xt)>suc,1,i):is"n"(<nofnt(xt)>suc,1)
3044 -5113
3045 xt@satz113a:=th3"l.imp"(is(<xt>suct,1t),is"n"(<nofnt(xt)>suc,1),<nofnt(xt)>ax3,[t:is(<xt>suct,1t)]t1".5113"(t)):nis(<xt>suct,1t)
3046 yt@[i:is(<xt>suct,<yt>suct)]
3047 +*5113
3048 i"nt"@t2:=isnint(<nofnt(xt)>suc,<nofnt(yt)>suc,i):is"n"(<nofnt(xt)>suc,<nofnt(yt)>suc)
3049 -5113
3050 i@satz113b:=isntin(xt,yt,<t2".5113"><nofnt(yt)><nofnt(xt)>ax4):is(xt,yt)
3051 @[st:set(natt)]
3052 cond1:=in(1t,st):'prop'
3053 cond2:=all([x:natt]imp(in(x,st),in(<x>suct,st))):'prop'
3054 [c1:cond1][c2:cond2]
3055 +*5113
3056 c2@[x:nat]
3057 prop1:=in(ntofn(x),st):'prop'
3058 [p:prop1(x)]
3059 t3:=<ntofn(x)>c2:imp(in(ntofn(x),st),in(<ntofn(x)>suct,st))
3060 t4:=mp(in(ntofn(x),st),in(<ntofn(x)>suct,st),p,t3):in(<ntofn(x)>suct,st)
3061 t5:=isp(nat,[t:nat]in(ntofn(<t>suc),st),nofnt(ntofn(x)),x,t4,isnnt2(x)):prop1(<x>suc)
3062 -5113
3063 c2@[xt:natt]
3064 +*5113
3065 xt@t6:=induction([t:nat]prop1(t),c1,[t:nat][u:prop1(t)]t5(t,u),nofnt(xt)):in(ntofn(nofnt(xt)),st)
3066 -5113
3067 xt@satz113c:=isp(natt,[t:natt]in(t,st),ntofn(nofnt(xt)),xt,t6".5113",isntn2(xt)):in(xt,st)
3068 @ax3t:=[x:natt]satz113a(x):[x:natt]nis(<x>suct,1t)
3069 ax4t:=[x:natt][y:natt][u:is(<x>suct,<y>suct)]satz113b(x,y,u):[x:natt][y:natt][u:is(<x>suct,<y>suct)]is(x,y)
3070 ax5t:=[s:set(natt)][u:cond1(s)][v:cond2(s)][x:natt]satz113c(s,u,v,x):[s:set(natt)][u:cond1(s)][v:cond2(s)][x:natt]in(x,s)
3071 yt@[n:nis(xt,yt)]
3072 +51
3073 t1:=th3"l.imp"(is"n"(nofnt(xt),nofnt(yt)),is(xt,yt),n,[t:is"n"(nofnt(xt),nofnt(yt))]isntin(xt,yt,t)):nis"n"(nofnt(xt),nofnt(yt))
3074 t2:=satz1(nofnt(xt),nofnt(yt),t1):nis"n"(<nofnt(xt)>suc,<nofnt(yt)>suc)
3075 -51
3076 satz1:=th3"l.imp"(is(<xt>suct,<yt>suct),is"n"(<nofnt(xt)>suc,<nofnt(yt)>suc),t2".51",[t:is(<xt>suct,<yt>suct)]isnint(<nofnt(xt)>suc,<nofnt(yt)>suc,t)):nis(<xt>suct,<yt>suct)
3077 +54
3078 xt@x:=nofnt(xt):nat
3079 [ft:[x:natt]natt]
3080 prop1t:=all([t:natt]is(<<t>suct>ft,<<t>ft>suct)):'prop'
3081 prop2t:=and(is(<1t>ft,<xt>suct),prop1t):'prop'
3082 xt@[f:[x:nat]nat]
3083 prop1:=all"n"([t:nat]is"n"(<<t>suc>f,<<t>f>suc)):'prop'
3084 prop2:=and(is"n"(<1>f,<x>suc),prop1):'prop'
3085 ft@g:=[t:nat]nofnt(<ntofn(t)>ft):[x:nat]nat
3086 [p:prop2t]
3087 t1:=ande1(is(<1t>ft,<xt>suct),prop1t,p):is(<1t>ft,<xt>suct)
3088 t2:=tris(nat,<1>g,nofnt(<xt>suct),<x>suc,isnten(<1t>ft,<xt>suct,t1),isnnt2(<x>suc)):is"n"(<1>g,<x>suc)
3089 t3:=ande2(is(<1t>ft,<xt>suct),prop1t,p):prop1t
3090 [u:nat]
3091 ut:=ntofn(u):natt
3092 t4:=isf(nat,nat,suc,u,nofnt(ut),isnnt1(u)):is"n"(<u>suc,<nofnt(ut)>suc)
3093 t5:=isf(nat,nat,g,<u>suc,<nofnt(ut)>suc,t4):is"n"(<<u>suc>g,nofnt(<<ut>suct>ft))
3094 t6:=<ut>t3:is(<<ut>suct>ft,<<ut>ft>suct)
3095 t7:=isnten(<<ut>suct>ft,<<ut>ft>suct,t6):is"n"(nofnt(<<ut>suct>ft),nofnt(<<ut>ft>suct))
3096 t8:=isnnt2(<<u>g>suc):is"n"(nofnt(<<ut>ft>suct),<<u>g>suc)
3097 t9:=tr3is(nat,<<u>suc>g,nofnt(<<ut>suct>ft),nofnt(<<ut>ft>suct),<<u>g>suc,t5,t7,t8):is"n"(<<u>suc>g,<<u>g>suc)
3098 p@t10:=[u:nat]t9(u):prop1(g)
3099 t11:=andi(is"n"(<1>g,<x>suc),prop1(g),t2,t10):prop2(g)
3100 xt@[a:[t:natt]natt][b:[t:natt]natt][pa:prop2t(a)][pb:prop2t(b)]
3101 t12:=onee1([t:nat]nat,[u:[t:nat]nat]prop2(u),satz4(x)):amone([t:nat]nat,[u:[t:nat]nat]prop2(u))
3102 t13:=<t11(b,pb)><t11(a,pa)><g(b)><g(a)>t12:is"e"([t:nat]nat,g(a),g(b))
3103 [yt:natt]
3104 y:=nofnt(yt):nat
3105 t14:=fise(nat,nat,g(a),g(b),t13,y):is"n"(nofnt(<ntofn(y)>a),nofnt(<ntofn(y)>b))
3106 t15:=isntin(<ntofn(y)>a,<ntofn(y)>b,t14):is(<ntofn(y)>a,<ntofn(y)>b)
3107 t16:=tr3is(natt,<yt>a,<ntofn(y)>a,<ntofn(y)>b,<yt>b,isf(natt,natt,a,yt,ntofn(y),isntn1(yt)),t15,isf(natt,natt,b,ntofn(y),yt,isntn2(yt))):is(<yt>a,<yt>b)
3108 pb@t17:=fisi(natt,natt,a,b,[t:natt]t16(t)):is"e"([t:natt]natt,a,b)
3109 xt@t18:=[u:[t:natt]natt][v:[t:natt]natt][w:prop2t(u)][z:prop2t(v)]t17(u,v,w,z):amone([t:natt]natt,[u:[t:natt]natt]prop2t(u))
3110 t19:=onee2([t:nat]nat,[u:[t:nat]nat]prop2(u),satz4(x)):some"l"([t:nat]nat,[u:[t:nat]nat]prop2(u))
3111 f@gt:=[t:natt]ntofn(<nofnt(t)>f):[x:natt]natt
3112 [p:prop2]
3113 t20:=ande1(is"n"(<1>f,<x>suc),prop1,p):is"n"(<1>f,<x>suc)
3114 t21:=isf(nat,nat,f,nofnt(1t),1,isnnt2(1)):is"n"(<nofnt(1t)>f,<1>f)
3115 t22:=isnent(<nofnt(1t)>f,<x>suc,tris(nat,<nofnt(1t)>f,<1>f,<x>suc,t21,t20)):is(<1t>gt,<xt>suct)
3116 t23:=ande2(is"n"(<1>f,<x>suc),prop1,p):prop1
3117 [zt:natt]
3118 z:=nofnt(zt):nat
3119 t24:=isf(nat,nat,f,nofnt(<zt>suct),<z>suc,isnnt2(<z>suc)):is"n"(<nofnt(<zt>suct)>f,<<z>suc>f)
3120 t25:=<z>t23:is"n"(<<z>suc>f,<<z>f>suc)
3121 t26:=isf(nat,nat,suc,<z>f,nofnt(<zt>gt),isnnt1(<z>f)):is"n"(<<z>f>suc,<nofnt(<zt>gt)>suc)
3122 t27:=isnent(<nofnt(<zt>suct)>f,<nofnt(<zt>gt)>suc,tr3is(nat,<nofnt(<zt>suct)>f,<<z>suc>f,<<z>f>suc,<nofnt(<zt>gt)>suc,t24,t25,t26)):is(<<zt>suct>gt,<<zt>gt>suct)
3123 p@t28:=[u:natt]t27(u):prop1t(gt)
3124 t29:=andi(is(<1t>gt,<xt>suct),prop1t(gt),t22,t28):prop2t(gt)
3125 t30:=somei([t:natt]natt,[u:[t:natt]natt]prop2t(u),gt,t29):some"l"([t:natt]natt,[u:[t:natt]natt]prop2t(u))
3126 xt@t31:=someapp([t:nat]nat,[u:[t:nat]nat]prop2(u),t19,some"l"([t:natt]natt,[u:[t:natt]natt]prop2t(u)),[u:[t:nat]nat][v:prop2(u)]t30(u,v)):some"l"([t:natt]natt,[u:[t:natt]natt]prop2t(u))
3127 -54
3128 xt@satz4:=onei([t:natt]natt,[u:[t:natt]natt]prop2t".54"(u),t18".54",t31".54"):one"e"([t:natt]natt,[u:[t:natt]natt]and(is(<1t>u,<xt>suct),all([v:natt]is(<<v>suct>u,<<v>u>suct))))
3129 yt@pl:=ntofrt(pl"rt"(rtofnt(xt),rtofnt(yt)),satz112d(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt))):natt
3130 +*ii5
3131 yt@t28:=satz112c(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt)):inf(fr(pl"n"(nofnt(xt),nofnt(yt)),1),class(pl"rt"(rtofnt(xt),rtofnt(yt))))
3132 t29:=isi(pl"rt"(rtofnt(xt),rtofnt(yt)),rtofn(pl"n"(nofnt(xt),nofnt(yt))),fr(pl"n"(nofnt(xt),nofnt(yt)),1),fr(pl"n"(nofnt(xt),nofnt(yt)),1),t28,inclass(fr(pl"n"(nofnt(xt),nofnt(yt)),1)),refeq"n"(fr(pl"n"(nofnt(xt),nofnt(yt)),1))):is"rt"(pl"rt"(rtofnt(xt),rtofnt(yt)),rtofn(pl"n"(nofnt(xt),nofnt(yt))))
3133 -ii5
3134 yt@isplnt:=isrtent(pl"rt"(rtofnt(xt),rtofnt(yt)),satz112d(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt)),rtofn(pl"n"(nofnt(xt),nofnt(yt))),natrti"rt"(pl"n"(nofnt(xt),nofnt(yt))),t29".ii5"):is(pl(xt,yt),ntofn(pl"n"(nofnt(xt),nofnt(yt))))
3135 isntpl:=symis(natt,pl(xt,yt),ntofn(pl"n"(nofnt(xt),nofnt(yt))),isplnt):is(ntofn(pl"n"(nofnt(xt),nofnt(yt))),pl(xt,yt))
3136 ispln:=tris(nat,pl"n"(nofnt(xt),nofnt(yt)),nofnt(ntofn(pl"n"(nofnt(xt),nofnt(yt)))),nofnt(pl(xt,yt)),isnnt1(pl"n"(nofnt(xt),nofnt(yt))),isnten(ntofn(pl"n"(nofnt(xt),nofnt(yt))),pl(xt,yt),isntpl)):is"n"(pl"n"(nofnt(xt),nofnt(yt)),nofnt(pl(xt,yt)))
3137 isnpl:=symis(nat,pl"n"(nofnt(xt),nofnt(yt)),nofnt(pl(xt,yt)),ispln):is"n"(nofnt(pl(xt,yt)),pl"n"(nofnt(xt),nofnt(yt)))
3138 [zt:natt]
3139 +55
3140 t1:=ispl1"n"(nofnt(pl(xt,yt)),pl"n"(nofnt(xt),nofnt(yt)),nofnt(zt),isnpl(xt,yt)):is"n"(pl"n"(nofnt(pl(xt,yt)),nofnt(zt)),pl"n"(pl"n"(nofnt(xt),nofnt(yt)),nofnt(zt)))
3141 t2:=ispl2"n"(pl"n"(nofnt(yt),nofnt(zt)),nofnt(pl(yt,zt)),nofnt(xt),ispln(yt,zt)):is"n"(pl"n"(nofnt(xt),pl"n"(nofnt(yt),nofnt(zt))),pl"n"(nofnt(xt),nofnt(pl(yt,zt))))
3142 t3:=tr3is(nat,pl"n"(nofnt(pl(xt,yt)),nofnt(zt)),pl"n"(pl"n"(nofnt(xt),nofnt(yt)),nofnt(zt)),pl"n"(nofnt(xt),pl"n"(nofnt(yt),nofnt(zt))),pl"n"(nofnt(xt),nofnt(pl(yt,zt))),t1,satz5(nofnt(xt),nofnt(yt),nofnt(zt)),t2):is"n"(pl"n"(nofnt(pl(xt,yt)),nofnt(zt)),pl"n"(nofnt(xt),nofnt(pl(yt,zt))))
3143 -55
3144 satz5:=tr3is(natt,pl(pl(xt,yt),zt),ntofn(pl"n"(nofnt(pl(xt,yt)),nofnt(zt))),ntofn(pl"n"(nofnt(xt),nofnt(pl(yt,zt)))),pl(xt,pl(yt,zt)),isplnt(pl(xt,yt),zt),isnent(pl"n"(nofnt(pl(xt,yt)),nofnt(zt)),pl"n"(nofnt(xt),nofnt(pl(yt,zt))),t3".55"),isntpl(xt,pl(yt,zt))):is(pl(pl(xt,yt),zt),pl(xt,pl(yt,zt)))
3145 diffprop:=is(xt,pl(yt,zt)):'prop'
3146 [d:diffprop]
3147 diffprope:=tris(nat,nofnt(xt),nofnt(pl(yt,zt)),pl"n"(nofnt(yt),nofnt(zt)),isnten(xt,pl(yt,zt),d),isnpl(yt,zt)):diffprop"n"(nofnt(xt),nofnt(yt),nofnt(zt))
3148 zt@[d:diffprop"n"(nofnt(xt),nofnt(yt),nofnt(zt))]
3149 +*ii5
3150 d@t30:=tris(nat,nofnt(xt),pl"n"(nofnt(yt),nofnt(zt)),nofnt(pl(yt,zt)),d,ispln(yt,zt)):is"n"(nofnt(xt),nofnt(pl(yt,zt)))
3151 -ii5
3152 d@diffpropi:=isntin(xt,pl(yt,zt),t30".ii5"):diffprop
3153 +59
3154 yt@it:=is(xt,yt):'prop'
3155 iit:=some([u:natt]diffprop(xt,yt,u)):'prop'
3156 iiit:=some([u:natt]diffprop(yt,xt,u)):'prop'
3157 i:=is"n"(nofnt(xt),nofnt(yt)):'prop'
3158 ii:=some"n"([u:nat]diffprop"n"(nofnt(xt),nofnt(yt),u)):'prop'
3159 iii:=some"n"([u:nat]diffprop"n"(nofnt(yt),nofnt(xt),u)):'prop'
3160 [one:i]
3161 t1:=or3i1(it,iit,iiit,isntin(xt,yt,one)):or3(it,iit,iiit)
3162 yt@[two:ii][v:nat][d:diffprop"n"(nofnt(xt),nofnt(yt),v)]
3163 t2:=isp(nat,[u:nat]diffprop"n"(nofnt(xt),nofnt(yt),u),v,nofnt(ntofn(v)),d,isnnt1(v)):diffprop"n"(nofnt(xt),nofnt(yt),nofnt(ntofn(v)))
3164 t3:=somei(natt,[u:natt]diffprop(xt,yt,u),ntofn(v),diffpropi(xt,yt,ntofn(v),t2)):iit
3165 two@t4:=someapp(nat,[u:nat]diffprop"n"(nofnt(xt),nofnt(yt),u),two,iit,[u:nat][v:diffprop"n"(nofnt(xt),nofnt(yt),u)]t3(u,v)):iit
3166 t5:=or3i2(it,iit,iiit,t4):or3(it,iit,iiit)
3167 yt@[three:iii]
3168 t6:=or3i3(it,iit,iiit,t4(yt,xt,three)):or3(it,iit,iiit)
3169 yt@t7:=or3app(i,ii,iii,or3(it,iit,iiit),satz9a(nofnt(xt),nofnt(yt)),[u:i]t1(u),[u:ii]t5(u),[u:iii]t6(u)):or3(it,iit,iiit)
3170 [onet:it]
3171 t8:=isnten(xt,yt,onet):i
3172 yt@[twot:iit][vt:natt][d:diffprop(xt,yt,vt)]
3173 t9:=somei(nat,[u:nat]diffprop"n"(nofnt(xt),nofnt(yt),u),nofnt(vt),diffprope(xt,yt,vt,d)):ii
3174 twot@t10:=someapp(natt,[u:natt]diffprop(xt,yt,u),twot,ii,[u:natt][v:diffprop(xt,yt,u)]t9(u,v)):ii
3175 yt@[threet:iiit]
3176 t11:=t10(yt,xt,threet):iii
3177 yt@t12:=satz9b(nofnt(xt),nofnt(yt)):ec3(i,ii,iii)
3178 onet@t13:=ec3e12(i,ii,iii,t12,t8):not(ii)
3179 t14:=th3"l.imp"(iit,ii,t13,[x:iit]t10(x)):not(iit)
3180 yt@t15:=th1"l.ec"(it,iit,[x:it]t14(x)):ec(it,iit)
3181 twot@t16:=ec3e23(i,ii,iii,t12,t10):not(iii)
3182 t17:=th3"l.imp"(iiit,iii,t16,[x:iiit]t11(x)):not(iiit)
3183 yt@t18:=th1"l.ec"(iit,iiit,[x:iit]t17(x)):ec(iit,iiit)
3184 threet@t19:=ec3e31(i,ii,iii,t12,t11):not(i)
3185 t20:=th3"l.imp"(it,i,t19,[x:it]t8(x)):not(it)
3186 yt@t21:=th1"l.ec"(iiit,it,[x:iiit]t20(x)):ec(iiit,it)
3187 t22:=th6"l.ec3"(it,iit,iiit,t15,t18,t21):ec3(it,iit,iiit)
3188 -59
3189 yt@satz9:=orec3i(it".59",iit".59",iiit".59",t7".59",t22".59"):orec3(is(xt,yt),some([u:natt]is(xt,pl(yt,u))),some([u:natt]is(yt,pl(xt,u))))
3190 more:=more"rt"(rtofnt(xt),rtofnt(yt)):'prop'
3191 [m:more(xt,yt)]
3192 +*ii5
3193 m@t31:=moree(rtofnt(xt),rtofnt(yt),fr(nofnt(xt),1),fr(nofnt(yt),1),inclassn(rtofnt(xt),natrti(xt)),inclassn(rtofnt(yt),natrti(yt)),m):moref(fr(nofnt(xt),1),fr(nofnt(yt),1))
3194 -ii5
3195 m@moree:=satz111a(nofnt(xt),nofnt(yt),t31".ii5"):more"n"(nofnt(xt),nofnt(yt))
3196 yt@[m:more"n"(nofnt(xt),nofnt(yt))]
3197 +*ii5
3198 m@t32:=satz111d(nofnt(xt),nofnt(yt),m):moref(fr(nofnt(xt),1),fr(nofnt(yt),1))
3199 -ii5
3200 m@morei:=morei"rt"(rtofnt(xt),rtofnt(yt),fr(nofnt(xt),1),fr(nofnt(yt),1),inclassn(rtofnt(xt),natrti(xt)),inclassn(rtofnt(yt),natrti(yt)),t32".ii5"):more(xt,yt)
3201 yt@less:=less"rt"(rtofnt(xt),rtofnt(yt)):'prop'
3202 [l:less(xt,yt)]
3203 +*ii5
3204 l@t33:=lesse(rtofnt(xt),rtofnt(yt),fr(nofnt(xt),1),fr(nofnt(yt),1),inclassn(rtofnt(xt),natrti(xt)),inclassn(rtofnt(yt),natrti(yt)),l):lessf(fr(nofnt(xt),1),fr(nofnt(yt),1))
3205 -ii5
3206 l@lesse:=satz111c(nofnt(xt),nofnt(yt),t33".ii5"):less"n"(nofnt(xt),nofnt(yt))
3207 yt@[l:less"n"(nofnt(xt),nofnt(yt))]
3208 +*ii5
3209 l@t34:=satz111f(nofnt(xt),nofnt(yt),l):lessf(fr(nofnt(xt),1),fr(nofnt(yt),1))
3210 -ii5
3211 l@lessi:=lessi"rt"(rtofnt(xt),rtofnt(yt),fr(nofnt(xt),1),fr(nofnt(yt),1),inclassn(rtofnt(xt),natrti(xt)),inclassn(rtofnt(yt),natrti(yt)),t34".ii5"):less(xt,yt)
3212 yt@moreis:=moreis"rt"(rtofnt(xt),rtofnt(yt)):'prop'
3213 [m:moreis(xt,yt)]
3214 moreise:=th9"l.or"(more"rt"(rtofnt(xt),rtofnt(yt)),is"rt"(rtofnt(xt),rtofnt(yt)),more"n"(nofnt(xt),nofnt(yt)),is"n"(nofnt(xt),nofnt(yt)),m,[u:more"rt"(rtofnt(xt),rtofnt(yt))]moree(xt,yt,u),[u:is"rt"(rtofnt(xt),rtofnt(yt))]isrten(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt),u)):moreis"n"(nofnt(xt),nofnt(yt))
3215 yt@[m:moreis"n"(nofnt(xt),nofnt(yt))]
3216 moreisi:=th9"l.or"(more"n"(nofnt(xt),nofnt(yt)),is"n"(nofnt(xt),nofnt(yt)),more"rt"(rtofnt(xt),rtofnt(yt)),is"rt"(rtofnt(xt),rtofnt(yt)),m,[u:more"n"(nofnt(xt),nofnt(yt))]morei(xt,yt,u),[u:is"n"(nofnt(xt),nofnt(yt))]isrtin(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt),u)):moreis(xt,yt)
3217 yt@lessis:=lessis"rt"(rtofnt(xt),rtofnt(yt)):'prop'
3218 [l:lessis(xt,yt)]
3219 lessise:=th9"l.or"(less"rt"(rtofnt(xt),rtofnt(yt)),is"rt"(rtofnt(xt),rtofnt(yt)),less"n"(nofnt(xt),nofnt(yt)),is"n"(nofnt(xt),nofnt(yt)),l,[u:less"rt"(rtofnt(xt),rtofnt(yt))]lesse(xt,yt,u),[u:is"rt"(rtofnt(xt),rtofnt(yt))]isrten(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt),u)):lessis"n"(nofnt(xt),nofnt(yt))
3220 yt@[l:lessis"n"(nofnt(xt),nofnt(yt))]
3221 lessisi:=th9"l.or"(less"n"(nofnt(xt),nofnt(yt)),is"n"(nofnt(xt),nofnt(yt)),less"rt"(rtofnt(xt),rtofnt(yt)),is"rt"(rtofnt(xt),rtofnt(yt)),l,[u:less"n"(nofnt(xt),nofnt(yt))]lessi(xt,yt,u),[u:is"n"(nofnt(xt),nofnt(yt))]isrtin(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt),u)):lessis(xt,yt)
3222 zt@[l:less(xt,yt)][k:less(yt,zt)]
3223 +515
3224 t1:=satz15(nofnt(xt),nofnt(yt),nofnt(zt),lesse(xt,yt,l),lesse(yt,zt,k)):less"n"(nofnt(xt),nofnt(zt))
3225 -515
3226 satz15:=lessi(xt,zt,t1".515"):less(xt,zt)
3227 zt@[ut:natt][m:more(xt,yt)][n:more(zt,ut)]
3228 +521
3229 t1:=satz21(nofnt(xt),nofnt(yt),nofnt(zt),nofnt(ut),moree(xt,yt,m),moree(zt,ut,n)):more"n"(pl"n"(nofnt(xt),nofnt(zt)),pl"n"(nofnt(yt),nofnt(ut)))
3230 t2:=ismore12"n"(pl"n"(nofnt(xt),nofnt(zt)),nofnt(pl(xt,zt)),pl"n"(nofnt(yt),nofnt(ut)),nofnt(pl(yt,ut)),ispln(xt,zt),ispln(yt,ut),t1):more"n"(nofnt(pl(xt,zt)),nofnt(pl(yt,ut)))
3231 -521
3232 satz21:=morei(pl(xt,zt),pl(yt,ut),t2".521"):more(pl(xt,zt),pl(yt,ut))
3233 @[p:[x:natt]'prop'][n:natt]
3234 lb:=all([x:natt]imp(<x>p,lessis(n,x))):'prop'
3235 min:=and(lb,<n>p):'prop'
3236 p@[s:some(p)]
3237 +527
3238 q:=[x:nat]<ntofn(x)>p:[x:nat]'prop'
3239 [n:natt][np:<n>p]
3240 t1:=isp(natt,p,n,ntofn(nofnt(n)),np,isntn1(n)):<nofnt(n)>q
3241 t2:=somei(nat,q,nofnt(n),t1):some"n"(q)
3242 s@t3:=someapp(natt,p,s,some"n"(q),[u:natt][v:<u>p]t2(u,v)):some"n"(q)
3243 t4:=satz27(q,t3):some"n"([x:nat]min"n"(q,x))
3244 [m:nat][mqm:min"n"(q,m)]
3245 t5:=ande1(lb"n"(q,m),<m>q,mqm):lb"n"(q,m)
3246 [n:nat][nq:<n>q]
3247 t6:=mp(<n>q,lessis"n"(m,n),nq,<n>t5):lessis"n"(m,n)
3248 mqm@[n:natt][np:<n>p]
3249 t7:=t6(nofnt(n),t1(n,np)):lessis"n"(m,nofnt(n))
3250 t8:=islessis1"n"(m,nofnt(ntofn(m)),nofnt(n),isnnt1(m),t7):lessis"n"(nofnt(ntofn(m)),nofnt(n))
3251 t9:=lessisi(ntofn(m),n,t8):lessis(ntofn(m),n)
3252 n@t10:=[u:<n>p]t9(u):imp(<n>p,lessis(ntofn(m),n))
3253 mqm@t11:=[x:natt]t10(x):lb(ntofn(m))
3254 t12:=ande2(lb"n"(q,m),<m>q,mqm):<ntofn(m)>p
3255 t13:=andi(lb(ntofn(m)),<ntofn(m)>p,t11,t12):min(ntofn(m))
3256 t14:=somei(natt,[x:natt]min(x),ntofn(m),t13):some([x:natt]min(x))
3257 -527
3258 satz27:=someapp(nat,[x:nat]min"n"(q".527",x),t4".527",some([x:natt]min(x)),[x:nat][y:min"n"(q".527",x)]t14".527"(x,y)):some([x:natt]min(p,x))
3259 -nt
3260 @1rt:=rtofn(1):rat
3261 x0@[x:frac][xix0:inf(x,class(x0))]
3262 +*ii5
3263 xix0@t35:=tr3eq(tf(x,fr(1,1)),fr(ts"n"(num(x),1),ts"n"(den(x),1)),fr(num(x),den(x)),x,tfeq1a(x,1,1),eqnd(ts"n"(num(x),1),ts"n"(den(x),1),num(x),den(x),satz28a(num(x)),satz28a(den(x))),refeq1(fr(num(x),den(x)),x,fris(x))):eq"n"(tf(x,fr(1,1)),x)
3264 t36:=isi(ts(x0,1rt),x0,tf(x,fr(1,1)),x,tict(x0,1rt,x,fr(1,1),xix0,inclass(fr(1,1))),xix0,t35):is(ts(x0,1rt),x0)
3265 -ii5
3266 x0@example1a:=ratapp1(x0,is(ts(x0,1rt),x0),[x:frac][xi:inf(x,class(x0))]t36".ii5"(x,xi)):is(ts(x0,1rt),x0)
3267 example1b:=symis(rat,ts(x0,1rt),x0,example1a):is(x0,ts(x0,1rt))
3268 example1c:=tris(rat,ts(1rt,x0),ts(x0,1rt),x0,comts(1rt,x0),example1a):is(ts(1rt,x0),x0)
3269 example1d:=symis(rat,ts(1rt,x0),x0,example1c):is(x0,ts(1rt,x0))
3270 @[x:frac]
3271 +5114
3272 t1:=tr3eq(tf(fr(den(x),1),x),fr(ts"n"(den(x),num(x)),ts"n"(1,den(x))),fr(ts"n"(num(x),den(x)),ts"n"(1,den(x))),fr(num(x),1),tfeq2a(x,den(x),1),eqn(ts"n"(den(x),num(x)),ts"n"(1,den(x)),ts"n"(num(x),den(x)),comts"n"(den(x),num(x))),satz40c(num(x),1,den(x))):eq"n"(tf(fr(den(x),1),x),fr(num(x),1))
3273 -5114
3274 satz114:=isi(ts(rtofn(den(x)),ratof(x)),rtofn(num(x)),tf(fr(den(x),1),x),fr(num(x),1),tict(rtofn(den(x)),ratof(x),fr(den(x),1),x,inclass(fr(den(x),1)),inclass(x)),inclass(fr(num(x),1)),t1".5114"):is(ts(rtofn(den(x)),ratof(x)),rtofn(num(x)))
3275 @[x1:nat][x2:nat]
3276 satz114a:=tr3is(rat,ts(rtofn(x2),ratof(fr(x1,x2))),ts(rtofn(den(fr(x1,x2))),ratof(fr(x1,x2))),rtofn(num(fr(x1,x2))),rtofn(x1),ists1(rtofn(x2),rtofn(den(fr(x1,x2))),ratof(fr(x1,x2)),isnert(x2,den(fr(x1,x2)),isden(x1,x2))),satz114(fr(x1,x2)),isnert(num(fr(x1,x2)),x1,numis(x1,x2))):is(ts(rtofn(x2),ratof(fr(x1,x2))),rtofn(x1))
3277 x0@[y0:rat]
3278 ov:=ind(rat,[t:rat]is(ts(y0,t),x0),satz110(x0,y0)):rat
3279 satz110c:=oneax(rat,[t:rat]is(ts(y0,t),x0),satz110(x0,y0)):is(ts(y0,ov(x0,y0)),x0)
3280 satz110d:=symis(rat,ts(y0,ov(x0,y0)),x0,satz110c):is(x0,ts(y0,ov(x0,y0)))
3281 satz110e:=tris(rat,ts(ov(x0,y0),y0),ts(y0,ov(x0,y0)),x0,comts(ov(x0,y0),y0),satz110c):is(ts(ov(x0,y0),y0),x0)
3282 satz110f:=symis(rat,ts(ov(x0,y0),y0),x0,satz110e):is(x0,ts(ov(x0,y0),y0))
3283 [v0:rat][i:is(ts(y0,v0),x0)]
3284 satz110g:=satz110b(x0,y0,v0,ov(x0,y0),i,satz110c):is(v0,ov(x0,y0))
3285 x2@satz114b:=satz110b(rtofn(x1),rtofn(x2),ratof(fr(x1,x2)),ov(rtofn(x1),rtofn(x2)),satz114a,satz110c(rtofn(x1),rtofn(x2))):is(ratof(fr(x1,x2)),ov(rtofn(x1),rtofn(x2)))
3286 satz114c:=symis(rat,ratof(fr(x1,x2)),ov(rtofn(x1),rtofn(x2)),satz114b):is(ov(rtofn(x1),rtofn(x2)),ratof(fr(x1,x2)))
3287 +5115
3288 y0@t1:=satz89(ov(y0,x0)):some([t:rat]more(t,ov(y0,x0)))
3289 [u0:rat][m:more(u0,ov(y0,x0))][u:frac][uiu0:inf(u,class(u0))]
3290 z:=num(u):nat
3291 v:=den(u):nat
3292 t2:=ismore1(u0,ov(rtofn(z),rtofn(v)),ov(y0,x0),tris(rat,u0,ratof(fr(z,v)),ov(rtofn(z),rtofn(v)),isi(u0,ratof(fr(z,v)),u,fr(z,v),uiu0,inclass(fr(z,v)),refeq1(u,fr(z,v),isfr(u))),satz114b(z,v)),m):more(ov(rtofn(z),rtofn(v)),ov(y0,x0))
3293 t3:=moreisi(rtofn(v),1rt,fr(v,1),fr(1,1),inclass(fr(v,1)),inclass(fr(1,1)),th9"l.or"(more"n"(v,1),is"n"(v,1),moref(fr(v,1),fr(1,1)),eq"n"(fr(v,1),fr(1,1)),satz24(v),[t:more"n"(v,1)]satz111d(v,1,t),[t:is"n"(v,1)]satz111e(v,1,t))):moreis(rtofn(v),1rt)
3294 t4:=tr3is(rat,ts(rtofn(z),x0),ts(x0,rtofn(z)),ts(x0,ts(ov(rtofn(z),rtofn(v)),rtofn(v))),ts(ts(x0,ov(rtofn(z),rtofn(v))),rtofn(v)),comts(rtofn(z),x0),ists2(rtofn(z),ts(ov(rtofn(z),rtofn(v)),rtofn(v)),x0,satz110f(rtofn(z),rtofn(v))),assts2(x0,ov(rtofn(z),rtofn(v)),rtofn(v))):is(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),rtofn(v)))
3295 [n:more(rtofn(v),1rt)]
3296 t5:=ismore1(ts(ts(x0,ov(rtofn(z),rtofn(v))),rtofn(v)),ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt),symis(rat,ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),rtofn(v)),t4),satz105d(rtofn(v),1rt,ts(x0,ov(rtofn(z),rtofn(v))),n)):more(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt))
3297 t6:=moreisi1(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt),t5):moreis(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt))
3298 uiu0@[i:is(rtofn(v),1rt)]
3299 t7:=moreisi2(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt),tris(rat,ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),rtofn(v)),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt),t4,ists2(rtofn(v),1rt,ts(x0,ov(rtofn(z),rtofn(v))),i))):moreis(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt))
3300 uiu0@t8:=orapp(more(rtofn(v),1rt),is(rtofn(v),1rt),moreis(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt)),t3,[t:more(rtofn(v),1rt)]t6(t),[t:is(rtofn(v),1rt)]t7(t)):moreis(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt))
3301 t9:=ismore12(ts(x0,ov(rtofn(z),rtofn(v))),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt),ts(x0,ov(y0,x0)),y0,example1b(ts(x0,ov(rtofn(z),rtofn(v)))),satz110c(y0,x0),satz105d(ov(rtofn(z),rtofn(v)),ov(y0,x0),x0,t2)):more(ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt),y0)
3302 t10:=satz87c(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt),y0,t8,t9):more(ts(rtofn(z),x0),y0)
3303 t11:=somei(nat,[t:nat]more(ts(rtofn(t),x0),y0),z,t10):some"n"([t:nat]more(ts(rtofn(t),x0),y0))
3304 m@t12:=ratapp1(u0,some"n"([t:nat]more(ts(rtofn(t),x0),y0)),[u:frac][ui:inf(u,class(u0))]t11(u,ui)):some"n"([t:nat]more(ts(rtofn(t),x0),y0))
3305 -5115
3306 y0@satz115:=someapp(rat,[t:rat]more(t,ov(y0,x0)),t1".5115",some"n"([t:nat]more(ts(rtofn(t),x0),y0)),[t:rat][u:more(t,ov(y0,x0))]t12".5115"(t,u)):some"n"([t:nat]more(ts(rtofn(t),x0),y0))
3307 +*5115
3308 uiu0@t13:=andi(natrt(rtofn(z)),more(ts(rtofn(z),x0),y0),natrti(z),t10):and(natrt(rtofn(z)),more(ts(rtofn(z),x0),y0))
3309 t14:=somei(rat,[t:rat]and(natrt(t),more(ts(t,x0),y0)),rtofn(z),t13):some([t:rat]and(natrt(t),more(ts(t,x0),y0)))
3310 m@t15:=ratapp1(u0,some([t:rat]and(natrt(t),more(ts(t,x0),y0))),[u:frac][ui:inf(u,class(u0))]t14(u,ui)):some([t:rat]and(natrt(t),more(ts(t,x0),y0)))
3311 -5115
3312 y0@satz115a:=someapp(rat,[t:rat]more(t,ov(y0,x0)),t1".5115",some([t:rat]and(natrt(t),more(ts(t,x0),y0))),[t:rat][u:more(t,ov(y0,x0))]t15".5115"(t,u)):some([t:rat]and(natrt(t),more(ts(t,x0),y0)))
3313 @[s:set(rat)]
3314 cutprop1a:=nonempty(rat,s):'prop'
3315 cutprop1b:=not(all([x:rat]in(x,s))):'prop'
3316 cutprop1:=and(cutprop1a,cutprop1b):'prop'
3317 [x0:rat]
3318 cutprop2a:=all([x:rat]imp(not(in(x,s)),less(x0,x))):'prop'
3319 s@cutprop2:=all([x:rat]imp(in(x,s),cutprop2a(x))):'prop'
3320 x0@[y0:rat]
3321 +iii1
3322 ubprop:=imp(in(y0,s),moreis(x0,y0)):'prop'
3323 -iii1
3324 x0@ub:=all([x:rat]ubprop".iii1"(x0,x)):'prop'
3325 max:=and(ub(x0),in(x0,s)):'prop'
3326 s@cutprop3:=not(some([x:rat]max(s,x))):'prop'
3327 cutprop:=and3(cutprop1,cutprop2,cutprop3):'prop'
3328 +*iii1
3329 y0@lbprop:=imp(in(y0,s),lessis(x0,y0)):'prop'
3330 -iii1
3331 x0@lb:=all([x:rat]lbprop".iii1"(x0,x)):'prop'
3332 min:=and(lb(x0),in(x0,s)):'prop'
3333 @cut:=ot(set(rat),[x:set(rat)]cutprop(x)):'type'
3334 [ksi:cut]
3335 lcl:=in"e"(set(rat),[x:set(rat)]cutprop(x),ksi):set(rat)
3336 [x0:rat]
3337 lrt:=in(x0,lcl(ksi)):'prop'
3338 urt:=not(in(x0,lcl(ksi))):'prop'
3339 ksi@clcl:=inp(set(rat),[x:set(rat)]cutprop(x),ksi):cutprop(lcl(ksi))
3340 clcl1:=and3e1(cutprop1(lcl),cutprop2(lcl),cutprop3(lcl),clcl):cutprop1(lcl(ksi))
3341 clcl2:=and3e2(cutprop1(lcl),cutprop2(lcl),cutprop3(lcl),clcl):cutprop2(lcl(ksi))
3342 clcl3:=and3e3(cutprop1(lcl),cutprop2(lcl),cutprop3(lcl),clcl):cutprop3(lcl(ksi))
3343 clcl1a:=ande1(cutprop1a(lcl),cutprop1b(lcl),clcl1):cutprop1a(lcl(ksi))
3344 clcl1b:=ande2(cutprop1a(lcl),cutprop1b(lcl),clcl1):cutprop1b(lcl(ksi))
3345 [p:'prop'][p1:[x:rat][t:lrt(ksi,x)]p]
3346 cutapp1a:=nonemptyapp(rat,lcl,clcl1a,p,p1):p
3347 +*iii1
3348 ksi@t1:=th1"l.some"(rat,[x:rat]lrt(ksi,x),clcl1b):some([x:rat]urt(ksi,x))
3349 -iii1
3350 p@[p1:[x:rat][t:urt(ksi,x)]p]
3351 cutapp1b:=someapp(rat,[x:rat]urt(ksi,x),t1".iii1",p,p1):p
3352 ksi@[x0:rat][lx:lrt(ksi,x0)]
3353 +*iii1
3354 lx@t2:=mp(lrt(ksi,x0),cutprop2a(lcl,x0),lx,<x0>clcl2):cutprop2a(lcl,x0)
3355 -iii1
3356 lx@[y0:rat][uy:urt(ksi,y0)]
3357 cutapp2a:=mp(urt(ksi,y0),less(x0,y0),uy,<y0>t2".iii1"):less(x0,y0)
3358 cutapp2b:=satz83(x0,y0,cutapp2a):more(y0,x0)
3359 +*iii1
3360 lx@t3:=th4"l.some"(rat,[x:rat]max(lcl,x),clcl3,x0):not(max(lcl,x0))
3361 t4:=th4"l.and"(ub(lcl,x0),lrt(ksi,x0),t3,lx):not(ub(lcl,x0))
3362 t5:=th1"l.some"(rat,[x:rat]ubprop(lcl,x0,x),t4):some([x:rat]not(ubprop(lcl,x0,x)))
3363 -iii1
3364 lx@[p:'prop'][p1:[y:rat][t:lrt(ksi,y)][u:less(x0,y)]p][y0:rat][n:not(ubprop".iii1"(lcl,x0,y0))]
3365 +*iii1
3366 n@t6:=th5"l.imp"(lrt(ksi,y0),moreis(x0,y0),n):lrt(ksi,y0)
3367 t7:=th6"l.imp"(lrt(ksi,y0),moreis(x0,y0),n):not(moreis(x0,y0))
3368 t8:=satz81j(x0,y0,t7):less(x0,y0)
3369 t9:=<t8><t6><y0>p1:p
3370 -iii1
3371 p1@cutapp3:=someapp(rat,[y:rat]not(ubprop".iii1"(lcl,x0,y)),t5".iii1",p,[y:rat][t:not(ubprop".iii1"(lcl,x0,y))]t9".iii1"(y,t)):p
3372 s@[x0:rat][i:in(x0,s)][y0:rat][n:not(in(y0,s))]
3373 +*iii1
3374 n@t10:=andi(cutprop1a,cutprop1b,nonemptyi(rat,s,x0,i),th1"l.all"(rat,[x:rat]in(x,s),y0,n)):cutprop1
3375 -iii1
3376 s@[n:[x:rat]not(max(s,x))]
3377 +*iii1
3378 n@t11:=th5"l.some"(rat,[x:rat]max(s,x),n):cutprop3
3379 -iii1
3380 s@[x0:rat][i:in(x0,s)][y0:rat][n:not(in(y0,s))][l:[x:rat][t:in(x,s)][y:rat][u:not(in(y,s))]less(x,y)][m:[x:rat]not(max(s,x))]
3381 cut1:=and3i(cutprop1,cutprop2,cutprop3,t10".iii1"(x0,i,y0,n),l,t11".iii1"(m)):cutprop(s)
3382 +rp
3383 ksi@[eta:cut]
3384 is:=is"e"(cut,ksi,eta):'prop'
3385 nis:=not(is(ksi,eta)):'prop'
3386 [i:is(ksi,eta)]
3387 ise:=isini(set(rat),[x:set(rat)]cutprop(x),ksi,eta,i):is"e"(set(rat),lcl(ksi),lcl(eta))
3388 [x0:rat][lx:lrt(ksi,x0)]
3389 ise1:=issete1(rat,lcl(ksi),lcl(eta),ise,x0,lx):lrt(eta,x0)
3390 eta@[i:is"e"(set(rat),lcl(ksi),lcl(eta))]
3391 isi:=isine(set(rat),[x:set(rat)]cutprop(x),ksi,eta,i):is(ksi,eta)
3392 eta@[l:[x:rat][t:lrt(ksi,x)]lrt(eta,x)][k:[x:rat][t:lrt(eta,x)]lrt(ksi,x)]
3393 isi1:=isi(isseti(rat,lcl(ksi),lcl(eta),l,k)):is(ksi,eta)
3394 @[s:set(rat)][cs:cutprop(s)]
3395 cutof:=out(set(rat),[x:set(rat)]cutprop(x),s,cs):cut
3396 [x0:rat][i:in(x0,s)]
3397 ine:=issete1(rat,s,lcl(cutof(s,cs)),isinout(set(rat),[x:set(rat)]cutprop(x),s,cs),x0,i):lrt(cutof(s,cs),x0)
3398 x0@[lx:lrt(cutof(s,cs),x0)]
3399 ini:=issete2(rat,s,lcl(cutof(s,cs)),isinout(set(rat),[x:set(rat)]cutprop(x),s,cs),x0,lx):in(x0,s)
3400 s@[t:set(rat)][cs:cutprop(s)][ct:cutprop(t)][i:[x:rat][u:in(x,s)]in(x,t)][j:[x:rat][u:in(x,t)]in(x,s)]
3401 isi2:=isouti(set(rat),[x:set(rat)]cutprop(x),s,cs,t,ct,isseti(rat,s,t,i,j)):is(cutof(s,cs),cutof(t,ct))
3402 @[p:[x:cut]'prop']
3403 all:=all"l"(cut,p):'prop'
3404 some:=some"l"(cut,p):'prop'
3405 one:=one"e"(cut,p):'prop'
3406 ksi@satz116:=refis(cut,ksi):is(ksi,ksi)
3407 eta@[i:is(ksi,eta)]
3408 satz117:=symis(cut,ksi,eta,i):is(eta,ksi)
3409 eta@[zeta:cut][i:is(ksi,eta)][j:is(eta,zeta)]
3410 satz118:=tris(cut,ksi,eta,zeta,i,j):is(ksi,zeta)
3411 +1119
3412 @[x0:rat][y0:rat][m:more(x0,y0)]
3413 t1:=ec3e23(is"rt"(x0,y0),more(x0,y0),less(x0,y0),satz81b(x0,y0),m):not(less(x0,y0))
3414 -1119
3415 ksi@[x0:rat][ux:urt(ksi,x0)][y0:rat][m:more(y0,x0)]
3416 satz119:=th3"l.imp"(lrt(ksi,y0),less(y0,x0),t1".1119"(y0,x0,m),[t:lrt(ksi,y0)]cutapp2a(y0,t,x0,ux)):urt(ksi,y0)
3417 y0@[l:less(x0,y0)]
3418 satz119a:=satz119(satz83(x0,y0,l)):urt(ksi,y0)
3419 +1120
3420 @[x0:rat][y0:rat][l:less(x0,y0)]
3421 t1:=ec3e32(is"rt"(x0,y0),more(x0,y0),less(x0,y0),satz81b(x0,y0),l):not(more(x0,y0))
3422 -1120
3423 ksi@[x0:rat][lx:lrt(ksi,x0)][y0:rat][l:less(y0,x0)]
3424 satz120:=th7"l.imp"(lrt(ksi,y0),more(y0,x0),t1".1120"(y0,x0,l),[t:urt(ksi,y0)]cutapp2b(x0,lx,y0,t)):lrt(ksi,y0)
3425 y0@[m:more(x0,y0)]
3426 satz120a:=satz120(satz82(x0,y0,m)):lrt(ksi,y0)
3427 -rp
3428 s@[i:[x:rat][t:in(x,s)][y:rat][u:less(y,x)]in(y,s)][x0:rat][j:in(x0,s)][y0:rat][n:not(in(y0,s))]
3429 +*iii1
3430 n@t12:=<y0><j><x0>i:[u:less(y0,x0)]in(y0,s)
3431 t13:=th3"l.imp"(less(y0,x0),in(y0,s),n,t12):not(less(y0,x0))
3432 t14:=satz81f(y0,x0,t13):moreis(y0,x0)
3433 -iii1
3434 n@[k:is(y0,x0)]
3435 +*iii1
3436 k@t15:=isp1(rat,[x:rat]in(x,s),x0,y0,j,k):in(y0,s)
3437 n@t16:=th3"l.imp"(is(y0,x0),in(y0,s),n,[t:is(y0,x0)]t15(t)):nis(y0,x0)
3438 t17:=ore1(more(y0,x0),is(y0,x0),t14,t16):more(y0,x0)
3439 t18:=satz82(y0,x0,t17):less(x0,y0)
3440 i@t19:=[x:rat][t:in(x,s)][y:rat][u:not(in(y,s))]t18(x,t,y,u):cutprop2
3441 -iii1
3442 s@[s1:[x:rat][t:in(x,s)]some([y:rat]and(in(y,s),more(y,x)))][x0:rat][i:in(x0,s)]
3443 +*iii1
3444 i@t20:=<i><x0>s1:some([y:rat]and(in(y,s),more(y,x0)))
3445 -iii1
3446 i@[y0:rat][a:and(in(y0,s),more(y0,x0))]
3447 +*iii1
3448 a@t21:=ande1(in(y0,s),more(y0,x0),a):in(y0,s)
3449 t22:=ande2(in(y0,s),more(y0,x0),a):more(y0,x0)
3450 t23:=satz81g(y0,x0,t22):not(lessis(y0,x0))
3451 t24:=th3"l.imp"(moreis(x0,y0),lessis(y0,x0),t23,[t:moreis(x0,y0)]satz84(x0,y0,t)):not(moreis(x0,y0))
3452 t25:=th4"l.imp"(in(y0,s),moreis(x0,y0),t21,t24):not(ubprop(x0,y0))
3453 t26:=th1"l.all"(rat,[y:rat]ubprop(x0,y),y0,t25):not(ub(s,x0))
3454 t27:=th1"l.and"(ub(s,x0),in(x0,s),t26):not(max(s,x0))
3455 i@t28:=someapp(rat,[y:rat]and(in(y,s),more(y,x0)),t20,not(max(s,x0)),[y:rat][t:and(in(y,s),more(y,x0))]t27(y,t)):not(max(s,x0))
3456 -iii1
3457 x0@[n:not(in(x0,s))]
3458 +*iii1
3459 n@t29:=th2"l.and"(ub(x0),in(x0,s),n):not(max(s,x0))
3460 x0@t30:=th1"l.imp"(in(x0,s),not(max(s,x0)),[u:in(x0,s)]t28(u),[u:not(in(x0,s))]t29(u)):not(max(s,x0))
3461 s1@t31:=t11([x:rat]t30(x)):cutprop3
3462 -iii1
3463 s@[x0:rat][i:in(x0,s)][y0:rat][n:not(in(y0,s))][j:[x:rat][t:in(x,s)][y:rat][u:less(y,x)]in(y,s)][s1:[x:rat][t:in(x,s)]some([y:rat]and(in(y,s),more(y,x)))]
3464 cut2:=and3i(cutprop1,cutprop2,cutprop3,t10".iii1"(x0,i,y0,n),t19".iii1"(j),t31".iii1"(s1)):cutprop(s)
3465 +*rp
3466 eta@more:=some"rt"([x:rat]and(lrt(ksi,x),urt(eta,x))):'prop'
3467 [m:more(ksi,eta)][p:'prop'][p1:[x:rat][t:lrt(ksi,x)][u:urt(eta,x)]p][x0:rat][a:and(lrt(ksi,x0),urt(eta,x0))]
3468 +iii2
3469 t1:=ande1(lrt(ksi,x0),urt(eta,x0),a):lrt(ksi,x0)
3470 t2:=ande2(lrt(ksi,x0),urt(eta,x0),a):urt(eta,x0)
3471 t3:=<t2><t1><x0>p1:p
3472 -iii2
3473 p1@moreapp:=someapp(rat,[x:rat]and(lrt(ksi,x),urt(eta,x)),m,p,[x:rat][t:and(lrt(ksi,x),urt(eta,x))]t3".iii2"(x,t)):p
3474 eta@less:=some"rt"([x:rat]and(urt(ksi,x),lrt(eta,x))):'prop'
3475 [l:less(ksi,eta)][p:'prop'][p1:[x:rat][t:urt(ksi,x)][u:lrt(eta,x)]p][x0:rat][a:and(urt(ksi,x0),lrt(eta,x0))]
3476 +*iii2
3477 a@t4:=ande1(urt(ksi,x0),lrt(eta,x0),a):urt(ksi,x0)
3478 t5:=ande2(urt(ksi,x0),lrt(eta,x0),a):lrt(eta,x0)
3479 t6:=<t5><t4><x0>p1:p
3480 -iii2
3481 p1@lessapp:=someapp(rat,[x:rat]and(urt(ksi,x),lrt(eta,x)),l,p,[x:rat][t:and(urt(ksi,x),lrt(eta,x))]t6".iii2"(x,t)):p
3482 eta@[m:more(ksi,eta)]
3483 +2121
3484 [x0:rat][lx:lrt(ksi,x0)][ux:urt(eta,x0)]
3485 t1:=andi(urt(eta,x0),lrt(ksi,x0),ux,lx):and(urt(eta,x0),lrt(ksi,x0))
3486 t2:=somei(rat,[x:rat]and(urt(eta,x),lrt(ksi,x)),x0,t1):less(eta,ksi)
3487 -2121
3488 satz121:=moreapp(m,less(eta,ksi),[x:rat][t:lrt(ksi,x)][u:urt(eta,x)]t2".2121"(x,t,u)):less(eta,ksi)
3489 eta@[l:less(ksi,eta)]
3490 +2122
3491 [x0:rat][ux:urt(ksi,x0)][lx:lrt(eta,x0)]
3492 t1:=andi(lrt(eta,x0),urt(ksi,x0),lx,ux):and(lrt(eta,x0),urt(ksi,x0))
3493 t2:=somei(rat,[x:rat]and(lrt(eta,x),urt(ksi,x)),x0,t1):more(eta,ksi)
3494 -2122
3495 satz122:=lessapp(l,more(eta,ksi),[x:rat][t:urt(ksi,x)][u:lrt(eta,x)]t2".2122"(x,t,u)):more(eta,ksi)
3496 +2123
3497 eta@[m:more(ksi,eta)][x0:rat][lx:lrt(ksi,x0)][ux:urt(eta,x0)]
3498 t1:=th3"st.isset"(rat,lcl(ksi),lcl(eta),x0,lx,ux):not(is"e"(set(rat),lcl(ksi),lcl(eta)))
3499 t2:=th3"l.imp"(is(ksi,eta),is"e"(set(rat),lcl(ksi),lcl(eta)),t1,[t:is(ksi,eta)]ise(ksi,eta,t)):not(is(ksi,eta))
3500 m@t3:=moreapp(ksi,eta,m,not(is(ksi,eta)),[x:rat][t:lrt(ksi,x)][u:urt(eta,x)]t2(x,t,u)):not(is(ksi,eta))
3501 eta@t4:=th2"l.ec"(is(ksi,eta),more(ksi,eta),[t:more(ksi,eta)]t3(t)):ec(is(ksi,eta),more(ksi,eta))
3502 [l:less(ksi,eta)][x0:rat][ux:urt(ksi,x0)][lx:lrt(eta,x0)]
3503 t5:=th4"st.isset"(rat,lcl(eta),lcl(ksi),x0,lx,ux):not(is"e"(set(rat),lcl(ksi),lcl(eta)))
3504 t6:=th3"l.imp"(is(ksi,eta),is"e"(set(rat),lcl(ksi),lcl(eta)),t5,[t:is(ksi,eta)]ise(ksi,eta,t)):not(is(ksi,eta))
3505 l@t7:=lessapp(ksi,eta,l,not(is(ksi,eta)),[x:rat][t:urt(ksi,x)][u:lrt(eta,x)]t6(x,t,u)):not(is(ksi,eta))
3506 eta@t8:=th1"l.ec"(less(ksi,eta),is(ksi,eta),[t:less(ksi,eta)]t7(t)):ec(less(ksi,eta),is(ksi,eta))
3507 m@[l:less(ksi,eta)][x0:rat][lx:lrt(ksi,x0)][ux:urt(eta,x0)][y0:rat][uy:urt(ksi,y0)][ly:lrt(eta,y0)]
3508 t9:=cutapp2a(ksi,x0,lx,y0,uy):less"rt"(x0,y0)
3509 t10:=cutapp2b(eta,y0,ly,x0,ux):more"rt"(x0,y0)
3510 t11:=mp(less"rt"(x0,y0),con,t9,ec3e23(is"rt"(x0,y0),more"rt"(x0,y0),less"rt"(x0,y0),satz81b(x0,y0),t10)):con
3511 ux@t12:=lessapp(ksi,eta,l,con,[x:rat][t:urt(ksi,x)][u:lrt(eta,x)]t11(x,t,u)):con
3512 l@t13:=moreapp(ksi,eta,m,con,[x:rat][t:lrt(ksi,x)][u:urt(eta,x)]t12(x,t,u)):con
3513 m@t14:=[t:less(ksi,eta)]t13(t):not(less(ksi,eta))
3514 eta@t15:=th1"l.ec"(more(ksi,eta),less(ksi,eta),[t:more(ksi,eta)]t14(t)):ec(more(ksi,eta),less(ksi,eta))
3515 a:=th6"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),t4,t15,t8):ec3(is(ksi,eta),more(ksi,eta),less(ksi,eta))
3516 [i:is(ksi,eta)]
3517 t16:=or3i1(is(ksi,eta),more(ksi,eta),less(ksi,eta),i):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta))
3518 eta@[n:nis(ksi,eta)]
3519 t17:=th3"l.imp"(is"e"(set(rat),lcl(ksi),lcl(eta)),is(ksi,eta),n,[t:is"e"(set(rat),lcl(ksi),lcl(eta))]isi(ksi,eta,t)):not(is"e"(set(rat),lcl(ksi),lcl(eta)))
3520 t18:=th5"st.isset"(rat,lcl(ksi),lcl(eta),t17):or(more(ksi,eta),more(eta,ksi))
3521 t19:=th8"l.or"(more(ksi,eta),more(eta,ksi),less(ksi,eta),t18,[t:more(eta,ksi)]satz121(eta,ksi,t)):or(more(ksi,eta),less(ksi,eta))
3522 t20:=th7"l.or3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),t19):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta))
3523 eta@b:=th1"l.imp"(is(ksi,eta),or3(is(ksi,eta),more(ksi,eta),less(ksi,eta)),[t:is(ksi,eta)]t16(t),[t:nis(ksi,eta)]t20(t)):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta))
3524 -2123
3525 eta@satz123:=orec3i(is(ksi,eta),more(ksi,eta),less(ksi,eta),b".2123",a".2123"):orec3(is(ksi,eta),more(ksi,eta),less(ksi,eta))
3526 satz123a:=orec3e1(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta))
3527 satz123b:=orec3e2(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123):ec3(is(ksi,eta),more(ksi,eta),less(ksi,eta))
3528 moreis:=or(more(ksi,eta),is(ksi,eta)):'prop'
3529 lessis:=or(less(ksi,eta),is(ksi,eta)):'prop'
3530 [m:moreis(ksi,eta)]
3531 satz124:=th9"l.or"(more(ksi,eta),is(ksi,eta),less(eta,ksi),is(eta,ksi),m,[t:more(ksi,eta)]satz121(t),[t:is(ksi,eta)]symis(cut,ksi,eta,t)):lessis(eta,ksi)
3532 eta@[l:lessis(ksi,eta)]
3533 satz125:=th9"l.or"(less(ksi,eta),is(ksi,eta),more(eta,ksi),is(eta,ksi),l,[t:less(ksi,eta)]satz122(t),[t:is(ksi,eta)]symis(cut,ksi,eta,t)):moreis(eta,ksi)
3534 zeta@[i:is(ksi,eta)][m:more(ksi,zeta)]
3535 ismore1:=isp(cut,[u:cut]more(u,zeta),ksi,eta,m,i):more(eta,zeta)
3536 i@[m:more(zeta,ksi)]
3537 ismore2:=isp(cut,[u:cut]more(zeta,u),ksi,eta,m,i):more(zeta,eta)
3538 i@[l:less(ksi,zeta)]
3539 isless1:=isp(cut,[u:cut]less(u,zeta),ksi,eta,l,i):less(eta,zeta)
3540 i@[l:less(zeta,ksi)]
3541 isless2:=isp(cut,[u:cut]less(zeta,u),ksi,eta,l,i):less(zeta,eta)
3542 i@[m:moreis(ksi,zeta)]
3543 ismoreis1:=isp(cut,[u:cut]moreis(u,zeta),ksi,eta,m,i):moreis(eta,zeta)
3544 i@[m:moreis(zeta,ksi)]
3545 ismoreis2:=isp(cut,[u:cut]moreis(zeta,u),ksi,eta,m,i):moreis(zeta,eta)
3546 i@[l:lessis(ksi,zeta)]
3547 islessis1:=isp(cut,[u:cut]lessis(u,zeta),ksi,eta,l,i):lessis(eta,zeta)
3548 i@[l:lessis(zeta,ksi)]
3549 islessis2:=isp(cut,[u:cut]lessis(zeta,u),ksi,eta,l,i):lessis(zeta,eta)
3550 eta@[i:is(ksi,eta)]
3551 moreisi2:=ori2(more(ksi,eta),is(ksi,eta),i):moreis(ksi,eta)
3552 lessisi2:=ori2(less(ksi,eta),is(ksi,eta),i):lessis(ksi,eta)
3553 eta@[m:more(ksi,eta)]
3554 moreisi1:=ori1(more(ksi,eta),is(ksi,eta),m):moreis(ksi,eta)
3555 eta@[l:less(ksi,eta)]
3556 lessisi1:=ori1(less(ksi,eta),is(ksi,eta),l):lessis(ksi,eta)
3557 zeta@[upsilon:cut][i:is(ksi,eta)][j:is(zeta,upsilon)][m:more(ksi,zeta)]
3558 ismore12:=ismore2(zeta,upsilon,eta,j,ismore1(ksi,eta,zeta,i,m)):more(eta,upsilon)
3559 j@[l:less(ksi,zeta)]
3560 isless12:=isless2(zeta,upsilon,eta,j,isless1(ksi,eta,zeta,i,l)):less(eta,upsilon)
3561 j@[m:moreis(ksi,zeta)]
3562 ismoreis12:=ismoreis2(zeta,upsilon,eta,j,ismoreis1(ksi,eta,zeta,i,m)):moreis(eta,upsilon)
3563 j@[l:lessis(ksi,zeta)]
3564 islessis12:=islessis2(zeta,upsilon,eta,j,islessis1(ksi,eta,zeta,i,l)):lessis(eta,upsilon)
3565 eta@[m:moreis(ksi,eta)]
3566 satz123c:=th7"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123b,comor(more(ksi,eta),is(ksi,eta),m)):not(less(ksi,eta))
3567 eta@[l:lessis(ksi,eta)]
3568 satz123d:=th9"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123b,l):not(more(ksi,eta))
3569 eta@[n:not(more(ksi,eta))]
3570 satz123e:=th2"l.or3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123a,n):lessis(ksi,eta)
3571 eta@[n:not(less(ksi,eta))]
3572 satz123f:=comor(is(ksi,eta),more(ksi,eta),th3"l.or3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123a,n)):moreis(ksi,eta)
3573 eta@[m:more(ksi,eta)]
3574 satz123g:=th3"l.or"(less(ksi,eta),is(ksi,eta),ec3e23(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123b,m),ec3e21(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123b,m)):not(lessis(ksi,eta))
3575 eta@[l:less(ksi,eta)]
3576 satz123h:=th3"l.or"(more(ksi,eta),is(ksi,eta),ec3e32(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123b,l),ec3e31(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123b,l)):not(moreis(ksi,eta))
3577 eta@[n:not(moreis(ksi,eta))]
3578 satz123j:=or3e3(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123a,th5"l.or"(more(ksi,eta),is(ksi,eta),n),th4"l.or"(more(ksi,eta),is(ksi,eta),n)):less(ksi,eta)
3579 eta@[n:not(lessis(ksi,eta))]
3580 satz123k:=or3e2(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123a,th4"l.or"(less(ksi,eta),is(ksi,eta),n),th5"l.or"(less(ksi,eta),is(ksi,eta),n)):more(ksi,eta)
3581 zeta@[l:less(ksi,eta)][k:less(eta,zeta)]
3582 +2126
3583 [x0:rat][ux:urt(ksi,x0)][lx:lrt(eta,x0)][y0:rat][uy:urt(eta,y0)][ly:lrt(zeta,y0)]
3584 t1:=cutapp2a(eta,x0,lx,y0,uy):less"rt"(x0,y0)
3585 t2:=satz119a(ksi,x0,ux,y0,t1):urt(ksi,y0)
3586 t3:=andi(urt(ksi,y0),lrt(zeta,y0),t2,ly):and(urt(ksi,y0),lrt(zeta,y0))
3587 t4:=somei(rat,[x:rat]and(urt(ksi,x),lrt(zeta,x)),y0,t3):less(ksi,zeta)
3588 lx@t5:=lessapp(eta,zeta,k,less(ksi,zeta),[x:rat][t:urt(eta,x)][u:lrt(zeta,x)]t4(x,t,u)):less(ksi,zeta)
3589 -2126
3590 satz126:=lessapp(ksi,eta,l,less(ksi,zeta),[x:rat][t:urt(ksi,x)][u:lrt(eta,x)]t5".2126"(x,t,u)):less(ksi,zeta)
3591 trless:=satz126:less(ksi,zeta)
3592 zeta@[m:more(ksi,eta)][n:more(eta,zeta)]
3593 trmore:=satz122(zeta,ksi,satz126(zeta,eta,ksi,satz121(eta,zeta,n),satz121(ksi,eta,m))):more(ksi,zeta)
3594 zeta@[l:lessis(ksi,eta)][k:less(eta,zeta)]
3595 satz127a:=orapp(less(ksi,eta),is(ksi,eta),less(ksi,zeta),l,[u:less(ksi,eta)]trless(u,k),[u:is(ksi,eta)]isless1(eta,ksi,zeta,symis(cut,ksi,eta,u),k)):less(ksi,zeta)
3596 zeta@[l:less(ksi,eta)][k:lessis(eta,zeta)]
3597 satz127b:=orapp(less(eta,zeta),is(eta,zeta),less(ksi,zeta),k,[u:less(eta,zeta)]trless(l,u),[u:is(eta,zeta)]isless2(eta,zeta,ksi,u,l)):less(ksi,zeta)
3598 zeta@[m:moreis(ksi,eta)][n:more(eta,zeta)]
3599 satz127c:=satz122(zeta,ksi,satz127b(zeta,eta,ksi,satz121(eta,zeta,n),satz124(ksi,eta,m))):more(ksi,zeta)
3600 zeta@[m:more(ksi,eta)][n:moreis(eta,zeta)]
3601 satz127d:=satz122(zeta,ksi,satz127a(zeta,eta,ksi,satz124(eta,zeta,n),satz121(ksi,eta,m))):more(ksi,zeta)
3602 zeta@[l:lessis(ksi,eta)][k:lessis(eta,zeta)]
3603 +2128
3604 [i:is(ksi,eta)][j:is(eta,zeta)]
3605 t1:=lessisi2(ksi,zeta,tris(cut,ksi,eta,zeta,i,j)):lessis(ksi,zeta)
3606 i@[j:less(eta,zeta)]
3607 t2:=lessisi1(ksi,zeta,satz127a(l,j)):lessis(ksi,zeta)
3608 i@t3:=orapp(less(eta,zeta),is(eta,zeta),lessis(ksi,zeta),k,[t:less(eta,zeta)]t2(t),[t:is(eta,zeta)]t1(t)):lessis(ksi,zeta)
3609 k@[j:less(ksi,eta)]
3610 t4:=lessisi1(ksi,zeta,satz127b(j,k)):lessis(ksi,zeta)
3611 -2128
3612 satz128:=orapp(less(ksi,eta),is(ksi,eta),lessis(ksi,zeta),l,[t:less(ksi,eta)]t4".2128"(t),[t:is(ksi,eta)]t3".2128"(t)):lessis(ksi,zeta)
3613 trlessis:=satz128:lessis(ksi,zeta)
3614 zeta@[m:moreis(ksi,eta)][n:moreis(eta,zeta)]
3615 trmoreis:=satz125(zeta,ksi,satz128(zeta,eta,ksi,satz124(eta,zeta,n),satz124(ksi,eta,m))):moreis(ksi,zeta)
3616 eta@[z0:rat][x0:rat][y0:rat]
3617 sumprop1:=and3(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0))):'prop'
3618 z0@sumprop:=some"rt"([x:rat]some"rt"([y:rat]sumprop1(z0,x,y))):'prop'
3619 eta@sum:=setof(rat,[z:rat]sumprop(z)):set(rat)
3620 x0@[lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,pl(x0,y0))]
3621 +iii3
3622 t1:=and3i(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0)),lx,ly,i):sumprop1(z0,x0,y0)
3623 t2:=somei(rat,[y:rat]sumprop1(z0,x0,y),y0,t1):some"rt"([y:rat]sumprop1(z0,x0,y))
3624 t3:=somei(rat,[x:rat]some"rt"([y:rat]sumprop1(z0,x,y)),x0,t2):sumprop(z0)
3625 -iii3
3626 sum1:=estii(rat,[z:rat]sumprop(z),z0,t3".iii3"):in(z0,sum)
3627 z0@[i:in(z0,sum)][p:'prop'][p1:[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,pl(x,y))]p]
3628 +*iii3
3629 p1@t4:=estie(rat,[z:rat]sumprop(z),z0,i):sumprop(z0)
3630 -iii3
3631 p1@[x0:rat][px:some"rt"([y:rat]sumprop1(z0,x0,y))][y0:rat][py:sumprop1(z0,x0,y0)]
3632 +*iii3
3633 py@t5:=and3e1(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0)),py):lrt(ksi,x0)
3634 t6:=and3e2(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0)),py):lrt(eta,y0)
3635 t7:=and3e3(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0)),py):is"rt"(z0,pl(x0,y0))
3636 t8:=<t7><t6><y0><t5><x0>p1:p
3637 px@t9:=someapp(rat,[y:rat]sumprop1(z0,x0,y),px,p,[y:rat][t:sumprop1(z0,x0,y)]t8(y,t)):p
3638 -iii3
3639 p1@sumapp:=someapp(rat,[x:rat]some"rt"([y:rat]sumprop1(z0,x,y)),t4".iii3",p,[x:rat][t:some"rt"([y:rat]sumprop1(z0,x,y))]t9".iii3"(x,t)):p
3640 eta@[x1:rat][ux:urt(ksi,x1)][y1:rat][uy:urt(eta,y1)]
3641 +3129
3642 [z0:rat][i:in(z0,sum)][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][j:is"rt"(z0,pl(x0,y0))]
3643 t1:=cutapp2a(ksi,x0,lx,x1,ux):less"rt"(x0,x1)
3644 t2:=cutapp2a(eta,y0,ly,y1,uy):less"rt"(y0,y1)
3645 t3:=isless1"rt"(pl(x0,y0),z0,pl(x1,y1),symis(rat,z0,pl(x0,y0),j),satz98a(x0,x1,y0,y1,t1,t2)):less"rt"(z0,pl(x1,y1))
3646 t4:=ec3e31(is"rt"(z0,pl(x1,y1)),more"rt"(z0,pl(x1,y1)),less"rt"(z0,pl(x1,y1)),satz81b(z0,pl(x1,y1)),t3):nis"rt"(z0,pl(x1,y1))
3647 i@t5:=sumapp(ksi,eta,z0,i,nis"rt"(z0,pl(x1,y1)),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,pl(x,y))]t4(x,t,y,u,v)):nis"rt"(z0,pl(x1,y1))
3648 -3129
3649 satz129a:=th3"l.imp"(in(pl(x1,y1),sum),nis"rt"(pl(x1,y1),pl(x1,y1)),weli(is"rt"(pl(x1,y1),pl(x1,y1)),refis(rat,pl(x1,y1))),[t:in(pl(x1,y1),sum)]t5".3129"(pl(x1,y1),t)):not(in(pl(x1,y1),sum))
3650 +*3129
3651 eta@[u0:rat][i:in(u0,sum)][z0:rat][l:less"rt"(z0,u0)][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][j:is"rt"(u0,pl(x0,y0))]
3652 t6:=isless2"rt"(u0,pl(x0,y0),z0,j,l):less"rt"(z0,pl(x0,y0))
3653 z1:=ov(z0,pl(x0,y0)):rat
3654 t7:=isless12"rt"(z0,ts(z1,pl(x0,y0)),pl(x0,y0),ts(1rt,pl(x0,y0)),satz110f(z0,pl(x0,y0)),example1d(pl(x0,y0)),t6):less"rt"(ts(z1,pl(x0,y0)),ts(1rt,pl(x0,y0)))
3655 t8:=satz106c(z1,1rt,pl(x0,y0),t7):less"rt"(z1,1rt)
3656 t9:=isless2"rt"(ts(x0,1rt),x0,ts(x0,z1),example1a(x0),satz105f(z1,1rt,x0,t8)):less"rt"(ts(x0,z1),x0)
3657 t10:=isless2"rt"(ts(y0,1rt),y0,ts(y0,z1),example1a(y0),satz105f(z1,1rt,y0,t8)):less"rt"(ts(y0,z1),y0)
3658 t11:=satz120(ksi,x0,lx,ts(x0,z1),t9):lrt(ksi,ts(x0,z1))
3659 t12:=satz120(eta,y0,ly,ts(y0,z1),t10):lrt(eta,ts(y0,z1))
3660 t13:=tris(rat,pl(ts(x0,z1),ts(y0,z1)),ts(pl(x0,y0),z1),z0,distpt1(x0,y0,z1),satz110c(z0,pl(x0,y0))):is"rt"(pl(ts(x0,z1),ts(y0,z1)),z0)
3661 t14:=symis(rat,pl(ts(x0,z1),ts(y0,z1)),z0,t13):is"rt"(z0,pl(ts(x0,z1),ts(y0,z1)))
3662 t15:=sum1(ksi,eta,z0,ts(x0,z1),t11,ts(y0,z1),t12,t14):in(z0,sum)
3663 l@t16:=sumapp(ksi,eta,u0,i,in(z0,sum),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(u0,pl(x,y))]t15(x,t,y,u,v)):in(z0,sum)
3664 eta@[z0:rat][i:in(z0,sum)][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][j:is"rt"(z0,pl(x0,y0))][x1:rat][lx1:lrt(ksi,x1)][l:less"rt"(x0,x1)]
3665 t17:=sum1(ksi,eta,pl(x1,y0),x1,lx1,y0,ly,refis(rat,pl(x1,y0))):in(pl(x1,y0),sum)
3666 t18:=satz96a(x1,x0,y0,satz83(x0,x1,l)):more"rt"(pl(x1,y0),pl(x0,y0))
3667 t19:=ismore2"rt"(pl(x0,y0),z0,pl(x1,y0),symis(rat,z0,pl(x0,y0),j),t18):more"rt"(pl(x1,y0),z0)
3668 t20:=andi(in(pl(x1,y0),sum),more"rt"(pl(x1,y0),z0),t17,t19):and(in(pl(x1,y0),sum),more"rt"(pl(x1,y0),z0))
3669 t21:=somei(rat,[y:rat]and(in(y,sum),more"rt"(y,z0)),pl(x1,y0),t20):some"rt"([y:rat]and(in(y,sum),more"rt"(y,z0)))
3670 j@t22:=cutapp3(ksi,x0,lx,some"rt"([y:rat]and(in(y,sum),more"rt"(y,z0))),[x:rat][t:lrt(ksi,x)][u:less"rt"(x0,x)]t21(x,t,u)):some"rt"([y:rat]and(in(y,sum),more"rt"(y,z0)))
3671 i@t23:=sumapp(ksi,eta,z0,i,some"rt"([y:rat]and(in(y,sum),more"rt"(y,z0))),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,pl(x,y))]t22(x,t,y,u,v)):some"rt"([y:rat]and(in(y,sum),more"rt"(y,z0)))
3672 eta@[x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][x1:rat][ux:urt(ksi,x1)][y1:rat][uy:urt(eta,y1)]
3673 t24:=cut2(sum,pl(x0,y0),sum1(ksi,eta,pl(x0,y0),x0,lx,y0,ly,refis(rat,pl(x0,y0))),pl(x1,y1),satz129a(x1,ux,y1,uy),[x:rat][t:in(x,sum)][y:rat][u:less"rt"(y,x)]t16(x,t,y,u),[x:rat][t:in(x,sum)]t23(x,t)):cutprop(sum)
3674 ux@t25:=cutapp1b(eta,cutprop(sum),[y:rat][t:urt(eta,y)]t24(y,t)):cutprop(sum)
3675 ly@t26:=cutapp1b(ksi,cutprop(sum),[x:rat][t:urt(ksi,x)]t25(x,t)):cutprop(sum)
3676 lx@t27:=cutapp1a(eta,cutprop(sum),[y:rat][t:lrt(eta,y)]t26(y,t)):cutprop(sum)
3677 -3129
3678 eta@satz129:=cutapp1a(ksi,cutprop(sum),[x:rat][t:lrt(ksi,x)]t27".3129"(x,t)):cutprop(sum)
3679 pl:=cutof(sum,satz129):cut
3680 [z0:rat][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,pl"rt"(x0,y0))]
3681 lrtpl:=ine(sum,satz129,z0,sum1(z0,x0,lx,y0,ly,i)):lrt(pl(ksi,eta),z0)
3682 eta@[z0:rat][x0:rat][ux:urt(ksi,x0)][y0:rat][uy:urt(eta,y0)][i:is"rt"(z0,pl"rt"(x0,y0))]
3683 +*iii3
3684 i@t10:=isp1(rat,[x:rat]not(in(x,sum)),pl"rt"(x0,y0),z0,satz129a(x0,ux,y0,uy),i):not(in(z0,sum))
3685 -iii3
3686 i@urtpl:=th3"l.imp"(lrt(pl(ksi,eta),z0),in(z0,sum),t10".iii3",[t:lrt(pl(ksi,eta),z0)]ini(sum,satz129,z0,t)):urt(pl(ksi,eta),z0)
3687 eta@[z0:rat][lz:lrt(pl(ksi,eta),z0)][p:'prop'][p1:[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,pl"rt"(x,y))]p]
3688 +*iii3
3689 p1@t11:=ini(sum,satz129,z0,lz):in(z0,sum)
3690 -iii3
3691 p1@plapp:=sumapp(z0,t11".iii3",p,p1):p
3692 zeta@[i:is(ksi,eta)]
3693 ispl1:=isf(cut,cut,[u:cut]pl(u,zeta),ksi,eta,i):is(pl(ksi,zeta),pl(eta,zeta))
3694 ispl2:=isf(cut,cut,[u:cut]pl(zeta,u),ksi,eta,i):is(pl(zeta,ksi),pl(zeta,eta))
3695 upsilon@[i:is(ksi,eta)][j:is(zeta,upsilon)]
3696 ispl12:=tris(cut,pl(ksi,zeta),pl(eta,zeta),pl(eta,upsilon),ispl1(i),ispl2(zeta,upsilon,eta,j)):is(pl(ksi,zeta),pl(eta,upsilon))
3697 +3130
3698 eta@[z0:rat][lz:lrt(pl(ksi,eta),z0)][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,pl"rt"(x0,y0))]
3699 t1:=tris(rat,z0,pl"rt"(x0,y0),pl"rt"(y0,x0),i,compl(x0,y0)):is"rt"(z0,pl"rt"(y0,x0))
3700 t2:=lrtpl(eta,ksi,z0,y0,ly,x0,lx,t1):lrt(pl(eta,ksi),z0)
3701 lz@t3:=plapp(ksi,eta,z0,lz,lrt(pl(eta,ksi),z0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,pl"rt"(x,y))]t2(x,t,y,u,v)):lrt(pl(eta,ksi),z0)
3702 -3130
3703 eta@satz130:=isi1(pl(ksi,eta),pl(eta,ksi),[x:rat][t:lrt(pl(ksi,eta),x)]t3".3130"(x,t),[x:rat][t:lrt(pl(eta,ksi),x)]t3".3130"(eta,ksi,x,t)):is(pl(ksi,eta),pl(eta,ksi))
3704 compl:=satz130:is(pl(ksi,eta),pl(eta,ksi))
3705 +3131
3706 zeta@[u0:rat][lu:lrt(pl(pl(ksi,eta),zeta),u0)][v0:rat][lv:lrt(pl(ksi,eta),v0)][z0:rat][lz:lrt(zeta,z0)][i:is"rt"(u0,pl"rt"(v0,z0))][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][j:is"rt"(v0,pl"rt"(x0,y0))]
3707 t1:=tr3is(rat,u0,pl"rt"(v0,z0),pl"rt"(pl"rt"(x0,y0),z0),pl"rt"(x0,pl"rt"(y0,z0)),i,ispl1"rt"(v0,pl"rt"(x0,y0),z0,j),asspl1(x0,y0,z0)):is"rt"(u0,pl"rt"(x0,pl"rt"(y0,z0)))
3708 t2:=lrtpl(eta,zeta,pl"rt"(y0,z0),y0,ly,z0,lz,refis(rat,pl"rt"(y0,z0))):lrt(pl(eta,zeta),pl"rt"(y0,z0))
3709 t3:=lrtpl(ksi,pl(eta,zeta),u0,x0,lx,pl"rt"(y0,z0),t2,t1):lrt(pl(ksi,pl(eta,zeta)),u0)
3710 i@t4:=plapp(ksi,eta,v0,lv,lrt(pl(ksi,pl(eta,zeta)),u0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(v0,pl"rt"(x,y))]t3(x,t,y,u,v)):lrt(pl(ksi,pl(eta,zeta)),u0)
3711 lu@t5:=plapp(pl(ksi,eta),zeta,u0,lu,lrt(pl(ksi,pl(eta,zeta)),u0),[x:rat][t:lrt(pl(ksi,eta),x)][y:rat][u:lrt(zeta,y)][v:is"rt"(u0,pl"rt"(x,y))]t4(x,t,y,u,v)):lrt(pl(ksi,pl(eta,zeta)),u0)
3712 u0@[lu:lrt(pl(ksi,pl(eta,zeta)),u0)][x0:rat][lx:lrt(ksi,x0)][v0:rat][lv:lrt(pl(eta,zeta),v0)][i:is"rt"(u0,pl"rt"(x0,v0))][y0:rat][ly:lrt(eta,y0)][z0:rat][lz:lrt(zeta,z0)][j:is"rt"(v0,pl"rt"(y0,z0))]
3713 t6:=tr3is(rat,u0,pl"rt"(x0,v0),pl"rt"(x0,pl"rt"(y0,z0)),pl"rt"(pl"rt"(x0,y0),z0),i,ispl2"rt"(v0,pl"rt"(y0,z0),x0,j),asspl2(x0,y0,z0)):is"rt"(u0,pl"rt"(pl"rt"(x0,y0),z0))
3714 t7:=lrtpl(ksi,eta,pl"rt"(x0,y0),x0,lx,y0,ly,refis(rat,pl"rt"(x0,y0))):lrt(pl(ksi,eta),pl"rt"(x0,y0))
3715 t8:=lrtpl(pl(ksi,eta),zeta,u0,pl"rt"(x0,y0),t7,z0,lz,t6):lrt(pl(pl(ksi,eta),zeta),u0)
3716 i@t9:=plapp(eta,zeta,v0,lv,lrt(pl(pl(ksi,eta),zeta),u0),[x:rat][t:lrt(eta,x)][y:rat][u:lrt(zeta,y)][v:is"rt"(v0,pl"rt"(x,y))]t8(x,t,y,u,v)):lrt(pl(pl(ksi,eta),zeta),u0)
3717 lu@t10:=plapp(ksi,pl(eta,zeta),u0,lu,lrt(pl(pl(ksi,eta),zeta),u0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(pl(eta,zeta),y)][v:is"rt"(u0,pl"rt"(x,y))]t9(x,t,y,u,v)):lrt(pl(pl(ksi,eta),zeta),u0)
3718 -3131
3719 zeta@satz131:=isi1(pl(pl(ksi,eta),zeta),pl(ksi,pl(eta,zeta)),[x:rat][t:lrt(pl(pl(ksi,eta),zeta),x)]t5".3131"(x,t),[x:rat][t:lrt(pl(ksi,pl(eta,zeta)),x)]t10".3131"(x,t)):is(pl(pl(ksi,eta),zeta),pl(ksi,pl(eta,zeta)))
3720 asspl1:=satz131:is(pl(pl(ksi,eta),zeta),pl(ksi,pl(eta,zeta)))
3721 asspl2:=symis(cut,pl(pl(ksi,eta),zeta),pl(ksi,pl(eta,zeta)),satz131):is(pl(ksi,pl(eta,zeta)),pl(pl(ksi,eta),zeta))
3722 ksi@[a0:rat]
3723 +3132
3724 [x0:rat][y0:rat]
3725 prop1:=and(lrt(ksi,x0),urt(ksi,y0)):'prop'
3726 [p:prop1]
3727 t1:=cutapp2b(x0,ande1(lrt(ksi,x0),urt(ksi,y0),p),y0,ande2(lrt(ksi,x0),urt(ksi,y0),p)):more"rt"(y0,x0)
3728 prop2:=is"rt"(mn(y0,x0,t1),a0):'prop'
3729 y0@prop3:=and(prop1,[t:prop1]prop2(t)):'prop'
3730 a0@prop4:=some"rt"([x:rat]some"rt"([y:rat]prop3(x,y))):'prop'
3731 x0@[lx:lrt(ksi,x0)][y0:rat][uy:urt(ksi,y0)]
3732 t2:=cutapp2b(x0,lx,y0,uy):more"rt"(y0,x0)
3733 [n:nat][m:more"rt"(ts(rtofn(n),a0),mn(y0,x0,t2))]
3734 t3:=satz96d(ts(rtofn(n),a0),mn(y0,x0,t2),x0,m):more"rt"(pl"rt"(x0,ts(rtofn(n),a0)),pl"rt"(x0,mn(y0,x0,t2)))
3735 t4:=ismore2"rt"(pl"rt"(x0,mn(y0,x0,t2)),y0,pl"rt"(x0,ts(rtofn(n),a0)),satz101c(y0,x0,t2),t3):more"rt"(pl"rt"(x0,ts(rtofn(n),a0)),y0)
3736 t5:=satz119(y0,uy,pl"rt"(x0,ts(rtofn(n),a0)),t4):urt(pl"rt"(x0,ts(rtofn(n),a0)))
3737 t6:=somei(nat,[x:nat]urt(pl"rt"(x0,ts(rtofn(x),a0))),n,t5):some"n"([x:nat]urt(pl"rt"(x0,ts(rtofn(x),a0))))
3738 uy@t7:=someapp(nat,[x:nat]more"rt"(ts(rtofn(x),a0),mn(y0,x0,t2)),satz115(a0,mn(y0,x0,t2)),some"n"([x:nat]urt(pl"rt"(x0,ts(rtofn(x),a0)))),[x:nat][t:more"rt"(ts(rtofn(x),a0),mn(y0,x0,t2))]t6(x,t)):some"n"([x:nat]urt(pl"rt"(x0,ts(rtofn(x),a0))))
3739 [u:nat][m:min"n"([x:nat]urt(pl"rt"(x0,ts(rtofn(x),a0))),u)]
3740 t8:=ande1(lb"n"([x:nat]urt(pl"rt"(x0,ts(rtofn(x),a0))),u),urt(pl"rt"(x0,ts(rtofn(u),a0))),m):lb"n"([x:nat]urt(pl"rt"(x0,ts(rtofn(x),a0))),u)
3741 t9:=ande2(lb"n"([x:nat]urt(pl"rt"(x0,ts(rtofn(x),a0))),u),urt(pl"rt"(x0,ts(rtofn(u),a0))),m):urt(pl"rt"(x0,ts(rtofn(u),a0)))
3742 [i:is"n"(u,1)]
3743 u0:=pl"rt"(x0,a0):rat
3744 t10:=tr3is(rat,ts(rtofn(u),a0),ts(1rt,a0),ts(a0,1rt),a0,ists1(rtofn(u),1rt,a0,isnert(u,1,i)),comts(1rt,a0),example1a(a0)):is"rt"(ts(rtofn(u),a0),a0)
3745 t11:=isp(rat,[x:rat]urt(pl"rt"(x0,x)),ts(rtofn(u),a0),a0,t9,t10):urt(ksi,u0)
3746 t12:=andi(lrt(ksi,x0),urt(ksi,u0),lx,t11):prop1(x0,u0)
3747 [p:prop1(x0,u0)]
3748 t13:=symis(rat,a0,mn(u0,x0,t1(x0,u0,p)),satz101g(u0,x0,a0,t1(x0,u0,p),refis(rat,u0))):prop2(x0,u0,p)
3749 i@t14:=andi(prop1(x0,u0),[t:prop1(x0,u0)]prop2(x0,u0,t),t12,[t:prop1(x0,u0)]t13(t)):prop3(x0,u0)
3750 t15:=somei(rat,[y:rat]prop3(x0,y),u0,t14):some"rt"([y:rat]prop3(x0,y))
3751 t16:=somei(rat,[x:rat]some"rt"([y:rat]prop3(x,y)),x0,t15):prop4
3752 m@[o:more"n"(u,1)]
3753 t17:=morei(rtofn(u),1rt,fr(u,1),fr(1,1),inclass(fr(u,1)),inclass(fr(1,1)),satz111d(u,1,o)):more"rt"(rtofn(u),1rt)
3754 um10:=mn(rtofn(u),1rt,t17):rat
3755 t18:=satz112g(rtofn(u),natrti(u),1rt,natrti(1),t17):natrt(um10)
3756 um1:=nofrt(um10,t18):nat
3757 v0:=pl"rt"(x0,ts(um10,a0)):rat
3758 w0:=pl"rt"(x0,ts(rtofn(u),a0)):rat
3759 t19:=isless2"rt"(pl"rt"(um10,1rt),rtofn(u),um10,satz101e(rtofn(u),1rt,t17),satz94a(um10,1rt)):less"rt"(um10,rtofn(u))
3760 t20:=lesse(um10,rtofn(u),fr(um1,1),fr(u,1),inclassn(um10,t18),inclass(fr(u,1)),t19):lessf(fr(um1,1),fr(u,1))
3761 t21:=satz111c(um1,u,t20):less"n"(um1,u)
3762 t22:=th3"l.imp"(lessis"n"(u,um1),moreis"n"(um1,u),satz10h(um1,u,t21),[t:lessis"n"(u,um1)]satz14(u,um1,t)):not(lessis"n"(u,um1))
3763 t23:=et(lrt(pl"rt"(x0,ts(rtofn(um1),a0))),th3"l.imp"(urt(pl"rt"(x0,ts(rtofn(um1),a0))),lessis"n"(u,um1),t22,<um1>t8)):lrt(pl"rt"(x0,ts(rtofn(um1),a0)))
3764 t24:=isp1(rat,[x:rat]lrt(pl"rt"(x0,ts(x,a0))),rtofn(um1),um10,t23,isrtn1(um10,t18)):lrt(ksi,v0)
3765 t25:=andi(lrt(ksi,v0),urt(ksi,w0),t24,t9):prop1(v0,w0)
3766 t26:=tr3is(rat,pl"rt"(ts(um10,a0),a0),pl"rt"(ts(um10,a0),ts(1rt,a0)),ts(pl"rt"(um10,1rt),a0),ts(rtofn(u),a0),ispl2"rt"(a0,ts(1rt,a0),ts(um10,a0),example1d(a0)),distpt1(um10,1rt,a0),ists1(pl"rt"(um10,1rt),rtofn(u),a0,satz101e(rtofn(u),1rt,t17))):is"rt"(pl"rt"(ts(um10,a0),a0),ts(rtofn(u),a0))
3767 t27:=tris(rat,pl"rt"(v0,a0),pl"rt"(x0,pl"rt"(ts(um10,a0),a0)),w0,asspl1"rt"(x0,ts(um10,a0),a0),ispl2"rt"(pl"rt"(ts(um10,a0),a0),ts(rtofn(u),a0),x0,t26)):is"rt"(pl"rt"(v0,a0),w0)
3768 [p:prop1(v0,w0)]
3769 t28:=symis(rat,a0,mn(w0,v0,t1(v0,w0,p)),satz101g(w0,v0,a0,t1(v0,w0,p),t27)):prop2(v0,w0,p)
3770 o@t29:=andi(prop1(v0,w0),[t:prop1(v0,w0)]prop2(v0,w0,t),t25,[t:prop1(v0,w0)]t28(t)):prop3(v0,w0)
3771 t30:=somei(rat,[y:rat]prop3(v0,y),w0,t29):some"rt"([y:rat]prop3(v0,y))
3772 t31:=somei(rat,[x:rat]some"rt"([y:rat]prop3(x,y)),v0,t30):prop4
3773 m@t32:=orapp(more"n"(u,1),is"n"(u,1),prop4,satz24(u),[t:more"n"(u,1)]t31(t),[t:is"n"(u,1)]t16(t)):prop4
3774 uy@t34:=someapp(nat,[x:nat]min"n"([y:nat]urt(pl"rt"(x0,ts(rtofn(y),a0))),x),satz27([y:nat]urt(pl"rt"(x0,ts(rtofn(y),a0))),t7),prop4,[x:nat][t:min"n"([y:nat]urt(pl"rt"(x0,ts(rtofn(y),a0))),x)]t32(x,t)):prop4
3775 lx@t35:=cutapp1b(prop4,[y:rat][t:urt(ksi,y)]t34(y,t)):prop4
3776 -3132
3777 satz132:=cutapp1a(prop4".3132",[x:rat][t:lrt(ksi,x)]t35".3132"(x,t)):some"rt"([x:rat]some"rt"([y:rat]and(and(lrt(ksi,x),urt(ksi,y)),[t:and(lrt(ksi,x),urt(ksi,y))]is"rt"(mn(y,x,cutapp2b(x,ande1(lrt(ksi,x),urt(ksi,y),t),y,ande2(lrt(ksi,x),urt(ksi,y),t))),a0))))
3778 ksi@[p:'prop'][a0:rat][p1:[x:rat][t:lrt(ksi,x)][y:rat][u:urt(ksi,y)][v:is"rt"(mn(y,x,cutapp2b(x,t,y,u)),a0)]p]
3779 +*3132
3780 p1@[x0:rat][s:some"rt"([y:rat]prop3(a0,x0,y))][y0:rat][p3:prop3(a0,x0,y0)]
3781 t36:=ande1(prop1(a0,x0,y0),[t:prop1(a0,x0,y0)]prop2(a0,x0,y0,t),p3):prop1(a0,x0,y0)
3782 t37:=ande2"l.r"(prop1(a0,x0,y0),[t:prop1(a0,x0,y0)]prop2(a0,x0,y0,t),p3):prop2(a0,x0,y0,t36)
3783 t38:=ande1(lrt(ksi,x0),urt(ksi,y0),t36):lrt(ksi,x0)
3784 t39:=ande2(lrt(ksi,x0),urt(ksi,y0),t36):urt(ksi,y0)
3785 t40:=satz101g(y0,x0,mn(y0,x0,cutapp2b(x0,t38,y0,t39)),t1(a0,x0,y0,t36),satz101c(y0,x0,cutapp2b(x0,t38,y0,t39))):is"rt"(mn(y0,x0,cutapp2b(x0,t38,y0,t39)),mn(y0,x0,t1(a0,x0,y0,t36)))
3786 t41:=tris(rat,mn(y0,x0,cutapp2b(x0,t38,y0,t39)),mn(y0,x0,t1(a0,x0,y0,t36)),a0,t40,t37):is"rt"(mn(y0,x0,cutapp2b(x0,t38,y0,t39)),a0)
3787 t42:=<t41><t39><y0><t38><x0>p1:p
3788 s@t43:=someapp(rat,[y:rat]prop3(a0,x0,y),s,p,[y:rat][t:prop3(a0,x0,y)]t42(y,t)):p
3789 -3132
3790 p1@satz132app:=someapp(rat,[x:rat]some"rt"([y:rat]prop3".3132"(a0,x,y)),satz132(a0),p,[x:rat][t:some"rt"([y:rat]prop3".3132"(a0,x,y))]t43".3132"(x,t)):p
3791 +3133
3792 eta@[y0:rat][ly:lrt(eta,y0)][x0:rat][lx:lrt(ksi,x0)][u0:rat][uu:urt(ksi,u0)][i:is"rt"(mn(u0,x0,cutapp2b(x0,lx,u0,uu)),y0)]
3793 t1:=tris(rat,u0,pl"rt"(x0,mn(u0,x0,cutapp2b(x0,lx,u0,uu))),pl"rt"(x0,y0),satz101d(u0,x0,cutapp2b(x0,lx,u0,uu)),ispl2"rt"(mn(u0,x0,cutapp2b(x0,lx,u0,uu)),y0,x0,i)):is"rt"(u0,pl"rt"(x0,y0))
3794 t2:=lrtpl(ksi,eta,u0,x0,lx,y0,ly,t1):lrt(pl(ksi,eta),u0)
3795 t3:=andi(lrt(pl(ksi,eta),u0),urt(ksi,u0),t2,uu):and(lrt(pl(ksi,eta),u0),urt(ksi,u0))
3796 t4:=somei(rat,[x:rat]and(lrt(pl(ksi,eta),x),urt(ksi,x)),u0,t3):more(pl(ksi,eta),ksi)
3797 ly@t5:=satz132app(ksi,more(pl(ksi,eta),ksi),y0,[x:rat][t:lrt(ksi,x)][y:rat][u:urt(ksi,y)][v:is"rt"(mn(y,x,cutapp2b(x,t,y,u)),y0)]t4(x,t,y,u,v)):more(pl(ksi,eta),ksi)
3798 -3133
3799 eta@satz133:=cutapp1a(eta,more(pl(ksi,eta),ksi),[x:rat][t:lrt(eta,x)]t5".3133"(x,t)):more(pl(ksi,eta),ksi)
3800 satz133a:=satz121(pl(ksi,eta),ksi,satz133):less(ksi,pl(ksi,eta))
3801 zeta@[m:more(ksi,eta)]
3802 +3134
3803 [y0:rat][ly:lrt(ksi,y0)][uy:urt(eta,y0)][x0:rat][lx:lrt(ksi,x0)][l:less"rt"(y0,x0)]
3804 t1:=satz119a(eta,y0,uy,x0,l):urt(eta,x0)
3805 t2:=satz83(y0,x0,l):more"rt"(x0,y0)
3806 [u0:rat][lu:lrt(zeta,u0)][z0:rat][uz:urt(zeta,z0)][i:is"rt"(mn(z0,u0,cutapp2b(zeta,u0,lu,z0,uz)),mn(x0,y0,t2))]
3807 t3:=tris(rat,z0,pl"rt"(mn(z0,u0,cutapp2b(zeta,u0,lu,z0,uz)),u0),pl"rt"(mn(x0,y0,t2),u0),satz101f(z0,u0,cutapp2b(zeta,u0,lu,z0,uz)),ispl1"rt"(mn(z0,u0,cutapp2b(zeta,u0,lu,z0,uz)),mn(x0,y0,t2),u0,i)):is"rt"(z0,pl"rt"(mn(x0,y0,t2),u0))
3808 t4:=tr3is(rat,pl"rt"(y0,z0),pl"rt"(y0,pl"rt"(mn(x0,y0,t2),u0)),pl"rt"(pl"rt"(y0,mn(x0,y0,t2)),u0),pl"rt"(x0,u0),ispl2"rt"(z0,pl"rt"(mn(x0,y0,t2),u0),y0,t3),asspl2"rt"(y0,mn(x0,y0,t2),u0),ispl1"rt"(pl"rt"(y0,mn(x0,y0,t2)),x0,u0,satz101c(x0,y0,t2))):is"rt"(pl"rt"(y0,z0),pl"rt"(x0,u0))
3809 t5:=lrtpl(ksi,zeta,pl"rt"(y0,z0),x0,lx,u0,lu,t4):lrt(pl(ksi,zeta),pl"rt"(y0,z0))
3810 t6:=urtpl(eta,zeta,pl"rt"(y0,z0),y0,uy,z0,uz,refis(rat,pl"rt"(y0,z0))):urt(pl(eta,zeta),pl"rt"(y0,z0))
3811 t7:=andi(lrt(pl(ksi,zeta),pl"rt"(y0,z0)),urt(pl(eta,zeta),pl"rt"(y0,z0)),t5,t6):and(lrt(pl(ksi,zeta),pl"rt"(y0,z0)),urt(pl(eta,zeta),pl"rt"(y0,z0)))
3812 t8:=somei(rat,[x:rat]and(lrt(pl(ksi,zeta),x),urt(pl(eta,zeta),x)),pl"rt"(y0,z0),t7):more(pl(ksi,zeta),pl(eta,zeta))
3813 l@t9:=satz132app(zeta,more(pl(ksi,zeta),pl(eta,zeta)),mn(x0,y0,t2),[x:rat][t:lrt(zeta,x)][y:rat][u:urt(zeta,y)][v:is"rt"(mn(y,x,cutapp2b(zeta,x,t,y,u)),mn(x0,y0,t2))]t8(x,t,y,u,v)):more(pl(ksi,zeta),pl(eta,zeta))
3814 uy@t10:=cutapp3(ksi,y0,ly,more(pl(ksi,zeta),pl(eta,zeta)),[x:rat][t:lrt(ksi,x)][u:less"rt"(y0,x)]t9(x,t,u)):more(pl(ksi,zeta),pl(eta,zeta))
3815 -3134
3816 satz134:=moreapp(ksi,eta,m,more(pl(ksi,zeta),pl(eta,zeta)),[x:rat][t:lrt(ksi,x)][u:urt(eta,x)]t10".3134"(x,t,u)):more(pl(ksi,zeta),pl(eta,zeta))
3817 zeta@[m:more(ksi,eta)]
3818 satz135a:=satz134(m):more(pl(ksi,zeta),pl(eta,zeta))
3819 zeta@[i:is(ksi,eta)]
3820 satz135b:=ispl1(ksi,eta,zeta,i):is(pl(ksi,zeta),pl(eta,zeta))
3821 zeta@[l:less(ksi,eta)]
3822 satz135c:=satz121(pl(eta,zeta),pl(ksi,zeta),satz134(eta,ksi,zeta,satz122(ksi,eta,l))):less(pl(ksi,zeta),pl(eta,zeta))
3823 m@satz135d:=ismore12(pl(ksi,zeta),pl(zeta,ksi),pl(eta,zeta),pl(zeta,eta),compl(ksi,zeta),compl(eta,zeta),satz135a):more(pl(zeta,ksi),pl(zeta,eta))
3824 i@satz135e:=ispl2(ksi,eta,zeta,i):is(pl(zeta,ksi),pl(zeta,eta))
3825 l@satz135f:=isless12(pl(ksi,zeta),pl(zeta,ksi),pl(eta,zeta),pl(zeta,eta),compl(ksi,zeta),compl(eta,zeta),satz135c):less(pl(zeta,ksi),pl(zeta,eta))
3826 upsilon@[i:is(ksi,eta)][m:more(zeta,upsilon)]
3827 satz135g:=ismore2(pl(ksi,upsilon),pl(eta,upsilon),pl(ksi,zeta),ispl1(ksi,eta,upsilon,i),satz135d(zeta,upsilon,ksi,m)):more(pl(ksi,zeta),pl(eta,upsilon))
3828 satz135h:=ismore12(pl(ksi,zeta),pl(zeta,ksi),pl(eta,upsilon),pl(upsilon,eta),compl(ksi,zeta),compl(eta,upsilon),satz135g):more(pl(zeta,ksi),pl(upsilon,eta))
3829 i@[l:less(zeta,upsilon)]
3830 satz135j:=isless2(pl(ksi,upsilon),pl(eta,upsilon),pl(ksi,zeta),ispl1(ksi,eta,upsilon,i),satz135f(zeta,upsilon,ksi,l)):less(pl(ksi,zeta),pl(eta,upsilon))
3831 satz135k:=isless12(pl(ksi,zeta),pl(zeta,ksi),pl(eta,upsilon),pl(upsilon,eta),compl(ksi,zeta),compl(eta,upsilon),satz135j):less(pl(zeta,ksi),pl(upsilon,eta))
3832 +3136
3833 zeta@t1:=satz123a(ksi,eta):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta))
3834 t2:=satz123b(pl(ksi,zeta),pl(eta,zeta)):ec3(is(pl(ksi,zeta),pl(eta,zeta)),more(pl(ksi,zeta),pl(eta,zeta)),less(pl(ksi,zeta),pl(eta,zeta)))
3835 -3136
3836 zeta@[m:more(pl(ksi,zeta),pl(eta,zeta))]
3837 satz136a:=th11"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),is(pl(ksi,zeta),pl(eta,zeta)),more(pl(ksi,zeta),pl(eta,zeta)),less(pl(ksi,zeta),pl(eta,zeta)),t1".3136",t2".3136",[u:is(ksi,eta)]satz135b(ksi,eta,zeta,u),[u:more(ksi,eta)]satz135a(ksi,eta,zeta,u),[u:less(ksi,eta)]satz135c(ksi,eta,zeta,u),m):more(ksi,eta)
3838 zeta@[i:is(pl(ksi,zeta),pl(eta,zeta))]
3839 satz136b:=th10"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),is(pl(ksi,zeta),pl(eta,zeta)),more(pl(ksi,zeta),pl(eta,zeta)),less(pl(ksi,zeta),pl(eta,zeta)),t1".3136",t2".3136",[u:is(ksi,eta)]satz135b(ksi,eta,zeta,u),[u:more(ksi,eta)]satz135a(ksi,eta,zeta,u),[u:less(ksi,eta)]satz135c(ksi,eta,zeta,u),i):is(ksi,eta)
3840 zeta@[l:less(pl(ksi,zeta),pl(eta,zeta))]
3841 satz136c:=th12"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),is(pl(ksi,zeta),pl(eta,zeta)),more(pl(ksi,zeta),pl(eta,zeta)),less(pl(ksi,zeta),pl(eta,zeta)),t1".3136",t2".3136",[u:is(ksi,eta)]satz135b(ksi,eta,zeta,u),[u:more(ksi,eta)]satz135a(ksi,eta,zeta,u),[u:less(ksi,eta)]satz135c(ksi,eta,zeta,u),l):less(ksi,eta)
3842 zeta@[m:more(pl(zeta,ksi),pl(zeta,eta))]
3843 satz136d:=satz136a(ismore12(pl(zeta,ksi),pl(ksi,zeta),pl(zeta,eta),pl(eta,zeta),compl(zeta,ksi),compl(zeta,eta),m)):more(ksi,eta)
3844 zeta@[i:is(pl(zeta,ksi),pl(zeta,eta))]
3845 satz136e:=satz136b(tr3is(cut,pl(ksi,zeta),pl(zeta,ksi),pl(zeta,eta),pl(eta,zeta),compl(ksi,zeta),i,compl(zeta,eta))):is(ksi,eta)
3846 zeta@[l:less(pl(zeta,ksi),pl(zeta,eta))]
3847 satz136f:=satz136c(isless12(pl(zeta,ksi),pl(ksi,zeta),pl(zeta,eta),pl(eta,zeta),compl(zeta,ksi),compl(zeta,eta),l)):less(ksi,eta)
3848 upsilon@[m:more(ksi,eta)][n:more(zeta,upsilon)]
3849 +3137
3850 t1:=satz134(ksi,eta,zeta,m):more(pl(ksi,zeta),pl(eta,zeta))
3851 t2:=ismore12(pl(zeta,eta),pl(eta,zeta),pl(upsilon,eta),pl(eta,upsilon),compl(zeta,eta),compl(upsilon,eta),satz134(zeta,upsilon,eta,n)):more(pl(eta,zeta),pl(eta,upsilon))
3852 -3137
3853 satz137:=trmore(pl(ksi,zeta),pl(eta,zeta),pl(eta,upsilon),t1".3137",t2".3137"):more(pl(ksi,zeta),pl(eta,upsilon))
3854 upsilon@[l:less(ksi,eta)][k:less(zeta,upsilon)]
3855 satz137a:=satz121(pl(eta,upsilon),pl(ksi,zeta),satz137(eta,ksi,upsilon,zeta,satz122(ksi,eta,l),satz122(zeta,upsilon,k))):less(pl(ksi,zeta),pl(eta,upsilon))
3856 upsilon@[m:moreis(ksi,eta)][n:more(zeta,upsilon)]
3857 satz138a:=orapp(more(ksi,eta),is(ksi,eta),more(pl(ksi,zeta),pl(eta,upsilon)),m,[t:more(ksi,eta)]satz137(t,n),[t:is(ksi,eta)]satz135g(t,n)):more(pl(ksi,zeta),pl(eta,upsilon))
3858 upsilon@[m:more(ksi,eta)][n:moreis(zeta,upsilon)]
3859 satz138b:=orapp(more(zeta,upsilon),is(zeta,upsilon),more(pl(ksi,zeta),pl(eta,upsilon)),n,[t:more(zeta,upsilon)]satz137(m,t),[t:is(zeta,upsilon)]satz135h(zeta,upsilon,ksi,eta,t,m)):more(pl(ksi,zeta),pl(eta,upsilon))
3860 upsilon@[l:lessis(ksi,eta)][k:less(zeta,upsilon)]
3861 satz138c:=satz121(pl(eta,upsilon),pl(ksi,zeta),satz138a(eta,ksi,upsilon,zeta,satz125(ksi,eta,l),satz122(zeta,upsilon,k))):less(pl(ksi,zeta),pl(eta,upsilon))
3862 upsilon@[l:less(ksi,eta)][k:lessis(zeta,upsilon)]
3863 satz138d:=satz121(pl(eta,upsilon),pl(ksi,zeta),satz138b(eta,ksi,upsilon,zeta,satz122(ksi,eta,l),satz125(zeta,upsilon,k))):less(pl(ksi,zeta),pl(eta,upsilon))
3864 upsilon@[m:moreis(ksi,eta)][n:moreis(zeta,upsilon)]
3865 +3139
3866 [i:is(ksi,eta)][j:is(zeta,upsilon)]
3867 t1:=moreisi2(pl(ksi,zeta),pl(eta,upsilon),ispl12(ksi,eta,zeta,upsilon,i,j)):moreis(pl(ksi,zeta),pl(eta,upsilon))
3868 i@[o:more(zeta,upsilon)]
3869 t2:=moreisi1(pl(ksi,zeta),pl(eta,upsilon),satz138a(m,o)):moreis(pl(ksi,zeta),pl(eta,upsilon))
3870 i@t3:=orapp(more(zeta,upsilon),is(zeta,upsilon),moreis(pl(ksi,zeta),pl(eta,upsilon)),n,[t:more(zeta,upsilon)]t2(t),[t:is(zeta,upsilon)]t1(t)):moreis(pl(ksi,zeta),pl(eta,upsilon))
3871 n@[o:more(ksi,eta)]
3872 t4:=moreisi1(pl(ksi,zeta),pl(eta,upsilon),satz138b(o,n)):moreis(pl(ksi,zeta),pl(eta,upsilon))
3873 -3139
3874 satz139:=orapp(more(ksi,eta),is(ksi,eta),moreis(pl(ksi,zeta),pl(eta,upsilon)),m,[t:more(ksi,eta)]t4".3139"(t),[t:is(ksi,eta)]t3".3139"(t)):moreis(pl(ksi,zeta),pl(eta,upsilon))
3875 upsilon@[l:lessis(ksi,eta)][k:lessis(zeta,upsilon)]
3876 satz139a:=satz124(pl(eta,upsilon),pl(ksi,zeta),satz139(eta,ksi,upsilon,zeta,satz125(ksi,eta,l),satz125(zeta,upsilon,k))):lessis(pl(ksi,zeta),pl(eta,upsilon))
3877 eta@[l:lessis(ksi,eta)]
3878 +3140
3879 [phi:cut][i:is(pl(eta,phi),ksi)]
3880 t1:=ismore1(pl(eta,phi),ksi,eta,i,satz133(eta,phi)):more(ksi,eta)
3881 phi@t2:=th3"l.imp"(is(pl(eta,phi),ksi),more(ksi,eta),satz123d(ksi,eta,l),[t:is(pl(eta,phi),ksi)]t1(t)):nis(pl(eta,phi),ksi)
3882 -3140
3883 vorbemerkung140:=th5"l.some"(cut,[a:cut]is(pl(eta,a),ksi),[a:cut]t2".3140"(a)):not(some([a:cut]is(pl(eta,a),ksi)))
3884 eta@[phi:cut][psi:cut]
3885 +*3140
3886 psi@[m:more(phi,psi)]
3887 t3:=satz135d(phi,psi,eta,m):more(pl(eta,phi),pl(eta,psi))
3888 t4:=ec3e21(is(pl(eta,phi),pl(eta,psi)),more(pl(eta,phi),pl(eta,psi)),less(pl(eta,phi),pl(eta,psi)),satz123b(pl(eta,phi),pl(eta,psi)),t3):nis(pl(eta,phi),pl(eta,psi))
3889 psi@[l:less(phi,psi)]
3890 t5:=satz135f(phi,psi,eta,l):less(pl(eta,phi),pl(eta,psi))
3891 t6:=ec3e31(is(pl(eta,phi),pl(eta,psi)),more(pl(eta,phi),pl(eta,psi)),less(pl(eta,phi),pl(eta,psi)),satz123b(pl(eta,phi),pl(eta,psi)),t5):nis(pl(eta,phi),pl(eta,psi))
3892 psi@[n:nis(phi,psi)]
3893 t7:=th1"l.or3"(is(phi,psi),more(phi,psi),less(phi,psi),satz123a(phi,psi),n):or(more(phi,psi),less(phi,psi))
3894 t8:=orapp(more(phi,psi),less(phi,psi),nis(pl(eta,phi),pl(eta,psi)),t7,[t:more(phi,psi)]t4(t),[t:less(phi,psi)]t6(t)):nis(pl(eta,phi),pl(eta,psi))
3895 -3140
3896 psi@[i:is(pl(eta,phi),ksi)][j:is(pl(eta,psi),ksi)]
3897 satz140b:=th7"l.imp"(is(phi,psi),nis(pl(eta,phi),pl(eta,psi)),weli(is(pl(eta,phi),pl(eta,psi)),tris2(cut,pl(eta,phi),pl(eta,psi),ksi,i,j)),[t:nis(phi,psi)]t8".3140"(t)):is(phi,psi)
3898 eta@[z0:rat][x0:rat][y0:rat]
3899 diffprop1:=and(more"rt"(x0,y0),[t:more"rt"(x0,y0)]is"rt"(z0,mn(x0,y0,t))):'prop'
3900 diffprop2:=and3(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0)):'prop'
3901 z0@diffprop:=some"rt"([x:rat]some"rt"([y:rat]diffprop2(z0,x,y))):'prop'
3902 eta@diff:=setof(rat,[z:rat]diffprop(z)):set(rat)
3903 x0@[lx:lrt(ksi,x0)][y0:rat][uy:urt(eta,y0)][m:more"rt"(x0,y0)][i:is"rt"(z0,mn(x0,y0,m))]
3904 +*iii3
3905 i@[m1:more"rt"(x0,y0)]
3906 t11a:=tris(rat,z0,mn(x0,y0,m),mn(x0,y0,m1),i,satz101g(x0,y0,mn(x0,y0,m),m1,satz101c(x0,y0,m))):is"rt"(z0,mn(x0,y0,m1))
3907 i@t12:=andi(more"rt"(x0,y0),[t:more"rt"(x0,y0)]is"rt"(z0,mn(x0,y0,t)),m,[t:more"rt"(x0,y0)]t11a(t)):diffprop1(z0,x0,y0)
3908 t13:=and3i(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0),lx,uy,t12):diffprop2(z0,x0,y0)
3909 t14:=somei(rat,[y:rat]diffprop2(z0,x0,y),y0,t13):some"rt"([y:rat]diffprop2(z0,x0,y))
3910 t15:=somei(rat,[x:rat]some"rt"([y:rat]diffprop2(z0,x,y)),x0,t14):diffprop(z0)
3911 -iii3
3912 i@diff1:=estii(rat,[z:rat]diffprop(z),z0,t15".iii3"):in(z0,diff)
3913 z0@[i:in(z0,diff)][p:'prop'][p1:[x:rat][t:lrt(ksi,x)][y:rat][u:urt(eta,y)][v:more"rt"(x,y)][w:is"rt"(z0,mn(x,y,v))]p]
3914 +*iii3
3915 p1@t16:=estie(rat,[z:rat]diffprop(z),z0,i):diffprop(z0)
3916 -iii3
3917 p1@[x0:rat][px:some"rt"([y:rat]diffprop2(z0,x0,y))][y0:rat][py:diffprop2(z0,x0,y0)]
3918 +*iii3
3919 py@t17:=and3e1(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0),py):lrt(ksi,x0)
3920 t18:=and3e2(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0),py):urt(eta,y0)
3921 t19:=and3e3(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0),py):diffprop1(z0,x0,y0)
3922 t20:=ande1(more"rt"(x0,y0),[t:more"rt"(x0,y0)]is"rt"(z0,mn(x0,y0,t)),t19):more"rt"(x0,y0)
3923 t21:=ande2"l.r"(more"rt"(x0,y0),[t:more"rt"(x0,y0)]is"rt"(z0,mn(x0,y0,t)),t19):is"rt"(z0,mn(x0,y0,t20))
3924 t22:=<t21><t20><t18><y0><t17><x0>p1:p
3925 px@t23:=someapp(rat,[y:rat]diffprop2(z0,x0,y),px,p,[y:rat][t:diffprop2(z0,x0,y)]t22(y,t)):p
3926 -iii3
3927 p1@diffapp:=someapp(rat,[x:rat]some"rt"([y:rat]diffprop2(z0,x,y)),t16".iii3",p,[x:rat][t:some"rt"([y:rat]diffprop2(z0,x,y))]t23".iii3"(x,t)):p
3928 eta@[m:more(ksi,eta)]
3929 +*3140
3930 m"rp"@[y0:rat][ly:lrt(ksi,y0)][uy:urt(eta,y0)][x0:rat][lx:lrt(ksi,x0)][l:less"rt"(y0,x0)]
3931 t9:=t2"rp.3134"(eta,m,y0,ly,uy,x0,lx,l):more"rt"(x0,y0)
3932 t10:=diff1(mn(x0,y0,t9),x0,lx,y0,uy,t9,refis(rat,mn(x0,y0,t9))):in(mn(x0,y0,t9),diff)
3933 m"rp"@[x1:rat][ux:urt(ksi,x1)][z0:rat][i:in(z0,diff)][x0:rat][lx:lrt(ksi,x0)][y0:rat][uy:urt(eta,y0)][n:more"rt"(x0,y0)][j:is"rt"(z0,mn(x0,y0,n))]
3934 t11:=isless12"rt"(mn(x0,y0,n),z0,pl"rt"(mn(x0,y0,n),y0),x0,symis(rat,z0,mn(x0,y0,n),j),satz101e(x0,y0,n),satz94a(mn(x0,y0,n),y0)):less"rt"(z0,x0)
3935 t12:=trless"rt"(z0,x0,x1,t11,cutapp2a(ksi,x0,lx,x1,ux)):less"rt"(z0,x1)
3936 t13:=ec3e31(is"rt"(z0,x1),more"rt"(z0,x1),less"rt"(z0,x1),satz81b(z0,x1),t12):nis"rt"(z0,x1)
3937 i@t14:=diffapp(z0,i,nis"rt"(z0,x1),[x:rat][t:lrt(ksi,x)][y:rat][u:urt(eta,y)][v:more"rt"(x,y)][w:is"rt"(z0,mn(x,y,v))]t13(x,t,y,u,v,w)):nis"rt"(z0,x1)
3938 ux@t15:=th3"l.imp"(in(x1,diff),nis"rt"(x1,x1),weli(is"rt"(x1,x1),refis(rat,x1)),[t:in(x1,diff)]t14(x1,t)):not(in(x1,diff))
3939 m"rp"@[z0:rat][i:in(z0,diff)][u0:rat][l:less"rt"(u0,z0)][x0:rat][lx:lrt(ksi,x0)][y0:rat][uy:urt(eta,y0)][n:more"rt"(x0,y0)][j:is"rt"(z0,mn(x0,y0,n))]
3940 t16:=tris(rat,pl"rt"(z0,y0),pl"rt"(mn(x0,y0,n),y0),x0,ispl1"rt"(z0,mn(x0,y0,n),y0,j),satz101e(x0,y0,n)):is"rt"(pl"rt"(z0,y0),x0)
3941 x2:=pl"rt"(u0,y0):rat
3942 t17:=isless2"rt"(pl"rt"(z0,y0),x0,x2,t16,satz96c(u0,z0,y0,l)):less"rt"(x2,x0)
3943 t18:=satz120(ksi,x0,lx,x2,t17):lrt(ksi,x2)
3944 t19:=ismore1"rt"(pl"rt"(y0,u0),pl"rt"(u0,y0),y0,compl"rt"(y0,u0),satz94(y0,u0)):more"rt"(x2,y0)
3945 t20:=satz101g(x2,y0,u0,t19,compl"rt"(y0,u0)):is"rt"(u0,mn(x2,y0,t19))
3946 t21:=diff1(u0,x2,t18,y0,uy,t19,t20):in(u0,diff)
3947 l@t22:=diffapp(z0,i,in(u0,diff),[x:rat][t:lrt(ksi,x)][y:rat][u:urt(eta,y)][v:more"rt"(x,y)][w:is"rt"(z0,mn(x,y,v))]t21(x,t,y,u,v,w)):in(u0,diff)
3948 m"rp"@[z0:rat][i:in(z0,diff)][x0:rat][lx:lrt(ksi,x0)][y0:rat][uy:urt(eta,y0)][n:more"rt"(x0,y0)][j:is"rt"(z0,mn(x0,y0,n))][x3:rat][lx3:lrt(ksi,x3)][l:less"rt"(x0,x3)]
3949 t23:=satz83(x0,x3,l):more"rt"(x3,x0)
3950 t24:=trmore"rt"(x3,x0,y0,t23,n):more"rt"(x3,y0)
3951 t25:=ismore12"rt"(x3,pl"rt"(mn(x3,y0,t24),y0),x0,pl"rt"(mn(x0,y0,n),y0),satz101f(x3,y0,t24),satz101f(x0,y0,n),t23):more"rt"(pl"rt"(mn(x3,y0,t24),y0),pl"rt"(mn(x0,y0,n),y0))
3952 t26:=satz97a(mn(x3,y0,t24),mn(x0,y0,n),y0,t25):more"rt"(mn(x3,y0,t24),mn(x0,y0,n))
3953 t27:=ismore2"rt"(mn(x0,y0,n),z0,mn(x3,y0,t24),symis(rat,z0,mn(x0,y0,n),j),t26):more"rt"(mn(x3,y0,t24),z0)
3954 t28:=diff1(mn(x3,y0,t24),x3,lx3,y0,uy,t24,refis(rat,mn(x3,y0,t24))):in(mn(x3,y0,t24),diff)
3955 t29:=andi(in(mn(x3,y0,t24),diff),more"rt"(mn(x3,y0,t24),z0),t28,t27):and(in(mn(x3,y0,t24),diff),more"rt"(mn(x3,y0,t24),z0))
3956 t30:=somei(rat,[x:rat]and(in(x,diff),more"rt"(x,z0)),mn(x3,y0,t24),t29):some"rt"([x:rat]and(in(x,diff),more"rt"(x,z0)))
3957 j@t31:=cutapp3(ksi,x0,lx,some"rt"([x:rat]and(in(x,diff),more"rt"(x,z0))),[y:rat][t:lrt(ksi,y)][u:less"rt"(x0,y)]t30(y,t,u)):some"rt"([x:rat]and(in(x,diff),more"rt"(x,z0)))
3958 i@t32:=diffapp(z0,i,some"rt"([x:rat]and(in(x,diff),more"rt"(x,z0))),[x:rat][t:lrt(ksi,x)][y:rat][u:urt(eta,y)][v:more"rt"(x,y)][w:is"rt"(z0,mn(x,y,v))]t31(x,t,y,u,v,w)):some"rt"([x:rat]and(in(x,diff),more"rt"(x,z0)))
3959 m"rp"@[y0:rat][ly:lrt(ksi,y0)][uy:urt(eta,y0)][x0:rat][lx:lrt(ksi,x0)][l:less"rt"(y0,x0)][x1:rat][ux1:urt(ksi,x1)]
3960 t33:=cut2(diff,mn(x0,y0,t9(y0,ly,uy,x0,lx,l)),t10(y0,ly,uy,x0,lx,l),x1,t15(x1,ux1),[x:rat][t:in(x,diff)][y:rat][u:less"rt"(y,x)]t22(x,t,y,u),[x:rat][t:in(x,diff)]t32(x,t)):cutprop(diff)
3961 l@t34:=cutapp1b(ksi,cutprop(diff),[x:rat][t:urt(ksi,x)]t33(x,t)):cutprop(diff)
3962 uy@t35:=cutapp3(ksi,y0,ly,cutprop(diff),[x:rat][t:lrt(ksi,x)][u:less"rt"(y0,x)]t34(x,t,u)):cutprop(diff)
3963 -3140
3964 m@satz140h:=moreapp(ksi,eta,m,cutprop(diff),[x:rat][u:lrt(ksi,x)][v:urt(eta,x)]t35".3140"(x,u,v)):cutprop(diff)
3965 +*3140
3966 m"rp"@chi:=cutof(diff,satz140h):cut
3967 [z0:rat][lz:lrt(pl(eta,chi),z0)][y1:rat][ly:lrt(eta,y1)][u0:rat][lu:lrt(chi,u0)][i:is"rt"(z0,pl"rt"(y1,u0))][x0:rat][lx:lrt(ksi,x0)][y0:rat][uy:urt(eta,y0)][o:more"rt"(x0,y0)][j:is"rt"(u0,mn(x0,y0,o))]
3968 t36:=cutapp2b(eta,y1,ly,y0,uy):more"rt"(y0,y1)
3969 t37:=tr3is(rat,pl"rt"(pl"rt"(mn(x0,y0,o),y1),mn(y0,y1,t36)),pl"rt"(mn(x0,y0,o),pl"rt"(y1,mn(y0,y1,t36))),pl"rt"(mn(x0,y0,o),y0),x0,asspl1"rt"(mn(x0,y0,o),y1,mn(y0,y1,t36)),ispl2"rt"(pl"rt"(y1,mn(y0,y1,t36)),y0,mn(x0,y0,o),satz101c(y0,y1,t36)),satz101e(x0,y0,o)):is"rt"(pl"rt"(pl"rt"(mn(x0,y0,o),y1),mn(y0,y1,t36)),x0)
3970 t38:=isless2"rt"(pl"rt"(pl"rt"(mn(x0,y0,o),y1),mn(y0,y1,t36)),x0,pl"rt"(mn(x0,y0,o),y1),t37,satz94a(pl"rt"(mn(x0,y0,o),y1),mn(y0,y1,t36))):less"rt"(pl"rt"(mn(x0,y0,o),y1),x0)
3971 t39:=tr3is(rat,z0,pl"rt"(y1,u0),pl"rt"(u0,y1),pl"rt"(mn(x0,y0,o),y1),i,compl"rt"(y1,u0),ispl1"rt"(u0,mn(x0,y0,o),y1,j)):is"rt"(z0,pl"rt"(mn(x0,y0,o),y1))
3972 t40:=isless1"rt"(pl"rt"(mn(x0,y0,o),y1),z0,x0,symis(rat,z0,pl"rt"(mn(x0,y0,o),y1),t39),t38):less"rt"(z0,x0)
3973 t41:=satz120(ksi,x0,lx,z0,t40):lrt(ksi,z0)
3974 i@t42:=diffapp(u0,ini(diff,satz140h,u0,lu),lrt(ksi,z0),[x:rat][t:lrt(ksi,x)][y:rat][u:urt(eta,y)][v:more"rt"(x,y)][w:is"rt"(u0,mn(x,y,v))]t41(x,t,y,u,v,w)):lrt(ksi,z0)
3975 lz@a:=plapp(eta,chi,z0,lz,lrt(ksi,z0),[x:rat][t:lrt(eta,x)][y:rat][u:lrt(chi,y)][v:is"rt"(z0,pl"rt"(x,y))]t42(x,t,y,u,v)):lrt(ksi,z0)
3976 m"rp"@[y0:rat][ly:lrt(ksi,y0)][uy:urt(eta,y0)][x0:rat][lx:lrt(ksi,x0)][l:less"rt"(y0,x0)]
3977 t43:=satz83(y0,x0,l):more"rt"(x0,y0)
3978 [y1:rat][ly1:lrt(eta,y1)][y2:rat][uy2:urt(eta,y2)]
3979 t44:=cutapp2b(eta,y1,ly1,y2,uy2):more"rt"(y2,y1)
3980 [i:is"rt"(mn(y2,y1,t44),mn(x0,y0,t43))]
3981 t45:=cutapp2b(eta,y1,ly1,y0,uy):more"rt"(y0,y1)
3982 t46:=tris(rat,y2,pl"rt"(mn(y2,y1,t44),y1),pl"rt"(mn(x0,y0,t43),y1),satz101f(y2,y1,t44),ispl1"rt"(mn(y2,y1,t44),mn(x0,y0,t43),y1,i)):is"rt"(y2,pl"rt"(mn(x0,y0,t43),y1))
3983 t47:=tr4is(rat,pl"rt"(y2,mn(y0,y1,t45)),pl"rt"(pl"rt"(mn(x0,y0,t43),y1),mn(y0,y1,t45)),pl"rt"(mn(x0,y0,t43),pl"rt"(y1,mn(y0,y1,t45))),pl"rt"(mn(x0,y0,t43),y0),x0,ispl1"rt"(y2,pl"rt"(mn(x0,y0,t43),y1),mn(y0,y1,t45),t46),asspl1"rt"(mn(x0,y0,t43),y1,mn(y0,y1,t45)),ispl2"rt"(pl"rt"(y1,mn(y0,y1,t45)),y0,mn(x0,y0,t43),satz101c(y0,y1,t45)),satz101e(x0,y0,t43)):is"rt"(pl"rt"(y2,mn(y0,y1,t45)),x0)
3984 t48:=ismore1"rt"(pl"rt"(y2,mn(y0,y1,t45)),x0,y2,t47,satz94(y2,mn(y0,y1,t45))):more"rt"(x0,y2)
3985 t49:=satz101g(x0,y2,mn(y0,y1,t45),t48,t47):is"rt"(mn(y0,y1,t45),mn(x0,y2,t48))
3986 t50:=tris(rat,y0,pl"rt"(mn(y0,y1,t45),y1),pl"rt"(mn(x0,y2,t48),y1),satz101f(y0,y1,t45),ispl1"rt"(mn(y0,y1,t45),mn(x0,y2,t48),y1,t49)):is"rt"(y0,pl"rt"(mn(x0,y2,t48),y1))
3987 t51:=ine(diff,satz140h,mn(x0,y2,t48),diff1(mn(x0,y2,t48),x0,lx,y2,uy2,t48,refis(rat,mn(x0,y2,t48)))):lrt(chi,mn(x0,y2,t48))
3988 t52:=lrtpl(eta,chi,y0,y1,ly1,mn(x0,y2,t48),t51,tris(rat,y0,pl"rt"(mn(x0,y2,t48),y1),pl"rt"(y1,mn(x0,y2,t48)),t50,compl"rt"(mn(x0,y2,t48),y1))):lrt(pl(eta,chi),y0)
3989 l@t53:=satz132app(eta,lrt(pl(eta,chi),y0),mn(x0,y0,t43),[x:rat][t:lrt(eta,x)][y:rat][u:urt(eta,y)][v:is"rt"(mn(y,x,cutapp2b(eta,x,t,y,u)),mn(x0,y0,t43))]t52(x,t,y,u,v)):lrt(pl(eta,chi),y0)
3990 uy@t54:=cutapp3(ksi,y0,ly,lrt(pl(eta,chi),y0),[x:rat][t:lrt(ksi,x)][u:less"rt"(y0,x)]t53(x,t,u)):lrt(pl(eta,chi),y0)
3991 ly@[ly0:lrt(eta,y0)][y1:rat][ly1:lrt(ksi,y1)][uy1:urt(eta,y1)]
3992 t55:=t54(y1,ly1,uy1):lrt(pl(eta,chi),y1)
3993 t56:=satz120(pl(eta,chi),y1,t55,y0,cutapp2a(eta,y0,ly0,y1,uy1)):lrt(pl(eta,chi),y0)
3994 ly0@t57:=moreapp(ksi,eta,m,lrt(pl(eta,chi),y0),[x:rat][t:lrt(ksi,x)][u:urt(eta,x)]t56(x,t,u)):lrt(pl(eta,chi),y0)
3995 ly@b:=th1"l.imp"(lrt(eta,y0),lrt(pl(eta,chi),y0),[t:lrt(eta,y0)]t57(t),[t:urt(eta,y0)]t54(t)):lrt(pl(eta,chi),y0)
3996 m"rp"@t58:=isi1(pl(eta,chi),ksi,[x:rat][t:lrt(pl(eta,chi),x)]a(x,t),[x:rat][t:lrt(ksi,x)]b(x,t)):is(pl(eta,chi),ksi)
3997 -3140
3998 m@satz140a:=somei(cut,[a:cut]is(pl(eta,a),ksi),chi".3140",t58".3140"):some([a:cut]is(pl(eta,a),ksi))
3999 +*3140
4000 eta@t59:=[c:cut][d:cut][t:is(pl(eta,c),ksi)][u:is(pl(eta,d),ksi)]satz140b(c,d,t,u):amone(cut,[c:cut]is(pl(eta,c),ksi))
4001 -3140
4002 m@satz140:=onei(cut,[a:cut]is(pl(eta,a),ksi),t59".3140",satz140a):one([a:cut]is(pl(eta,a),ksi))
4003 mn:=ind(cut,[a:cut]is(pl(eta,a),ksi),satz140):cut
4004 satz140c:=oneax(cut,[a:cut]is(pl(eta,a),ksi),satz140):is(pl(eta,mn(ksi,eta,m)),ksi)
4005 satz140d:=symis(cut,pl(eta,mn(ksi,eta,m)),ksi,satz140c):is(ksi,pl(eta,mn(ksi,eta,m)))
4006 satz140e:=tris(cut,pl(mn(ksi,eta,m),eta),pl(eta,mn(ksi,eta,m)),ksi,compl(mn(ksi,eta,m),eta),satz140c):is(pl(mn(ksi,eta,m),eta),ksi)
4007 satz140f:=symis(cut,pl(mn(ksi,eta,m),eta),ksi,satz140e):is(ksi,pl(mn(ksi,eta,m),eta))
4008 eta@[phi:cut][m:more(ksi,eta)][i:is(pl(eta,phi),ksi)]
4009 satz140g:=satz140b(phi,mn(ksi,eta,m),i,satz140c(m)):is(phi,mn(ksi,eta,m))
4010 upsilon@[m:more(ksi,zeta)][n:more(eta,upsilon)][i:is(ksi,eta)][j:is(zeta,upsilon)]
4011 +*3140
4012 j"rp"@t60:=tr3is(cut,pl(upsilon,mn(ksi,zeta,m)),pl(zeta,mn(ksi,zeta,m)),ksi,eta,ispl1(upsilon,zeta,mn(ksi,zeta,m),symis(cut,zeta,upsilon,j)),satz140c(ksi,zeta,m),i):is(pl(upsilon,mn(ksi,zeta,m)),eta)
4013 -3140
4014 j@ismn12:=satz140g(eta,upsilon,mn(ksi,zeta,m),n,t60".3140"):is(mn(ksi,zeta,m),mn(eta,upsilon,n))
4015 zeta@[m:more(ksi,zeta)][n:more(eta,zeta)][i:is(ksi,eta)]
4016 ismn1:=ismn12(zeta,m,n,i,refis(cut,zeta)):is(mn(ksi,zeta,m),mn(eta,zeta,n))
4017 zeta@[m:more(zeta,ksi)][n:more(zeta,eta)][i:is(ksi,eta)]
4018 ismn2:=ismn12(zeta,zeta,ksi,eta,m,n,refis(cut,zeta),i):is(mn(zeta,ksi,m),mn(zeta,eta,n))
4019 eta@[z0:rat][x0:rat][y0:rat]
4020 prodprop1:=and3(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0))):'prop'
4021 z0@prodprop:=some"rt"([x:rat]some"rt"([y:rat]prodprop1(z0,x,y))):'prop'
4022 eta@prod:=setof(rat,[z:rat]prodprop(z)):set(rat)
4023 x0@[lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,ts(x0,y0))]
4024 +iii4
4025 t1:=and3i(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0)),lx,ly,i):prodprop1(z0,x0,y0)
4026 t2:=somei(rat,[y:rat]prodprop1(z0,x0,y),y0,t1):some"rt"([y:rat]prodprop1(z0,x0,y))
4027 t3:=somei(rat,[x:rat]some"rt"([y:rat]prodprop1(z0,x,y)),x0,t2):prodprop(z0)
4028 -iii4
4029 prod1:=estii(rat,[z:rat]prodprop(z),z0,t3".iii4"):in(z0,prod)
4030 z0@[i:in(z0,prod)][p:'prop'][p1:[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,ts(x,y))]p]
4031 +*iii4
4032 p1@t4:=estie(rat,[z:rat]prodprop(z),z0,i):prodprop(z0)
4033 -iii4
4034 p1@[x0:rat][px:some"rt"([y:rat]prodprop1(z0,x0,y))][y0:rat][py:prodprop1(z0,x0,y0)]
4035 +*iii4
4036 py@t5:=and3e1(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0)),py):lrt(ksi,x0)
4037 t6:=and3e2(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0)),py):lrt(eta,y0)
4038 t7:=and3e3(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0)),py):is"rt"(z0,ts(x0,y0))
4039 t8:=<t7><t6><y0><t5><x0>p1:p
4040 px@t9:=someapp(rat,[y:rat]prodprop1(z0,x0,y),px,p,[y:rat][t:prodprop1(z0,x0,y)]t8(y,t)):p
4041 -iii4
4042 p1@prodapp:=someapp(rat,[x:rat]some"rt"([y:rat]prodprop1(z0,x,y)),t4".iii4",p,[x:rat][t:some"rt"([y:rat]prodprop1(z0,x,y))]t9".iii4"(x,t)):p
4043 eta@[x1:rat][ux:urt(ksi,x1)][y1:rat][uy:urt(eta,y1)]
4044 +4141
4045 [z0:rat][i:in(z0,prod)][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][j:is"rt"(z0,ts(x0,y0))]
4046 t1:=cutapp2a(ksi,x0,lx,x1,ux):less"rt"(x0,x1)
4047 t2:=cutapp2a(eta,y0,ly,y1,uy):less"rt"(y0,y1)
4048 t3:=isless1"rt"(ts(x0,y0),z0,ts(x1,y1),symis(rat,z0,ts(x0,y0),j),satz107a(x0,x1,y0,y1,t1,t2)):less"rt"(z0,ts(x1,y1))
4049 t4:=ec3e31(is"rt"(z0,ts(x1,y1)),more"rt"(z0,ts(x1,y1)),less"rt"(z0,ts(x1,y1)),satz81b(z0,ts(x1,y1)),t3):nis"rt"(z0,ts(x1,y1))
4050 i@t5:=prodapp(ksi,eta,z0,i,nis"rt"(z0,ts(x1,y1)),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,ts(x,y))]t4(x,t,y,u,v)):nis"rt"(z0,ts(x1,y1))
4051 -4141
4052 satz141a:=th3"l.imp"(in(ts(x1,y1),prod),nis"rt"(ts(x1,y1),ts(x1,y1)),weli(is"rt"(ts(x1,y1),ts(x1,y1)),refis(rat,ts(x1,y1))),[t:in(ts(x1,y1),prod)]t5".4141"(ts(x1,y1),t)):not(in(ts(x1,y1),prod))
4053 -rp
4054 @[x0:rat][y0:rat]
4055 +4141
4056 v0:=ts(ov(1rt,y0),x0):rat
4057 t6:=tr3is(rat,ts(y0,v0),ts(ts(y0,ov(1rt,y0)),x0),ts(1rt,x0),x0,assts2(y0,ov(1rt,y0),x0),ists1(ts(y0,ov(1rt,y0)),1rt,x0,satz110c(1rt,y0)),example1c(x0)):is(ts(y0,v0),x0)
4058 -4141
4059 satz141b:=satz110g(x0,y0,v0".4141",t6".4141"):is(ts(ov(1rt,y0),x0),ov(x0,y0))
4060 satz141c:=symis(rat,ts(ov(1rt,y0),x0),ov(x0,y0),satz141b):is"rt"(ov(x0,y0),ts(ov(1rt,y0),x0))
4061 +*rp
4062 +*4141
4063 eta@[u0:rat][i:in(u0,prod)][z0:rat][l:less"rt"(z0,u0)][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][j:is"rt"(u0,ts(x0,y0))]
4064 t7:=isless2"rt"(u0,ts(x0,y0),z0,j,l):less"rt"(z0,ts(x0,y0))
4065 t8:=tr3is(rat,ts(ov(1rt,x0),ts(x0,y0)),ts(ts(ov(1rt,x0),x0),y0),ts(1rt,y0),y0,assts2(ov(1rt,x0),x0,y0),ists1(ts(ov(1rt,x0),x0),1rt,y0,satz110e(1rt,x0)),example1c(y0)):is"rt"(ts(ov(1rt,x0),ts(x0,y0)),y0)
4066 t9:=isless12"rt"(ts(ov(1rt,x0),z0),ov(z0,x0),ts(ov(1rt,x0),ts(x0,y0)),y0,satz141b(z0,x0),t8,satz105f(z0,ts(x0,y0),ov(1rt,x0),t7)):less"rt"(ov(z0,x0),y0)
4067 t10:=satz120(eta,y0,ly,ov(z0,x0),t9):lrt(eta,ov(z0,x0))
4068 t11:=prod1(z0,x0,lx,ov(z0,x0),t10,satz110d(z0,x0)):in(z0,prod)
4069 l@t12:=prodapp(u0,i,in(z0,prod),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(u0,ts(x,y))]t11(x,t,y,u,v)):in(z0,prod)
4070 eta@[z0:rat][i:in(z0,prod)][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][j:is"rt"(z0,ts(x0,y0))][x1:rat][lx1:lrt(ksi,x1)][l:less"rt"(x0,x1)]
4071 t13:=prod1(ts(x1,y0),x1,lx1,y0,ly,refis(rat,ts(x1,y0))):in(ts(x1,y0),prod)
4072 t14:=satz105a(x1,x0,y0,satz83(x0,x1,l)):more"rt"(ts(x1,y0),ts(x0,y0))
4073 t15:=ismore2"rt"(ts(x0,y0),z0,ts(x1,y0),symis(rat,z0,ts(x0,y0),j),t14):more"rt"(ts(x1,y0),z0)
4074 t16:=andi(in(ts(x1,y0),prod),more"rt"(ts(x1,y0),z0),t13,t15):and(in(ts(x1,y0),prod),more"rt"(ts(x1,y0),z0))
4075 t17:=somei(rat,[y:rat]and(in(y,prod),more"rt"(y,z0)),ts(x1,y0),t16):some"rt"([y:rat]and(in(y,prod),more"rt"(y,z0)))
4076 j@t18:=cutapp3(ksi,x0,lx,some"rt"([y:rat]and(in(y,prod),more"rt"(y,z0))),[x:rat][t:lrt(ksi,x)][u:less"rt"(x0,x)]t17(x,t,u)):some"rt"([y:rat]and(in(y,prod),more"rt"(y,z0)))
4077 i@t19:=prodapp(z0,i,some"rt"([y:rat]and(in(y,prod),more"rt"(y,z0))),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,ts(x,y))]t18(x,t,y,u,v)):some"rt"([y:rat]and(in(y,prod),more"rt"(y,z0)))
4078 eta@[x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][x1:rat][ux:urt(ksi,x1)][y1:rat][uy:urt(eta,y1)]
4079 t20:=cut2(prod,ts(x0,y0),prod1(ts(x0,y0),x0,lx,y0,ly,refis(rat,ts(x0,y0))),ts(x1,y1),satz141a(x1,ux,y1,uy),[x:rat][t:in(x,prod)][y:rat][u:less"rt"(y,x)]t12(x,t,y,u),[x:rat][t:in(x,prod)]t19(x,t)):cutprop(prod)
4080 ux@t21:=cutapp1b(eta,cutprop(prod),[y:rat][t:urt(eta,y)]t20(y,t)):cutprop(prod)
4081 ly@t22:=cutapp1b(ksi,cutprop(prod),[x:rat][t:urt(ksi,x)]t21(x,t)):cutprop(prod)
4082 lx@t23:=cutapp1a(eta,cutprop(prod),[y:rat][t:lrt(eta,y)]t22(y,t)):cutprop(prod)
4083 -4141
4084 eta@satz141:=cutapp1a(ksi,cutprop(prod),[x:rat][t:lrt(ksi,x)]t23".4141"(x,t)):cutprop(prod)
4085 ts:=cutof(prod,satz141):cut
4086 [z0:rat][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,ts"rt"(x0,y0))]
4087 lrtts:=ine(prod,satz141,z0,prod1(z0,x0,lx,y0,ly,i)):lrt(ts(ksi,eta),z0)
4088 eta@[z0:rat][x0:rat][ux:urt(ksi,x0)][y0:rat][uy:urt(eta,y0)][i:is"rt"(z0,ts"rt"(x0,y0))]
4089 +*iii4
4090 i@t10:=isp1(rat,[x:rat]not(in(x,prod)),ts"rt"(x0,y0),z0,satz141a(x0,ux,y0,uy),i):not(in(z0,prod))
4091 -iii4
4092 i@urtts:=th3"l.imp"(lrt(ts(ksi,eta),z0),in(z0,prod),t10".iii4",[t:lrt(ts(ksi,eta),z0)]ini(prod,satz141,z0,t)):urt(ts(ksi,eta),z0)
4093 eta@[z0:rat][lz:lrt(ts(ksi,eta),z0)][p:'prop'][p1:[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,ts"rt"(x,y))]p]
4094 +*iii4
4095 p1@t11:=ini(prod,satz141,z0,lz):in(z0,prod)
4096 -iii4
4097 p1@tsapp:=prodapp(z0,t11".iii4",p,p1):p
4098 zeta@[i:is(ksi,eta)]
4099 ists1:=isf(cut,cut,[u:cut]ts(u,zeta),ksi,eta,i):is(ts(ksi,zeta),ts(eta,zeta))
4100 ists2:=isf(cut,cut,[u:cut]ts(zeta,u),ksi,eta,i):is(ts(zeta,ksi),ts(zeta,eta))
4101 upsilon@[i:is(ksi,eta)][j:is(zeta,upsilon)]
4102 ists12:=tris(cut,ts(ksi,zeta),ts(eta,zeta),ts(eta,upsilon),ists1(i),ists2(zeta,upsilon,eta,j)):is(ts(ksi,zeta),ts(eta,upsilon))
4103 +4142
4104 eta@[z0:rat][lz:lrt(ts(ksi,eta),z0)][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,ts"rt"(x0,y0))]
4105 t1:=tris(rat,z0,ts"rt"(x0,y0),ts"rt"(y0,x0),i,comts(x0,y0)):is"rt"(z0,ts"rt"(y0,x0))
4106 t2:=lrtts(eta,ksi,z0,y0,ly,x0,lx,t1):lrt(ts(eta,ksi),z0)
4107 lz@t3:=tsapp(ksi,eta,z0,lz,lrt(ts(eta,ksi),z0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(z0,ts"rt"(x,y))]t2(x,t,y,u,v)):lrt(ts(eta,ksi),z0)
4108 -4142
4109 eta@satz142:=isi1(ts(ksi,eta),ts(eta,ksi),[x:rat][t:lrt(ts(ksi,eta),x)]t3".4142"(x,t),[x:rat][t:lrt(ts(eta,ksi),x)]t3".4142"(eta,ksi,x,t)):is(ts(ksi,eta),ts(eta,ksi))
4110 comts:=satz142:is(ts(ksi,eta),ts(eta,ksi))
4111 +4143
4112 zeta@[u0:rat][lu:lrt(ts(ts(ksi,eta),zeta),u0)][v0:rat][lv:lrt(ts(ksi,eta),v0)][z0:rat][lz:lrt(zeta,z0)][i:is"rt"(u0,ts"rt"(v0,z0))][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][j:is"rt"(v0,ts"rt"(x0,y0))]
4113 t1:=tr3is(rat,u0,ts"rt"(v0,z0),ts"rt"(ts"rt"(x0,y0),z0),ts"rt"(x0,ts"rt"(y0,z0)),i,ists1"rt"(v0,ts"rt"(x0,y0),z0,j),assts1(x0,y0,z0)):is"rt"(u0,ts"rt"(x0,ts"rt"(y0,z0)))
4114 t2:=lrtts(eta,zeta,ts"rt"(y0,z0),y0,ly,z0,lz,refis(rat,ts"rt"(y0,z0))):lrt(ts(eta,zeta),ts"rt"(y0,z0))
4115 t3:=lrtts(ksi,ts(eta,zeta),u0,x0,lx,ts"rt"(y0,z0),t2,t1):lrt(ts(ksi,ts(eta,zeta)),u0)
4116 i@t4:=tsapp(ksi,eta,v0,lv,lrt(ts(ksi,ts(eta,zeta)),u0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(v0,ts"rt"(x,y))]t3(x,t,y,u,v)):lrt(ts(ksi,ts(eta,zeta)),u0)
4117 lu@t5:=tsapp(ts(ksi,eta),zeta,u0,lu,lrt(ts(ksi,ts(eta,zeta)),u0),[x:rat][t:lrt(ts(ksi,eta),x)][y:rat][u:lrt(zeta,y)][v:is"rt"(u0,ts"rt"(x,y))]t4(x,t,y,u,v)):lrt(ts(ksi,ts(eta,zeta)),u0)
4118 u0@[lu:lrt(ts(ksi,ts(eta,zeta)),u0)][x0:rat][lx:lrt(ksi,x0)][v0:rat][lv:lrt(ts(eta,zeta),v0)][i:is"rt"(u0,ts"rt"(x0,v0))][y0:rat][ly:lrt(eta,y0)][z0:rat][lz:lrt(zeta,z0)][j:is"rt"(v0,ts"rt"(y0,z0))]
4119 t6:=tr3is(rat,u0,ts"rt"(x0,v0),ts"rt"(x0,ts"rt"(y0,z0)),ts"rt"(ts"rt"(x0,y0),z0),i,ists2"rt"(v0,ts"rt"(y0,z0),x0,j),assts2(x0,y0,z0)):is"rt"(u0,ts"rt"(ts"rt"(x0,y0),z0))
4120 t7:=lrtts(ksi,eta,ts"rt"(x0,y0),x0,lx,y0,ly,refis(rat,ts"rt"(x0,y0))):lrt(ts(ksi,eta),ts"rt"(x0,y0))
4121 t8:=lrtts(ts(ksi,eta),zeta,u0,ts"rt"(x0,y0),t7,z0,lz,t6):lrt(ts(ts(ksi,eta),zeta),u0)
4122 i@t9:=tsapp(eta,zeta,v0,lv,lrt(ts(ts(ksi,eta),zeta),u0),[x:rat][t:lrt(eta,x)][y:rat][u:lrt(zeta,y)][v:is"rt"(v0,ts"rt"(x,y))]t8(x,t,y,u,v)):lrt(ts(ts(ksi,eta),zeta),u0)
4123 lu@t10:=tsapp(ksi,ts(eta,zeta),u0,lu,lrt(ts(ts(ksi,eta),zeta),u0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(ts(eta,zeta),y)][v:is"rt"(u0,ts"rt"(x,y))]t9(x,t,y,u,v)):lrt(ts(ts(ksi,eta),zeta),u0)
4124 -4143
4125 zeta@satz143:=isi1(ts(ts(ksi,eta),zeta),ts(ksi,ts(eta,zeta)),[x:rat][t:lrt(ts(ts(ksi,eta),zeta),x)]t5".4143"(x,t),[x:rat][t:lrt(ts(ksi,ts(eta,zeta)),x)]t10".4143"(x,t)):is(ts(ts(ksi,eta),zeta),ts(ksi,ts(eta,zeta)))
4126 assts1:=satz143:is(ts(ts(ksi,eta),zeta),ts(ksi,ts(eta,zeta)))
4127 assts2:=symis(cut,ts(ts(ksi,eta),zeta),ts(ksi,ts(eta,zeta)),satz143):is(ts(ksi,ts(eta,zeta)),ts(ts(ksi,eta),zeta))
4128 +4144
4129 [u0:rat][lu:lrt(ts(ksi,pl(eta,zeta)),u0)][x0:rat][lx:lrt(ksi,x0)][v0:rat][lv:lrt(pl(eta,zeta),v0)][i:is"rt"(u0,ts"rt"(x0,v0))][y0:rat][ly:lrt(eta,y0)][z0:rat][lz:lrt(zeta,z0)][j:is"rt"(v0,pl"rt"(y0,z0))]
4130 t1:=tr3is(rat,u0,ts"rt"(x0,v0),ts"rt"(x0,pl"rt"(y0,z0)),pl"rt"(ts"rt"(x0,y0),ts"rt"(x0,z0)),i,ists2"rt"(v0,pl"rt"(y0,z0),x0,j),disttp2(x0,y0,z0)):is"rt"(u0,pl"rt"(ts"rt"(x0,y0),ts"rt"(x0,z0)))
4131 t2:=lrtts(ksi,eta,ts"rt"(x0,y0),x0,lx,y0,ly,refis(rat,ts"rt"(x0,y0))):lrt(ts(ksi,eta),ts"rt"(x0,y0))
4132 t3:=lrtts(ksi,zeta,ts"rt"(x0,z0),x0,lx,z0,lz,refis(rat,ts"rt"(x0,z0))):lrt(ts(ksi,zeta),ts"rt"(x0,z0))
4133 t4:=lrtpl(ts(ksi,eta),ts(ksi,zeta),u0,ts"rt"(x0,y0),t2,ts"rt"(x0,z0),t3,t1):lrt(pl(ts(ksi,eta),ts(ksi,zeta)),u0)
4134 i@t5:=plapp(eta,zeta,v0,lv,lrt(pl(ts(ksi,eta),ts(ksi,zeta)),u0),[x:rat][t:lrt(eta,x)][y:rat][u:lrt(zeta,y)][v:is"rt"(v0,pl"rt"(x,y))]t4(x,t,y,u,v)):lrt(pl(ts(ksi,eta),ts(ksi,zeta)),u0)
4135 lu@t6:=tsapp(ksi,pl(eta,zeta),u0,lu,lrt(pl(ts(ksi,eta),ts(ksi,zeta)),u0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(pl(eta,zeta),y)][v:is"rt"(u0,ts"rt"(x,y))]t5(x,t,y,u,v)):lrt(pl(ts(ksi,eta),ts(ksi,zeta)),u0)
4136 u0@[lu:lrt(pl(ts(ksi,eta),ts(ksi,zeta)),u0)][v0:rat][lv:lrt(ts(ksi,eta),v0)][w0:rat][lw:lrt(ts(ksi,zeta),w0)][i:is"rt"(u0,pl"rt"(v0,w0))][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][j:is"rt"(v0,ts"rt"(x0,y0))][x1:rat][lx1:lrt(ksi,x1)][z0:rat][lz:lrt(zeta,z0)][k:is"rt"(w0,ts"rt"(x1,z0))]
4137 t7:=tris(rat,u0,pl"rt"(v0,w0),pl"rt"(ts"rt"(x0,y0),ts"rt"(x1,z0)),i,ispl12"rt"(v0,ts"rt"(x0,y0),w0,ts"rt"(x1,z0),j,k)):is"rt"(u0,pl"rt"(ts"rt"(x0,y0),ts"rt"(x1,z0)))
4138 x2:=ite(moreis"rt"(x0,x1),rat,x0,x1):rat
4139 [m:moreis"rt"(x0,x1)]
4140 t8:=itet(moreis"rt"(x0,x1),rat,x0,x1,m):is"rt"(x2,x0)
4141 t9:=isp1(rat,[t:rat]lrt(ksi,t),x0,x2,lx,t8):lrt(ksi,x2)
4142 t10:=lessisi2"rt"(x0,x2,symis(rat,x2,x0,t8)):lessis"rt"(x0,x2)
4143 t11:=satz88(x1,x0,x2,satz84(x0,x1,m),t10):lessis"rt"(x1,x2)
4144 k@[n:not(moreis"rt"(x0,x1))]
4145 t12:=itef(moreis"rt"(x0,x1),rat,x0,x1,n):is"rt"(x2,x1)
4146 t13:=isp1(rat,[t:rat]lrt(ksi,t),x1,x2,lx1,t12):lrt(ksi,x2)
4147 t14:=lessisi2"rt"(x1,x2,symis(rat,x2,x1,t12)):lessis"rt"(x1,x2)
4148 t15:=lessisi1"rt"(x0,x2,satz87b(x0,x1,x2,satz81j(x0,x1,n),t14)):lessis"rt"(x0,x2)
4149 k@t16:=th1"l.imp"(moreis"rt"(x0,x1),lrt(ksi,x2),[t:moreis"rt"(x0,x1)]t9(t),[t:not(moreis"rt"(x0,x1))]t13(t)):lrt(ksi,x2)
4150 t17:=th1"l.imp"(moreis"rt"(x0,x1),lessis"rt"(x0,x2),[t:moreis"rt"(x0,x1)]t10(t),[t:not(moreis"rt"(x0,x1))]t15(t)):lessis"rt"(x0,x2)
4151 t18:=th1"l.imp"(moreis"rt"(x0,x1),lessis"rt"(x1,x2),[t:moreis"rt"(x0,x1)]t11(t),[t:not(moreis"rt"(x0,x1))]t14(t)):lessis"rt"(x1,x2)
4152 t19:=lrtpl(eta,zeta,pl"rt"(y0,z0),y0,ly,z0,lz,refis(rat,pl"rt"(y0,z0))):lrt(pl(eta,zeta),pl"rt"(y0,z0))
4153 t20:=lrtts(ksi,pl(eta,zeta),ts"rt"(x2,pl"rt"(y0,z0)),x2,t16,pl"rt"(y0,z0),t19,refis(rat,ts"rt"(x2,pl"rt"(y0,z0)))):lrt(ts(ksi,pl(eta,zeta)),ts"rt"(x2,pl"rt"(y0,z0)))
4154 t21:=satz109a(x0,x2,y0,y0,t17,lessisi2"rt"(y0,y0,refis(rat,y0))):lessis"rt"(ts"rt"(x0,y0),ts"rt"(x2,y0))
4155 t22:=satz109a(x1,x2,z0,z0,t18,lessisi2"rt"(z0,z0,refis(rat,z0))):lessis"rt"(ts"rt"(x1,z0),ts"rt"(x2,z0))
4156 t23:=islessis12"rt"(pl"rt"(ts"rt"(x0,y0),ts"rt"(x1,z0)),u0,pl"rt"(ts"rt"(x2,y0),ts"rt"(x2,z0)),ts"rt"(x2,pl"rt"(y0,z0)),symis(rat,u0,pl"rt"(ts"rt"(x0,y0),ts"rt"(x1,z0)),t7),distpt2(x2,y0,z0),satz100a(ts"rt"(x0,y0),ts"rt"(x2,y0),ts"rt"(x1,z0),ts"rt"(x2,z0),t21,t22)):lessis"rt"(u0,ts"rt"(x2,pl"rt"(y0,z0)))
4157 t24:=orapp(less"rt"(u0,ts"rt"(x2,pl"rt"(y0,z0))),is"rt"(u0,ts"rt"(x2,pl"rt"(y0,z0))),lrt(ts(ksi,pl(eta,zeta)),u0),t23,[t:less"rt"(u0,ts"rt"(x2,pl"rt"(y0,z0)))]satz120(ts(ksi,pl(eta,zeta)),ts"rt"(x2,pl"rt"(y0,z0)),t20,u0,t),[t:is"rt"(u0,ts"rt"(x2,pl"rt"(y0,z0)))]isp1(rat,[u:rat]lrt(ts(ksi,pl(eta,zeta)),u),ts"rt"(x2,pl"rt"(y0,z0)),u0,t20,t)):lrt(ts(ksi,pl(eta,zeta)),u0)
4158 j@t25:=tsapp(ksi,zeta,w0,lw,lrt(ts(ksi,pl(eta,zeta)),u0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(zeta,y)][v:is"rt"(w0,ts"rt"(x,y))]t24(x,t,y,u,v)):lrt(ts(ksi,pl(eta,zeta)),u0)
4159 i@t26:=tsapp(ksi,eta,v0,lv,lrt(ts(ksi,pl(eta,zeta)),u0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(eta,y)][v:is"rt"(v0,ts"rt"(x,y))]t25(x,t,y,u,v)):lrt(ts(ksi,pl(eta,zeta)),u0)
4160 lu@t27:=plapp(ts(ksi,eta),ts(ksi,zeta),u0,lu,lrt(ts(ksi,pl(eta,zeta)),u0),[x:rat][t:lrt(ts(ksi,eta),x)][y:rat][u:lrt(ts(ksi,zeta),y)][v:is"rt"(u0,pl"rt"(x,y))]t26(x,t,y,u,v)):lrt(ts(ksi,pl(eta,zeta)),u0)
4161 -4144
4162 satz144:=isi1(ts(ksi,pl(eta,zeta)),pl(ts(ksi,eta),ts(ksi,zeta)),[x:rat][t:lrt(ts(ksi,pl(eta,zeta)),x)]t6".4144"(x,t),[x:rat][t:lrt(pl(ts(ksi,eta),ts(ksi,zeta)),x)]t27".4144"(x,t)):is(ts(ksi,pl(eta,zeta)),pl(ts(ksi,eta),ts(ksi,zeta)))
4163 disttp1:=tr3is(cut,ts(pl(ksi,eta),zeta),ts(zeta,pl(ksi,eta)),pl(ts(zeta,ksi),ts(zeta,eta)),pl(ts(ksi,zeta),ts(eta,zeta)),comts(pl(ksi,eta),zeta),satz144(zeta,ksi,eta),ispl12(ts(zeta,ksi),ts(ksi,zeta),ts(zeta,eta),ts(eta,zeta),comts(zeta,ksi),comts(zeta,eta))):is(ts(pl(ksi,eta),zeta),pl(ts(ksi,zeta),ts(eta,zeta)))
4164 disttp2:=satz144:is(ts(ksi,pl(eta,zeta)),pl(ts(ksi,eta),ts(ksi,zeta)))
4165 distpt1:=symis(cut,ts(pl(ksi,eta),zeta),pl(ts(ksi,zeta),ts(eta,zeta)),disttp1):is(pl(ts(ksi,zeta),ts(eta,zeta)),ts(pl(ksi,eta),zeta))
4166 distpt2:=symis(cut,ts(ksi,pl(eta,zeta)),pl(ts(ksi,eta),ts(ksi,zeta)),disttp2):is(pl(ts(ksi,eta),ts(ksi,zeta)),ts(ksi,pl(eta,zeta)))
4167 [m:more(ksi,eta)]
4168 +4145
4169 phi:=mn(ksi,eta,m):cut
4170 t1:=satz140d(ksi,eta,m):is(ksi,pl(eta,phi))
4171 t2:=tris(cut,ts(ksi,zeta),ts(pl(eta,phi),zeta),pl(ts(eta,zeta),ts(phi,zeta)),ists1(ksi,pl(eta,phi),zeta,t1),disttp1(eta,phi,zeta)):is(ts(ksi,zeta),pl(ts(eta,zeta),ts(phi,zeta)))
4172 -4145
4173 satz145a:=ismore1(pl(ts(eta,zeta),ts(phi".4145",zeta)),ts(ksi,zeta),ts(eta,zeta),symis(cut,ts(ksi,zeta),pl(ts(eta,zeta),ts(phi".4145",zeta)),t2".4145"),satz133(ts(eta,zeta),ts(phi".4145",zeta))):more(ts(ksi,zeta),ts(eta,zeta))
4174 zeta@[i:is(ksi,eta)]
4175 satz145b:=ists1(ksi,eta,zeta,i):is(ts(ksi,zeta),ts(eta,zeta))
4176 zeta@[l:less(ksi,eta)]
4177 satz145c:=satz121(ts(eta,zeta),ts(ksi,zeta),satz145a(eta,ksi,zeta,satz122(ksi,eta,l))):less(ts(ksi,zeta),ts(eta,zeta))
4178 m@satz145d:=ismore12(ts(ksi,zeta),ts(zeta,ksi),ts(eta,zeta),ts(zeta,eta),comts(ksi,zeta),comts(eta,zeta),satz145a):more(ts(zeta,ksi),ts(zeta,eta))
4179 i@satz145e:=ists2(ksi,eta,zeta,i):is(ts(zeta,ksi),ts(zeta,eta))
4180 l@satz145f:=isless12(ts(ksi,zeta),ts(zeta,ksi),ts(eta,zeta),ts(zeta,eta),comts(ksi,zeta),comts(eta,zeta),satz145c):less(ts(zeta,ksi),ts(zeta,eta))
4181 upsilon@[i:is(ksi,eta)][m:more(zeta,upsilon)]
4182 satz145g:=ismore2(ts(ksi,upsilon),ts(eta,upsilon),ts(ksi,zeta),ists1(ksi,eta,upsilon,i),satz145d(zeta,upsilon,ksi,m)):more(ts(ksi,zeta),ts(eta,upsilon))
4183 satz145h:=ismore12(ts(ksi,zeta),ts(zeta,ksi),ts(eta,upsilon),ts(upsilon,eta),comts(ksi,zeta),comts(eta,upsilon),satz145g):more(ts(zeta,ksi),ts(upsilon,eta))
4184 i@[l:less(zeta,upsilon)]
4185 satz145j:=isless2(ts(ksi,upsilon),ts(eta,upsilon),ts(ksi,zeta),ists1(ksi,eta,upsilon,i),satz145f(zeta,upsilon,ksi,l)):less(ts(ksi,zeta),ts(eta,upsilon))
4186 satz145k:=isless12(ts(ksi,zeta),ts(zeta,ksi),ts(eta,upsilon),ts(upsilon,eta),comts(ksi,zeta),comts(eta,upsilon),satz145j):less(ts(zeta,ksi),ts(upsilon,eta))
4187 +4146
4188 zeta@t1:=satz123a(ksi,eta):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta))
4189 t2:=satz123b(ts(ksi,zeta),ts(eta,zeta)):ec3(is(ts(ksi,zeta),ts(eta,zeta)),more(ts(ksi,zeta),ts(eta,zeta)),less(ts(ksi,zeta),ts(eta,zeta)))
4190 -4146
4191 zeta@[m:more(ts(ksi,zeta),ts(eta,zeta))]
4192 satz146a:=th11"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),is(ts(ksi,zeta),ts(eta,zeta)),more(ts(ksi,zeta),ts(eta,zeta)),less(ts(ksi,zeta),ts(eta,zeta)),t1".4146",t2".4146",[u:is(ksi,eta)]satz145b(ksi,eta,zeta,u),[u:more(ksi,eta)]satz145a(ksi,eta,zeta,u),[u:less(ksi,eta)]satz145c(ksi,eta,zeta,u),m):more(ksi,eta)
4193 zeta@[i:is(ts(ksi,zeta),ts(eta,zeta))]
4194 satz146b:=th10"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),is(ts(ksi,zeta),ts(eta,zeta)),more(ts(ksi,zeta),ts(eta,zeta)),less(ts(ksi,zeta),ts(eta,zeta)),t1".4146",t2".4146",[u:is(ksi,eta)]satz145b(ksi,eta,zeta,u),[u:more(ksi,eta)]satz145a(ksi,eta,zeta,u),[u:less(ksi,eta)]satz145c(ksi,eta,zeta,u),i):is(ksi,eta)
4195 zeta@[l:less(ts(ksi,zeta),ts(eta,zeta))]
4196 satz146c:=th12"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),is(ts(ksi,zeta),ts(eta,zeta)),more(ts(ksi,zeta),ts(eta,zeta)),less(ts(ksi,zeta),ts(eta,zeta)),t1".4146",t2".4146",[u:is(ksi,eta)]satz145b(ksi,eta,zeta,u),[u:more(ksi,eta)]satz145a(ksi,eta,zeta,u),[u:less(ksi,eta)]satz145c(ksi,eta,zeta,u),l):less(ksi,eta)
4197 zeta@[m:more(ts(zeta,ksi),ts(zeta,eta))]
4198 satz146d:=satz146a(ismore12(ts(zeta,ksi),ts(ksi,zeta),ts(zeta,eta),ts(eta,zeta),comts(zeta,ksi),comts(zeta,eta),m)):more(ksi,eta)
4199 zeta@[i:is(ts(zeta,ksi),ts(zeta,eta))]
4200 satz146e:=satz146b(tr3is(cut,ts(ksi,zeta),ts(zeta,ksi),ts(zeta,eta),ts(eta,zeta),comts(ksi,zeta),i,comts(zeta,eta))):is(ksi,eta)
4201 zeta@[l:less(ts(zeta,ksi),ts(zeta,eta))]
4202 satz146f:=satz146c(isless12(ts(zeta,ksi),ts(ksi,zeta),ts(zeta,eta),ts(eta,zeta),comts(zeta,ksi),comts(zeta,eta),l)):less(ksi,eta)
4203 upsilon@[m:more(ksi,eta)][n:more(zeta,upsilon)]
4204 +4147
4205 t1:=satz145a(ksi,eta,zeta,m):more(ts(ksi,zeta),ts(eta,zeta))
4206 t2:=ismore12(ts(zeta,eta),ts(eta,zeta),ts(upsilon,eta),ts(eta,upsilon),comts(zeta,eta),comts(upsilon,eta),satz145a(zeta,upsilon,eta,n)):more(ts(eta,zeta),ts(eta,upsilon))
4207 -4147
4208 satz147:=trmore(ts(ksi,zeta),ts(eta,zeta),ts(eta,upsilon),t1".4147",t2".4147"):more(ts(ksi,zeta),ts(eta,upsilon))
4209 upsilon@[l:less(ksi,eta)][k:less(zeta,upsilon)]
4210 satz147a:=satz121(ts(eta,upsilon),ts(ksi,zeta),satz147(eta,ksi,upsilon,zeta,satz122(ksi,eta,l),satz122(zeta,upsilon,k))):less(ts(ksi,zeta),ts(eta,upsilon))
4211 upsilon@[m:moreis(ksi,eta)][n:more(zeta,upsilon)]
4212 satz148a:=orapp(more(ksi,eta),is(ksi,eta),more(ts(ksi,zeta),ts(eta,upsilon)),m,[t:more(ksi,eta)]satz147(t,n),[t:is(ksi,eta)]satz145g(t,n)):more(ts(ksi,zeta),ts(eta,upsilon))
4213 upsilon@[m:more(ksi,eta)][n:moreis(zeta,upsilon)]
4214 satz148b:=orapp(more(zeta,upsilon),is(zeta,upsilon),more(ts(ksi,zeta),ts(eta,upsilon)),n,[t:more(zeta,upsilon)]satz147(m,t),[t:is(zeta,upsilon)]satz145h(zeta,upsilon,ksi,eta,t,m)):more(ts(ksi,zeta),ts(eta,upsilon))
4215 upsilon@[l:lessis(ksi,eta)][k:less(zeta,upsilon)]
4216 satz148c:=satz121(ts(eta,upsilon),ts(ksi,zeta),satz148a(eta,ksi,upsilon,zeta,satz125(ksi,eta,l),satz122(zeta,upsilon,k))):less(ts(ksi,zeta),ts(eta,upsilon))
4217 upsilon@[l:less(ksi,eta)][k:lessis(zeta,upsilon)]
4218 satz148d:=satz121(ts(eta,upsilon),ts(ksi,zeta),satz148b(eta,ksi,upsilon,zeta,satz122(ksi,eta,l),satz125(zeta,upsilon,k))):less(ts(ksi,zeta),ts(eta,upsilon))
4219 upsilon@[m:moreis(ksi,eta)][n:moreis(zeta,upsilon)]
4220 +4149
4221 [i:is(ksi,eta)][j:is(zeta,upsilon)]
4222 t1:=moreisi2(ts(ksi,zeta),ts(eta,upsilon),ists12(ksi,eta,zeta,upsilon,i,j)):moreis(ts(ksi,zeta),ts(eta,upsilon))
4223 i@[o:more(zeta,upsilon)]
4224 t2:=moreisi1(ts(ksi,zeta),ts(eta,upsilon),satz148a(m,o)):moreis(ts(ksi,zeta),ts(eta,upsilon))
4225 i@t3:=orapp(more(zeta,upsilon),is(zeta,upsilon),moreis(ts(ksi,zeta),ts(eta,upsilon)),n,[t:more(zeta,upsilon)]t2(t),[t:is(zeta,upsilon)]t1(t)):moreis(ts(ksi,zeta),ts(eta,upsilon))
4226 n@[o:more(ksi,eta)]
4227 t4:=moreisi1(ts(ksi,zeta),ts(eta,upsilon),satz148b(o,n)):moreis(ts(ksi,zeta),ts(eta,upsilon))
4228 -4149
4229 satz149:=orapp(more(ksi,eta),is(ksi,eta),moreis(ts(ksi,zeta),ts(eta,upsilon)),m,[t:more(ksi,eta)]t4".4149"(t),[t:is(ksi,eta)]t3".4149"(t)):moreis(ts(ksi,zeta),ts(eta,upsilon))
4230 upsilon@[l:lessis(ksi,eta)][k:lessis(zeta,upsilon)]
4231 satz149a:=satz124(ts(eta,upsilon),ts(ksi,zeta),satz149(eta,ksi,upsilon,zeta,satz125(ksi,eta,l),satz125(zeta,upsilon,k))):lessis(ts(ksi,zeta),ts(eta,upsilon))
4232 -rp
4233 @[r0:rat]
4234 ratset:=setof(rat,[x:rat]less(x,r0)):set(rat)
4235 +4150
4236 t1:=satz90(r0):some([x:rat]less(x,r0))
4237 [x0:rat][l:less(x0,r0)]
4238 t2:=estii(rat,[x:rat]less(x,r0),x0,l):in(x0,ratset)
4239 r0@t3:=ec3e13(is(r0,r0),more(r0,r0),less(r0,r0),satz81b(r0,r0),refis(rat,r0)):not(less(r0,r0))
4240 t4:=th3"l.imp"(in(r0,ratset),less(r0,r0),t3,[t:in(r0,ratset)]estie(rat,[x:rat]less(x,r0),r0,t)):not(in(r0,ratset))
4241 x0@[i:in(x0,ratset)]
4242 t5:=estie(rat,[x:rat]less(x,r0),x0,i):less(x0,r0)
4243 [x1:rat][k:less(x1,x0)]
4244 t6:=t2(x1,trless(x1,x0,r0,k,t5)):in(x1,ratset)
4245 i@t7:=satz91(x0,r0,t5):some([x:rat]and(less(x0,x),less(x,r0)))
4246 [x1:rat][a:and(less(x0,x1),less(x1,r0))]
4247 t8:=ande1(less(x0,x1),less(x1,r0),a):less(x0,x1)
4248 t9:=ande2(less(x0,x1),less(x1,r0),a):less(x1,r0)
4249 t10:=andi(in(x1,ratset),more(x1,x0),t2(x1,t9),satz83(x0,x1,t8)):and(in(x1,ratset),more(x1,x0))
4250 i@t11:=th6"l.some"(rat,[x:rat]and(less(x0,x),less(x,r0)),[x:rat]and(in(x,ratset),more(x,x0)),t7,[x:rat][t:and(less(x0,x),less(x,r0))]t10(x,t)):some([x:rat]and(in(x,ratset),more(x,x0)))
4251 l@t12:=cut2(ratset,x0,t2,r0,t4,[x:rat][t:in(x,ratset)][y:rat][u:less(y,x)]t6(x,t,y,u),[x:rat][t:in(x,ratset)]t11(x,t)):cutprop(ratset)
4252 -4150
4253 satz150:=someapp(rat,[x:rat]less(x,r0),t1".4150",cutprop(ratset),[x:rat][t:less(x,r0)]t12".4150"(x,t)):cutprop(ratset)
4254 +*rp
4255 r0@rpofrt:=cutof(ratset,satz150):cut
4256 [x0:rat][l:less"rt"(x0,r0)]
4257 lrtrpofrt:=ine(ratset,satz150,x0,t2"rt.4150"(x0,l)):lrt(rpofrt,x0)
4258 r0@[x0:rat][lx:lrt(rpofrt,x0)]
4259 lrtrpofrte:=t5"rt.4150"(x0,ini(ratset,satz150,x0,lx)):less"rt"(x0,r0)
4260 r0@[x0:rat][m:moreis"rt"(x0,r0)]
4261 +*iii4
4262 m@t12:=satz81c(x0,r0,m):not(less"rt"(x0,r0))
4263 -iii4
4264 m@urtrpofrt:=th3"l.imp"(lrt(rpofrt,x0),less"rt"(x0,r0),t12".iii4",[t:lrt(rpofrt,x0)]lrtrpofrte(x0,t)):urt(rpofrt,x0)
4265 @1rp:=rpofrt(1rt):cut
4266 +4151
4267 ksi@[z0:rat][lz:lrt(ts(ksi,1rp),z0)][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(1rp,y0)][i:is"rt"(z0,ts"rt"(x0,y0))]
4268 t1:=lrtrpofrte(1rt,y0,ly):less"rt"(y0,1rt)
4269 t2:=isless12"rt"(ts"rt"(x0,y0),z0,ts"rt"(x0,1rt),x0,symis(rat,z0,ts"rt"(x0,y0),i),example1a(x0),satz105f(y0,1rt,x0,t1)):less"rt"(z0,x0)
4270 t3:=satz120(ksi,x0,lx,z0,t2):lrt(ksi,z0)
4271 lz@t4:=tsapp(ksi,1rp,z0,lz,lrt(ksi,z0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(1rp,y)][v:is"rt"(z0,ts"rt"(x,y))]t3(x,t,y,u,v)):lrt(ksi,z0)
4272 ksi@[x0:rat][lx:lrt(ksi,x0)][x1:rat][lx1:lrt(ksi,x1)][l:less"rt"(x0,x1)]
4273 y1:=ts"rt"(ov(1rt,x1),x0):rat
4274 t5:=isless2"rt"(ts"rt"(ov(1rt,x1),x1),1rt,y1,satz110e(1rt,x1),satz105f(x0,x1,ov(1rt,x1),l)):less"rt"(y1,1rt)
4275 t6:=lrtrpofrt(1rt,y1,t5):lrt(1rp,y1)
4276 t7:=tr3is(rat,ts"rt"(x1,y1),ts"rt"(ts"rt"(x1,ov(1rt,x1)),x0),ts"rt"(1rt,x0),x0,assts2"rt"(x1,ov(1rt,x1),x0),ists1"rt"(ts"rt"(x1,ov(1rt,x1)),1rt,x0,satz110c(1rt,x1)),example1c(x0)):is"rt"(ts"rt"(x1,y1),x0)
4277 t8:=lrtts(ksi,1rp,x0,x1,lx1,y1,t6,symis(rat,ts"rt"(x1,y1),x0,t7)):lrt(ts(ksi,1rp),x0)
4278 lx@t9:=cutapp3(ksi,x0,lx,lrt(ts(ksi,1rp),x0),[y:rat][t:lrt(ksi,y)][u:less"rt"(x0,y)]t8(y,t,u)):lrt(ts(ksi,1rp),x0)
4279 -4151
4280 ksi@satz151:=isi1(ts(ksi,1rp),ksi,[x:rat][t:lrt(ts(ksi,1rp),x)]t4".4151"(x,t),[x:rat][t:lrt(ksi,x)]t9".4151"(x,t)):is(ts(ksi,1rp),ksi)
4281 satz151a:=symis(cut,ts(ksi,1rp),ksi,satz151):is(ksi,ts(ksi,1rp))
4282 satz151b:=tris(cut,ts(1rp,ksi),ts(ksi,1rp),ksi,comts(1rp,ksi),satz151):is(ts(1rp,ksi),ksi)
4283 satz151c:=symis(cut,ts(1rp,ksi),ksi,satz151b):is(ksi,ts(1rp,ksi))
4284 +4152
4285 [x0:rat][y0:rat]
4286 invprop1:=and(urt(ksi,y0),less"rt"(y0,x0)):'prop'
4287 ksi@[z0:rat][x0:rat]
4288 invprop2:=and3(urt(ksi,x0),some"rt"([x:rat]invprop1(x0,x)),is"rt"(z0,ov(1rt,x0))):'prop'
4289 z0@invprop:=some"rt"([x:rat]invprop2(z0,x)):'prop'
4290 ksi@inv:=setof(rat,[z:rat]invprop(z)):set(rat)
4291 x0@[ux:urt(ksi,x0)][y0:rat][uy:urt(ksi,y0)][l:less"rt"(y0,x0)][i:is"rt"(z0,ov(1rt,x0))]
4292 t1:=andi(urt(ksi,y0),less"rt"(y0,x0),uy,l):invprop1(x0,y0)
4293 t2:=somei(rat,[x:rat]invprop1(x0,x),y0,t1):some"rt"([x:rat]invprop1(x0,x))
4294 t3:=and3i(urt(ksi,x0),some"rt"([x:rat]invprop1(x0,x)),is"rt"(z0,ov(1rt,x0)),ux,t2,i):invprop2(z0,x0)
4295 t4:=somei(rat,[x:rat]invprop2(z0,x),x0,t3):invprop(z0)
4296 inv1:=estii(rat,[z:rat]invprop(z),z0,t4):in(z0,inv)
4297 z0@[i:in(z0,inv)][p:'prop'][p1:[x:rat][t:urt(ksi,x)][y:rat][u:urt(ksi,y)][v:less"rt"(y,x)][w:is"rt"(z0,ov(1rt,x))]p]
4298 t5:=estie(rat,[x:rat]invprop(x),z0,i):invprop(z0)
4299 [x0:rat][px:invprop2(z0,x0)]
4300 t6:=and3e1(urt(ksi,x0),some"rt"([x:rat]invprop1(x0,x)),is"rt"(z0,ov(1rt,x0)),px):urt(ksi,x0)
4301 t7:=and3e2(urt(ksi,x0),some"rt"([x:rat]invprop1(x0,x)),is"rt"(z0,ov(1rt,x0)),px):some"rt"([x:rat]invprop1(x0,x))
4302 t8:=and3e3(urt(ksi,x0),some"rt"([x:rat]invprop1(x0,x)),is"rt"(z0,ov(1rt,x0)),px):is"rt"(z0,ov(1rt,x0))
4303 [y0:rat][py:invprop1(x0,y0)]
4304 t9:=ande1(urt(ksi,y0),less"rt"(y0,x0),py):urt(ksi,y0)
4305 t10:=ande2(urt(ksi,y0),less"rt"(y0,x0),py):less"rt"(y0,x0)
4306 t11:=<t8><t10><t9><y0><t6><x0>p1:p
4307 px@t12:=someapp(rat,[x:rat]invprop1(x0,x),t7,p,[x:rat][t:invprop1(x0,x)]t11(x,t)):p
4308 p1@invapp:=someapp(rat,[x:rat]invprop2(z0,x),t5,p,[x:rat][t:invprop2(z0,x)]t12(x,t)):p
4309 ksi@[x0:rat][ux:urt(ksi,x0)]
4310 2x0:=pl"rt"(x0,x0):rat
4311 t13:=satz94a(x0,x0):less"rt"(x0,2x0)
4312 t14:=satz119a(ksi,x0,ux,2x0,t13):urt(ksi,2x0)
4313 t15:=inv1(ov(1rt,2x0),2x0,t14,x0,ux,t13,refis(rat,ov(1rt,2x0))):in(ov(1rt,2x0),inv)
4314 ksi@[x1:rat][lx:lrt(ksi,x1)][x0:rat][ux:urt(ksi,x0)]
4315 t16:=th3"l.imp"(is"rt"(x0,x1),lrt(ksi,x0),ux,[t:is"rt"(x0,x1)]isp1(rat,[x:rat]lrt(ksi,x),x1,x0,lx,t)):nis"rt"(x0,x1)
4316 t17:=satz110e(1rt,x0):is"rt"(ts"rt"(ov(1rt,x0),x0),1rt)
4317 t18:=satz110e(1rt,x1):is"rt"(ts"rt"(ov(1rt,x1),x1),1rt)
4318 [i:is"rt"(ov(1rt,x0),ov(1rt,x1))]
4319 t19:=tris(rat,ts"rt"(ov(1rt,x0),x1),ts"rt"(ov(1rt,x1),x1),1rt,ists1"rt"(ov(1rt,x0),ov(1rt,x1),x1,i),t18):is"rt"(ts"rt"(ov(1rt,x0),x1),1rt)
4320 t20:=satz110b(1rt,ov(1rt,x0),x0,x1,t17,t19):is"rt"(x0,x1)
4321 ux@t21:=th3"l.imp"(is"rt"(ov(1rt,x0),ov(1rt,x1)),is"rt"(x0,x1),t16,[t:is"rt"(ov(1rt,x0),ov(1rt,x1))]t20(t)):nis"rt"(ov(1rt,x0),ov(1rt,x1))
4322 lx@[i:in(ov(1rt,x1),inv)]
4323 t22:=invapp(ov(1rt,x1),i,con,[x:rat][t:urt(ksi,x)][y:rat][u:urt(ksi,y)][v:less"rt"(y,x)][w:is"rt"(ov(1rt,x1),ov(1rt,x))]<symis(rat,ov(1rt,x1),ov(1rt,x),w)>t21(x,t)):con
4324 lx@t23:=[t:in(ov(1rt,x1),inv)]t22(t):not(in(ov(1rt,x1),inv))
4325 ksi@[z0:rat][i:in(z0,inv)][u0:rat][l:less"rt"(u0,z0)][x0:rat][ux:urt(ksi,x0)][j:is"rt"(z0,ov(1rt,x0))]
4326 t24:=isless2"rt"(z0,ov(1rt,x0),u0,j,l):less"rt"(u0,ov(1rt,x0))
4327 t25:=tris(rat,ts"rt"(ov(1rt,x0),x0),1rt,ts"rt"(u0,ov(1rt,u0)),satz110e(1rt,x0),satz110d(1rt,u0)):is"rt"(ts"rt"(ov(1rt,x0),x0),ts"rt"(u0,ov(1rt,u0)))
4328 t26:=isless2"rt"(ts"rt"(ov(1rt,x0),x0),ts"rt"(u0,ov(1rt,u0)),ts"rt"(u0,x0),t25,satz105c(u0,ov(1rt,x0),x0,t24)):less"rt"(ts"rt"(u0,x0),ts"rt"(u0,ov(1rt,u0)))
4329 t27:=isless12"rt"(ts"rt"(u0,x0),ts"rt"(x0,u0),ts"rt"(u0,ov(1rt,u0)),ts"rt"(ov(1rt,u0),u0),comts"rt"(u0,x0),comts"rt"(u0,ov(1rt,u0)),t26):less"rt"(ts"rt"(x0,u0),ts"rt"(ov(1rt,u0),u0))
4330 t28:=satz106c(x0,ov(1rt,u0),u0,t27):less"rt"(x0,ov(1rt,u0))
4331 t29:=satz119a(x0,ux,ov(1rt,u0),t28):urt(ksi,ov(1rt,u0))
4332 t30:=satz110e(1rt,u0):is"rt"(ts"rt"(ov(1rt,u0),u0),1rt)
4333 t31:=satz110g(1rt,ov(1rt,u0),u0,t30):is"rt"(u0,ov(1rt,ov(1rt,u0)))
4334 t32:=inv1(u0,ov(1rt,u0),t29,x0,ux,t28,t31):in(u0,inv)
4335 l@t33:=invapp(z0,i,in(u0,inv),[x:rat][t:urt(ksi,x)][y:rat][u:urt(ksi,y)][v:less"rt"(y,x)][w:is"rt"(z0,ov(1rt,x))]t32(x,t,w)):in(u0,inv)
4336 i@[x0:rat][ux:urt(ksi,x0)][x1:rat][ux1:urt(ksi,x1)][l:less"rt"(x1,x0)][j:is"rt"(z0,ov(1rt,x0))]
4337 t34:=satz91(x1,x0,l):some"rt"([x:rat]and(less"rt"(x1,x),less"rt"(x,x0)))
4338 [x2:rat][a:and(less"rt"(x1,x2),less"rt"(x2,x0))]
4339 t35:=ande1(less"rt"(x1,x2),less"rt"(x2,x0),a):less"rt"(x1,x2)
4340 t36:=satz119a(ksi,x1,ux1,x2,t35):urt(ksi,x2)
4341 t37:=inv1(ov(1rt,x2),x2,t36,x1,ux1,t35,refis(rat,ov(1rt,x2))):in(ov(1rt,x2),inv)
4342 t38:=ande2(less"rt"(x1,x2),less"rt"(x2,x0),a):less"rt"(x2,x0)
4343 t39:=tris(rat,ts"rt"(x0,ov(1rt,x0)),1rt,ts"rt"(x2,ov(1rt,x2)),satz110c(1rt,x0),satz110d(1rt,x2)):is"rt"(ts"rt"(x0,ov(1rt,x0)),ts"rt"(x2,ov(1rt,x2)))
4344 t40:=isless2"rt"(ts"rt"(x0,ov(1rt,x0)),ts"rt"(x2,ov(1rt,x2)),ts"rt"(x2,ov(1rt,x0)),t39,satz105c(x2,x0,ov(1rt,x0),t38)):less"rt"(ts"rt"(x2,ov(1rt,x0)),ts"rt"(x2,ov(1rt,x2)))
4345 t41:=isless12"rt"(ts"rt"(x2,ov(1rt,x0)),ts"rt"(ov(1rt,x0),x2),ts"rt"(x2,ov(1rt,x2)),ts"rt"(ov(1rt,x2),x2),comts"rt"(x2,ov(1rt,x0)),comts"rt"(x2,ov(1rt,x2)),t40):less"rt"(ts"rt"(ov(1rt,x0),x2),ts"rt"(ov(1rt,x2),x2))
4346 t42:=satz106c(ov(1rt,x0),ov(1rt,x2),x2,t41):less"rt"(ov(1rt,x0),ov(1rt,x2))
4347 t43:=ismore2"rt"(ov(1rt,x0),z0,ov(1rt,x2),symis(rat,z0,ov(1rt,x0),j),satz83(ov(1rt,x0),ov(1rt,x2),t42)):more"rt"(ov(1rt,x2),z0)
4348 t44:=andi(in(ov(1rt,x2),inv),more"rt"(ov(1rt,x2),z0),t37,t43):and(in(ov(1rt,x2),inv),more"rt"(ov(1rt,x2),z0))
4349 t45:=somei(rat,[x:rat]and(in(x,inv),more"rt"(x,z0)),ov(1rt,x2),t44):some"rt"([x:rat]and(in(x,inv),more"rt"(x,z0)))
4350 j@t46:=someapp(rat,[x:rat]and(less"rt"(x1,x),less"rt"(x,x0)),t34,some"rt"([x:rat]and(in(x,inv),more"rt"(x,z0))),[x:rat][t:and(less"rt"(x1,x),less"rt"(x,x0))]t45(x,t)):some"rt"([x:rat]and(in(x,inv),more"rt"(x,z0)))
4351 i@t47:=invapp(z0,i,some"rt"([x:rat]and(in(x,inv),more"rt"(x,z0))),[x:rat][t:urt(ksi,x)][y:rat][u:urt(ksi,y)][v:less"rt"(y,x)][w:is"rt"(z0,ov(1rt,x))]t46(x,t,y,u,v,w)):some"rt"([x:rat]and(in(x,inv),more"rt"(x,z0)))
4352 ksi@[x0:rat][lx:lrt(ksi,x0)][y0:rat][uy:urt(ksi,y0)]
4353 t48:=cut2(inv,ov(1rt,pl"rt"(y0,y0)),t15(y0,uy),ov(1rt,x0),t23(x0,lx),[x:rat][t:in(x,inv)][y:rat][u:less"rt"(y,x)]t33(x,t,y,u),[x:rat][t:in(x,inv)]t47(x,t)):cutprop(inv)
4354 lx@t49:=cutapp1b(ksi,cutprop(inv),[x:rat][t:urt(ksi,x)]t48(x,t)):cutprop(inv)
4355 ksi@t50:=cutapp1a(ksi,cutprop(inv),[x:rat][t:lrt(ksi,x)]t49(x,t)):cutprop(inv)
4356 chi:=cutof(inv,t50):cut
4357 [z0:rat][lz:lrt(ts(ksi,chi),z0)][x0:rat][lx:lrt(ksi,x0)][u0:rat][lu:lrt(chi,u0)][i:is"rt"(z0,ts"rt"(x0,u0))][x1:rat][ux:urt(ksi,x1)][j:is"rt"(u0,ov(1rt,x1))]
4358 t51:=tris(rat,z0,ts"rt"(x0,u0),ts"rt"(x0,ov(1rt,x1)),i,ists2"rt"(u0,ov(1rt,x1),x0,j)):is"rt"(z0,ts"rt"(x0,ov(1rt,x1)))
4359 t52:=cutapp2a(ksi,x0,lx,x1,ux):less"rt"(x0,x1)
4360 t53:=isless12"rt"(ts"rt"(x0,ov(1rt,x1)),z0,ts"rt"(x1,ov(1rt,x1)),1rt,symis(rat,z0,ts"rt"(x0,ov(1rt,x1)),t51),satz110c(1rt,x1),satz105c(x0,x1,ov(1rt,x1),t52)):less"rt"(z0,1rt)
4361 t54:=lrtrpofrt(1rt,z0,t53):lrt(1rp,z0)
4362 i@r1:=ini(inv,t50,u0,lu):in(u0,inv)
4363 r2:=invapp(u0,r1,lrt(1rp,z0),[x:rat][t:urt(ksi,x)][y:rat][u:urt(ksi,y)][v:less"rt"(y,x)][w:is"rt"(u0,ov(1rt,x))]t54(x,t,w)):lrt(1rp,z0)
4364 lz@r3:=tsapp(ksi,chi,z0,lz,lrt(1rp,z0),[x:rat][t:lrt(ksi,x)][y:rat][u:lrt(chi,y)][v:is"rt"(z0,ts"rt"(x,y))]r2(x,t,y,u,v)):lrt(1rp,z0)
4365 ksi@[u0:rat][lu:lrt(1rp,u0)]
4366 t55:=lrtrpofrte(1rt,u0,lu):less"rt"(u0,1rt)
4367 t56:=satz83(u0,1rt,t55):more"rt"(1rt,u0)
4368 [x0:rat][lx:lrt(ksi,x0)][x1:rat][lx1:lrt(ksi,x1)][x2:rat][ux2:urt(ksi,x2)]
4369 t57:=cutapp2b(x1,lx1,x2,ux2):more"rt"(x2,x1)
4370 [i:is"rt"(mn"rt"(x2,x1,t57),ts"rt"(mn"rt"(1rt,u0,t56),x0))]
4371 t58:=cutapp2a(x0,lx,x2,ux2):less"rt"(x0,x2)
4372 t59:=satz105f(x0,x2,mn"rt"(1rt,u0,t56),t58):less"rt"(ts"rt"(mn"rt"(1rt,u0,t56),x0),ts"rt"(mn"rt"(1rt,u0,t56),x2))
4373 t60:=isless1"rt"(ts"rt"(mn"rt"(1rt,u0,t56),x0),mn"rt"(x2,x1,t57),ts"rt"(mn"rt"(1rt,u0,t56),x2),symis(rat,mn"rt"(x2,x1,t57),ts"rt"(mn"rt"(1rt,u0,t56),x0),i),t59):less"rt"(mn"rt"(x2,x1,t57),ts"rt"(mn"rt"(1rt,u0,t56),x2))
4374 t61:=tr4is(rat,pl"rt"(ts"rt"(mn"rt"(1rt,u0,t56),x2),ts"rt"(u0,x2)),ts"rt"(pl"rt"(mn"rt"(1rt,u0,t56),u0),x2),ts"rt"(1rt,x2),x2,pl"rt"(mn"rt"(x2,x1,t57),x1),distpt1"rt"(mn"rt"(1rt,u0,t56),u0,x2),ists1"rt"(pl"rt"(mn"rt"(1rt,u0,t56),u0),1rt,x2,satz101e(1rt,u0,t56)),example1c(x2),satz101f(x2,x1,t57)):is"rt"(pl"rt"(ts"rt"(mn"rt"(1rt,u0,t56),x2),ts"rt"(u0,x2)),pl"rt"(mn"rt"(x2,x1,t57),x1))
4375 t62:=satz96c(mn"rt"(x2,x1,t57),ts"rt"(mn"rt"(1rt,u0,t56),x2),ts"rt"(u0,x2),t60):less"rt"(pl"rt"(mn"rt"(x2,x1,t57),ts"rt"(u0,x2)),pl"rt"(ts"rt"(mn"rt"(1rt,u0,t56),x2),ts"rt"(u0,x2)))
4376 t63:=isless2"rt"(pl"rt"(ts"rt"(mn"rt"(1rt,u0,t56),x2),ts"rt"(u0,x2)),pl"rt"(mn"rt"(x2,x1,t57),x1),pl"rt"(mn"rt"(x2,x1,t57),ts"rt"(u0,x2)),t61,t62):less"rt"(pl"rt"(mn"rt"(x2,x1,t57),ts"rt"(u0,x2)),pl"rt"(mn"rt"(x2,x1,t57),x1))
4377 t64:=isless12"rt"(pl"rt"(mn"rt"(x2,x1,t57),ts"rt"(u0,x2)),pl"rt"(ts"rt"(u0,x2),mn"rt"(x2,x1,t57)),pl"rt"(mn"rt"(x2,x1,t57),x1),pl"rt"(x1,mn"rt"(x2,x1,t57)),compl"rt"(mn"rt"(x2,x1,t57),ts"rt"(u0,x2)),compl"rt"(mn"rt"(x2,x1,t57),x1),t63):less"rt"(pl"rt"(ts"rt"(u0,x2),mn"rt"(x2,x1,t57)),pl"rt"(x1,mn"rt"(x2,x1,t57)))
4378 t65:=satz97c(ts"rt"(u0,x2),x1,mn"rt"(x2,x1,t57),t64):less"rt"(ts"rt"(u0,x2),x1)
4379 t66:=tr3is(rat,ts"rt"(ov(1rt,u0),ts"rt"(u0,x2)),ts"rt"(ts"rt"(ov(1rt,u0),u0),x2),ts"rt"(1rt,x2),x2,assts2"rt"(ov(1rt,u0),u0,x2),ists1"rt"(ts"rt"(ov(1rt,u0),u0),1rt,x2,satz110e(1rt,u0)),example1c(x2)):is"rt"(ts"rt"(ov(1rt,u0),ts"rt"(u0,x2)),x2)
4380 t67:=isless12"rt"(ts"rt"(ov(1rt,u0),ts"rt"(u0,x2)),x2,ts"rt"(ov(1rt,u0),x1),ov(x1,u0),t66,satz141b(x1,u0),satz105f(ts"rt"(u0,x2),x1,ov(1rt,u0),t65)):less"rt"(x2,ov(x1,u0))
4381 t68:=satz119a(x2,ux2,ov(x1,u0),t67):urt(ksi,ov(x1,u0))
4382 t69:=satz110e(x1,u0):is"rt"(ts"rt"(ov(x1,u0),u0),x1)
4383 t70:=tr3is(rat,u0,ov(x1,ov(x1,u0)),ts"rt"(ov(1rt,ov(x1,u0)),x1),ts"rt"(x1,ov(1rt,ov(x1,u0))),satz110g(x1,ov(x1,u0),u0,t69),satz141c(x1,ov(x1,u0)),comts"rt"(ov(1rt,ov(x1,u0)),x1)):is"rt"(u0,ts"rt"(x1,ov(1rt,ov(x1,u0))))
4384 t71:=inv1(ov(1rt,ov(x1,u0)),ov(x1,u0),t68,x2,ux2,t67,refis(rat,ov(1rt,ov(x1,u0)))):in(ov(1rt,ov(x1,u0)),inv)
4385 t72:=ine(inv,t50,ov(1rt,ov(x1,u0)),t71):lrt(chi,ov(1rt,ov(x1,u0)))
4386 t73:=lrtts(ksi,chi,u0,x1,lx1,ov(1rt,ov(x1,u0)),t72,t70):lrt(ts(ksi,chi),u0)
4387 lx@t74:=satz132app(ksi,lrt(ts(ksi,chi),u0),ts"rt"(mn"rt"(1rt,u0,t56),x0),[x:rat][t:lrt(ksi,x)][y:rat][u:urt(ksi,y)][v:is"rt"(mn"rt"(y,x,cutapp2b(x,t,y,u)),ts"rt"(mn"rt"(1rt,u0,t56),x0))]t73(x,t,y,u,v)):lrt(ts(ksi,chi),u0)
4388 lu@t75:=cutapp1a(ksi,lrt(ts(ksi,chi),u0),[x:rat][t:lrt(ksi,x)]t74(x,t)):lrt(ts(ksi,chi),u0)
4389 ksi@t76:=isi1(ts(ksi,chi),1rp,[x:rat][t:lrt(ts(ksi,chi),x)]r3(x,t),[x:rat][t:lrt(1rp,x)]t75(x,t)):is(ts(ksi,chi),1rp)
4390 -4152
4391 satz152:=somei(cut,[t:cut]is(ts(ksi,t),1rp),chi".4152",t76".4152"):some([c:cut]is(ts(ksi,c),1rp))
4392 eta@[phi:cut][psi:cut]
4393 +4153
4394 [m:more(phi,psi)]
4395 t1:=satz145d(phi,psi,eta,m):more(ts(eta,phi),ts(eta,psi))
4396 t2:=ec3e21(is(ts(eta,phi),ts(eta,psi)),more(ts(eta,phi),ts(eta,psi)),less(ts(eta,phi),ts(eta,psi)),satz123b(ts(eta,phi),ts(eta,psi)),t1):nis(ts(eta,phi),ts(eta,psi))
4397 psi@[l:less(phi,psi)]
4398 t3:=satz145f(phi,psi,eta,l):less(ts(eta,phi),ts(eta,psi))
4399 t4:=ec3e31(is(ts(eta,phi),ts(eta,psi)),more(ts(eta,phi),ts(eta,psi)),less(ts(eta,phi),ts(eta,psi)),satz123b(ts(eta,phi),ts(eta,psi)),t3):nis(ts(eta,phi),ts(eta,psi))
4400 psi@[n:nis(phi,psi)]
4401 t5:=th1"l.or3"(is(phi,psi),more(phi,psi),less(phi,psi),satz123a(phi,psi),n):or(more(phi,psi),less(phi,psi))
4402 t6:=orapp(more(phi,psi),less(phi,psi),nis(ts(eta,phi),ts(eta,psi)),t5,[t:more(phi,psi)]t2(t),[t:less(phi,psi)]t4(t)):nis(ts(eta,phi),ts(eta,psi))
4403 -4153
4404 [i:is(ts(eta,phi),ksi)][j:is(ts(eta,psi),ksi)]
4405 satz153b:=th7"l.imp"(is(phi,psi),nis(ts(eta,phi),ts(eta,psi)),weli(is(ts(eta,phi),ts(eta,psi)),tris2(cut,ts(eta,phi),ts(eta,psi),ksi,i,j)),[t:nis(phi,psi)]t6".4153"(t)):is(phi,psi)
4406 +*4153
4407 eta@[tau:cut][i:is(ts(eta,tau),1rp)]
4408 chi:=ts(tau,ksi):cut
4409 t7:=tr3is(cut,ts(eta,chi),ts(ts(eta,tau),ksi),ts(1rp,ksi),ksi,assts2(eta,tau,ksi),ists1(ts(eta,tau),1rp,ksi,i),satz151b(ksi)):is(ts(eta,chi),ksi)
4410 t8:=somei(cut,[c:cut]is(ts(eta,c),ksi),chi,t7):some([c:cut]is(ts(eta,c),ksi))
4411 -4153
4412 eta@satz153a:=someapp(cut,[c:cut]is(ts(eta,c),1rp),satz152(eta),some([c:cut]is(ts(eta,c),ksi)),[c:cut][t:is(ts(eta,c),1rp)]t8".4153"(c,t)):some([c:cut]is(ts(eta,c),ksi))
4413 +*4153
4414 eta@t9:=[c:cut][d:cut][t:is(ts(eta,c),ksi)][u:is(ts(eta,d),ksi)]satz153b(c,d,t,u):amone(cut,[c:cut]is(ts(eta,c),ksi))
4415 -4153
4416 eta@satz153:=onei(cut,[c:cut]is(ts(eta,c),ksi),t9".4153",satz153a):one([c:cut]is(ts(eta,c),ksi))
4417 ov:=ind(cut,[a:cut]is(ts(eta,a),ksi),satz153):cut
4418 satz153c:=oneax(cut,[a:cut]is(ts(eta,a),ksi),satz153):is(ts(eta,ov(ksi,eta)),ksi)
4419 satz153d:=symis(cut,ts(eta,ov(ksi,eta)),ksi,satz153c):is(ksi,ts(eta,ov(ksi,eta)))
4420 satz153e:=tris(cut,ts(ov(ksi,eta),eta),ts(eta,ov(ksi,eta)),ksi,comts(ov(ksi,eta),eta),satz153c):is(ts(ov(ksi,eta),eta),ksi)
4421 satz153f:=symis(cut,ts(ov(ksi,eta),eta),ksi,satz153e):is(ksi,ts(ov(ksi,eta),eta))
4422 [phi:cut][i:is(ts(eta,phi),ksi)]
4423 satz153g:=satz153b(phi,ov(ksi,eta),i,satz153c):is(phi,ov(ksi,eta))
4424 @[ksi:cut]
4425 ratrp:=image(rat,cut,[x:rat]rpofrt(x),ksi):'prop'
4426 @[x0:rat]
4427 ratrpi:=imagei(rat,cut,[x:rat]rpofrt(x),x0):ratrp(rpofrt(x0))
4428 @[x:nat]
4429 rpofnt:=rpofrt(rtofn(x)):cut
4430 ksi@natrp:=image(nat,cut,[x:nat]rpofnt(x),ksi):'prop'
4431 x@natrpi:=imagei(nat,cut,[y:nat]rpofnt(y),x):natrp(rpofnt(x))
4432 ksi@[n:natrp(ksi)]
4433 +iii5
4434 [x:nat][i:is(ksi,rpofnt(x))]
4435 t1:=somei(rat,[y:rat]is(ksi,rpofrt(y)),rtofn(x),i):ratrp(ksi)
4436 -iii5
4437 lemmaiii5:=someapp(nat,[x:nat]is(ksi,rpofnt(x)),n,ratrp(ksi),[x:nat][t:is(ksi,rpofnt(x))]t1".iii5"(x,t)):ratrp(ksi)
4438 @[x0:rat][y0:rat][m:more"rt"(x0,y0)]
4439 +5154
4440 t1:=lrtrpofrt(x0,y0,satz82(x0,y0,m)):lrt(rpofrt(x0),y0)
4441 t2:=urtrpofrt(y0,y0,moreisi2"rt"(y0,y0,refis(rat,y0))):urt(rpofrt(y0),y0)
4442 t3:=andi(lrt(rpofrt(x0),y0),urt(rpofrt(y0),y0),t1,t2):and(lrt(rpofrt(x0),y0),urt(rpofrt(y0),y0))
4443 -5154
4444 satz154a:=somei(rat,[x:rat]and(lrt(rpofrt(x0),x),urt(rpofrt(y0),x)),y0,t3".5154"):more(rpofrt(x0),rpofrt(y0))
4445 y0@[i:is"rt"(x0,y0)]
4446 satz154b:=isf(rat,cut,[x:rat]rpofrt(x),x0,y0,i):is(rpofrt(x0),rpofrt(y0))
4447 y0@[l:less"rt"(x0,y0)]
4448 satz154c:=satz121(rpofrt(y0),rpofrt(x0),satz154a(y0,x0,satz83(x0,y0,l))):less(rpofrt(x0),rpofrt(y0))
4449 +*5154
4450 y0@t4:=satz81a(x0,y0):or3(is"rt"(x0,y0),more"rt"(x0,y0),less"rt"(x0,y0))
4451 t5:=satz123b(rpofrt(x0),rpofrt(y0)):ec3(is(rpofrt(x0),rpofrt(y0)),more(rpofrt(x0),rpofrt(y0)),less(rpofrt(x0),rpofrt(y0)))
4452 -5154
4453 y0@[m:more(rpofrt(x0),rpofrt(y0))]
4454 satz154d:=th11"l.ec3"(is"rt"(x0,y0),more"rt"(x0,y0),less"rt"(x0,y0),is(rpofrt(x0),rpofrt(y0)),more(rpofrt(x0),rpofrt(y0)),less(rpofrt(x0),rpofrt(y0)),t4".5154",t5".5154",[u:is"rt"(x0,y0)]satz154b(u),[u:more"rt"(x0,y0)]satz154a(u),[u:less"rt"(x0,y0)]satz154c(u),m):more"rt"(x0,y0)
4455 y0@[i:is(rpofrt(x0),rpofrt(y0))]
4456 satz154e:=th10"l.ec3"(is"rt"(x0,y0),more"rt"(x0,y0),less"rt"(x0,y0),is(rpofrt(x0),rpofrt(y0)),more(rpofrt(x0),rpofrt(y0)),less(rpofrt(x0),rpofrt(y0)),t4".5154",t5".5154",[u:is"rt"(x0,y0)]satz154b(u),[u:more"rt"(x0,y0)]satz154a(u),[u:less"rt"(x0,y0)]satz154c(u),i):is"rt"(x0,y0)
4457 y0@[l:less(rpofrt(x0),rpofrt(y0))]
4458 satz154f:=th12"l.ec3"(is"rt"(x0,y0),more"rt"(x0,y0),less"rt"(x0,y0),is(rpofrt(x0),rpofrt(y0)),more(rpofrt(x0),rpofrt(y0)),less(rpofrt(x0),rpofrt(y0)),t4".5154",t5".5154",[u:is"rt"(x0,y0)]satz154b(u),[u:more"rt"(x0,y0)]satz154a(u),[u:less"rt"(x0,y0)]satz154c(u),l):less"rt"(x0,y0)
4459 +*iii5
4460 @t2:=[x:rat][y:rat][t:is(rpofrt(x),rpofrt(y))]satz154e(x,y,t):injective(rat,cut,[x:rat]rpofrt(x))
4461 -iii5
4462 y0@[i:is"rt"(x0,y0)]
4463 isrterp:=satz154b(x0,y0,i):is(rpofrt(x0),rpofrt(y0))
4464 y0@[i:is(rpofrt(x0),rpofrt(y0))]
4465 isrtirp:=satz154e(x0,y0,i):is"rt"(x0,y0)
4466 ksi@[rtksi:ratrp(ksi)]
4467 rtofrp:=soft(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi):rat
4468 [eta:cut][rteta:ratrp(eta)][i:is(ksi,eta)]
4469 isrpert:=isinv(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi,eta,rteta,i):is"rt"(rtofrp(ksi,rtksi),rtofrp(eta,rteta))
4470 rteta@[i:is"rt"(rtofrp(ksi,rtksi),rtofrp(eta,rteta))]
4471 isrpirt:=isinve(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi,eta,rteta,i):is(ksi,eta)
4472 x0@isrtrp1:=isst1(rat,cut,[x:rat]rpofrt(x),t2".iii5",x0):is"rt"(x0,rtofrp(rpofrt(x0),ratrpi(x0)))
4473 isrtrp2:=isst2(rat,cut,[x:rat]rpofrt(x),t2".iii5",x0):is"rt"(rtofrp(rpofrt(x0),ratrpi(x0)),x0)
4474 rtksi@isrprt1:=ists1"e"(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi):is(ksi,rpofrt(rtofrp(ksi,rtksi)))
4475 isrprt2:=ists2"e"(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi):is(rpofrt(rtofrp(ksi,rtksi)),ksi)
4476 @[x:nat][y:nat][i:is"n"(x,y)]
4477 isnterp:=isf(nat,cut,[z:nat]rpofnt(z),x,y,i):is(rpofnt(x),rpofnt(y))
4478 y@[i:is(rpofnt(x),rpofnt(y))]
4479 isntirp:=isnirt(x,y,isrtirp(rtofn(x),rtofn(y),i)):is"n"(x,y)
4480 +*iii5
4481 @t3:=[x:nat][y:nat][t:is(rpofnt(x),rpofnt(y))]isntirp(x,y,t):injective(nat,cut,[x:nat]rpofnt(x))
4482 -iii5
4483 ksi@[ntksi:natrp(ksi)]
4484 ntofrp:=soft(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi):nat
4485 [eta:cut][nteta:natrp(eta)][i:is(ksi,eta)]
4486 isrpent:=isinv(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi,eta,nteta,i):is"n"(ntofrp(ksi,ntksi),ntofrp(eta,nteta))
4487 nteta@[i:is"n"(ntofrp(ksi,ntksi),ntofrp(eta,nteta))]
4488 isrpint:=isinve(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi,eta,nteta,i):is(ksi,eta)
4489 x@isntrp1:=isst1(nat,cut,[y:nat]rpofnt(y),t3".iii5",x):is"n"(x,ntofrp(rpofnt(x),natrpi(x)))
4490 isntrp2:=isst2(nat,cut,[y:nat]rpofnt(y),t3".iii5",x):is"n"(ntofrp(rpofnt(x),natrpi(x)),x)
4491 ntksi@isrpnt1:=ists1"e"(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi):is(ksi,rpofnt(ntofrp(ksi,ntksi)))
4492 isrpnt2:=ists2"e"(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi):is(rpofnt(ntofrp(ksi,ntksi)),ksi)
4493 @[x0:rat][y0:rat]
4494 +5155
4495 [z0:rat][lz:lrt(pl(rpofrt(x0),rpofrt(y0)),z0)][u0:rat][lu:lrt(rpofrt(x0),u0)][v0:rat][lv:lrt(rpofrt(y0),v0)][i:is"rt"(z0,pl"rt"(u0,v0))]
4496 t1:=lrtrpofrte(x0,u0,lu):less"rt"(u0,x0)
4497 t2:=lrtrpofrte(y0,v0,lv):less"rt"(v0,y0)
4498 t3:=satz98a(u0,x0,v0,y0,t1,t2):less"rt"(pl"rt"(u0,v0),pl"rt"(x0,y0))
4499 t4:=isless1"rt"(pl"rt"(u0,v0),z0,pl"rt"(x0,y0),symis(rat,z0,pl"rt"(u0,v0),i),t3):less"rt"(z0,pl"rt"(x0,y0))
4500 t5:=lrtrpofrt(pl"rt"(x0,y0),z0,t4):lrt(rpofrt(pl"rt"(x0,y0)),z0)
4501 lz@t6:=plapp(rpofrt(x0),rpofrt(y0),z0,lz,lrt(rpofrt(pl"rt"(x0,y0)),z0),[x:rat][t:lrt(rpofrt(x0),x)][y:rat][u:lrt(rpofrt(y0),y)][v:is"rt"(z0,pl"rt"(x,y))]t5(x,t,y,u,v)):lrt(rpofrt(pl"rt"(x0,y0)),z0)
4502 y0@[u0:rat][lu:lrt(rpofrt(pl"rt"(x0,y0)),u0)]
4503 t7:=lrtrpofrte(pl"rt"(x0,y0),u0,lu):less"rt"(u0,pl"rt"(x0,y0))
4504 u01:=ov"rt"(u0,pl"rt"(x0,y0)):rat
4505 t8:=isless12"rt"(u0,ts"rt"(u01,pl"rt"(x0,y0)),pl"rt"(x0,y0),ts"rt"(1rt,pl"rt"(x0,y0)),satz110f(u0,pl"rt"(x0,y0)),example1d(pl"rt"(x0,y0)),t7):less"rt"(ts"rt"(u01,pl"rt"(x0,y0)),ts"rt"(1rt,pl"rt"(x0,y0)))
4506 t9:=satz106c(u01,1rt,pl"rt"(x0,y0),t8):less"rt"(u01,1rt)
4507 t10:=tris(rat,u0,ts"rt"(pl"rt"(x0,y0),u01),pl"rt"(ts"rt"(x0,u01),ts"rt"(y0,u01)),satz110d(u0,pl"rt"(x0,y0)),disttp1"rt"(x0,y0,u01)):is"rt"(u0,pl"rt"(ts"rt"(x0,u01),ts"rt"(y0,u01)))
4508 y0@[l:less"rt"(y0,1rt)]
4509 t11:=isless12"rt"(ts"rt"(y0,x0),ts"rt"(x0,y0),ts"rt"(1rt,x0),x0,comts"rt"(y0,x0),example1c(x0),satz105c(y0,1rt,x0,l)):less"rt"(ts"rt"(x0,y0),x0)
4510 t12:=lrtrpofrt(x0,ts"rt"(x0,y0),t11):lrt(rpofrt(x0),ts"rt"(x0,y0))
4511 lu@t13:=lrtpl(rpofrt(x0),rpofrt(y0),u0,ts"rt"(x0,u01),t12(x0,u01,t9),ts"rt"(y0,u01),t12(y0,u01,t9),t10):lrt(pl(rpofrt(x0),rpofrt(y0)),u0)
4512 -5155
4513 satz155a:=isi1(rpofrt(pl"rt"(x0,y0)),pl(rpofrt(x0),rpofrt(y0)),[x:rat][t:lrt(rpofrt(pl"rt"(x0,y0)),x)]t13".5155"(x,t),[x:rat][t:lrt(pl(rpofrt(x0),rpofrt(y0)),x)]t6".5155"(x,t)):is(rpofrt(pl"rt"(x0,y0)),pl(rpofrt(x0),rpofrt(y0)))
4514 [m:more"rt"(x0,y0)]
4515 +*5155
4516 m@t14:=satz101f(x0,y0,m):is"rt"(x0,pl"rt"(mn"rt"(x0,y0,m),y0))
4517 t15:=tris(cut,rpofrt(x0),rpofrt(pl"rt"(mn"rt"(x0,y0,m),y0)),pl(rpofrt(mn"rt"(x0,y0,m)),rpofrt(y0)),isrterp(x0,pl"rt"(mn"rt"(x0,y0,m),y0),t14),satz155a(mn"rt"(x0,y0,m),y0)):is(rpofrt(x0),pl(rpofrt(mn"rt"(x0,y0,m)),rpofrt(y0)))
4518 t16:=tris2(cut,pl(rpofrt(y0),rpofrt(mn"rt"(x0,y0,m))),rpofrt(x0),pl(rpofrt(mn"rt"(x0,y0,m)),rpofrt(y0)),compl(rpofrt(y0),rpofrt(mn"rt"(x0,y0,m))),t15):is(pl(rpofrt(y0),rpofrt(mn"rt"(x0,y0,m))),rpofrt(x0))
4519 -5155
4520 m@satz155b:=satz140g(rpofrt(x0),rpofrt(y0),rpofrt(mn"rt"(x0,y0,m)),satz154a(x0,y0,m),t16".5155"):is(rpofrt(mn"rt"(x0,y0,m)),mn(rpofrt(x0),rpofrt(y0),satz154a(x0,y0,m)))
4521 +*5155
4522 y0@[z0:rat][lz:lrt(ts(rpofrt(x0),rpofrt(y0)),z0)][u0:rat][lu:lrt(rpofrt(x0),u0)][v0:rat][lv:lrt(rpofrt(y0),v0)][i:is"rt"(z0,ts"rt"(u0,v0))]
4523 t17:=lrtrpofrte(x0,u0,lu):less"rt"(u0,x0)
4524 t18:=lrtrpofrte(y0,v0,lv):less"rt"(v0,y0)
4525 t19:=satz107a(u0,x0,v0,y0,t17,t18):less"rt"(ts"rt"(u0,v0),ts"rt"(x0,y0))
4526 t20:=isless1"rt"(ts"rt"(u0,v0),z0,ts"rt"(x0,y0),symis(rat,z0,ts"rt"(u0,v0),i),t19):less"rt"(z0,ts"rt"(x0,y0))
4527 t21:=lrtrpofrt(ts"rt"(x0,y0),z0,t20):lrt(rpofrt(ts"rt"(x0,y0)),z0)
4528 lz@t22:=tsapp(rpofrt(x0),rpofrt(y0),z0,lz,lrt(rpofrt(ts"rt"(x0,y0)),z0),[x:rat][t:lrt(rpofrt(x0),x)][y:rat][u:lrt(rpofrt(y0),y)][v:is"rt"(z0,ts"rt"(x,y))]t21(x,t,y,u,v)):lrt(rpofrt(ts"rt"(x0,y0)),z0)
4529 y0@[u0:rat][lu:lrt(rpofrt(ts"rt"(x0,y0)),u0)]
4530 t23:=lrtrpofrte(ts"rt"(x0,y0),u0,lu):less"rt"(u0,ts"rt"(x0,y0))
4531 [u1:rat][a:and(less"rt"(u0,u1),less"rt"(u1,ts"rt"(x0,y0)))]
4532 t24:=ande1(less"rt"(u0,u1),less"rt"(u1,ts"rt"(x0,y0)),a):less"rt"(u0,u1)
4533 t25:=ande2(less"rt"(u0,u1),less"rt"(u1,ts"rt"(x0,y0)),a):less"rt"(u1,ts"rt"(x0,y0))
4534 t26:=isless12"rt"(u0,ts"rt"(ov"rt"(u0,u1),u1),u1,ts"rt"(1rt,u1),satz110f(u0,u1),example1d(u1),t24):less"rt"(ts"rt"(ov"rt"(u0,u1),u1),ts"rt"(1rt,u1))
4535 t27:=satz106c(ov"rt"(u0,u1),1rt,u1,t26):less"rt"(ov"rt"(u0,u1),1rt)
4536 t28:=isless1"rt"(u1,ts"rt"(ov"rt"(u1,y0),y0),ts"rt"(x0,y0),satz110f(u1,y0),t25):less"rt"(ts"rt"(ov"rt"(u1,y0),y0),ts"rt"(x0,y0))
4537 t29:=satz106c(ov"rt"(u1,y0),x0,y0,t28):less"rt"(ov"rt"(u1,y0),x0)
4538 t30:=tr3is(rat,u0,ts"rt"(u1,ov"rt"(u0,u1)),ts"rt"(ts"rt"(ov"rt"(u1,y0),y0),ov"rt"(u0,u1)),ts"rt"(ov"rt"(u1,y0),ts"rt"(y0,ov"rt"(u0,u1))),satz110d(u0,u1),ists1"rt"(u1,ts"rt"(ov"rt"(u1,y0),y0),ov"rt"(u0,u1),satz110f(u1,y0)),assts1"rt"(ov"rt"(u1,y0),y0,ov"rt"(u0,u1))):is"rt"(u0,ts"rt"(ov"rt"(u1,y0),ts"rt"(y0,ov"rt"(u0,u1))))
4539 t31:=lrtts(rpofrt(x0),rpofrt(y0),u0,ov"rt"(u1,y0),lrtrpofrt(x0,ov"rt"(u1,y0),t29),ts"rt"(y0,ov"rt"(u0,u1)),t12(y0,ov"rt"(u0,u1),t27),t30):lrt(ts(rpofrt(x0),rpofrt(y0)),u0)
4540 lu@t32:=someapp(rat,[x:rat]and(less"rt"(u0,x),less"rt"(x,ts"rt"(x0,y0))),satz91(u0,ts"rt"(x0,y0),t23),lrt(ts(rpofrt(x0),rpofrt(y0)),u0),[x:rat][t:and(less"rt"(u0,x),less"rt"(x,ts"rt"(x0,y0)))]t31(x,t)):lrt(ts(rpofrt(x0),rpofrt(y0)),u0)
4541 -5155
4542 y0@satz155c:=isi1(rpofrt(ts"rt"(x0,y0)),ts(rpofrt(x0),rpofrt(y0)),[x:rat][t:lrt(rpofrt(ts"rt"(x0,y0)),x)]t32".5155"(x,t),[x:rat][t:lrt(ts(rpofrt(x0),rpofrt(y0)),x)]t22".5155"(x,t)):is(rpofrt(ts"rt"(x0,y0)),ts(rpofrt(x0),rpofrt(y0)))
4543 +*5155
4544 y0@t33:=satz110f(x0,y0):is"rt"(x0,ts"rt"(ov"rt"(x0,y0),y0))
4545 t34:=tris(cut,rpofrt(x0),rpofrt(ts"rt"(ov"rt"(x0,y0),y0)),ts(rpofrt(ov"rt"(x0,y0)),rpofrt(y0)),isrterp(x0,ts"rt"(ov"rt"(x0,y0),y0),t33),satz155c(ov"rt"(x0,y0),y0)):is(rpofrt(x0),ts(rpofrt(ov"rt"(x0,y0)),rpofrt(y0)))
4546 t35:=tris2(cut,ts(rpofrt(y0),rpofrt(ov"rt"(x0,y0))),rpofrt(x0),ts(rpofrt(ov"rt"(x0,y0)),rpofrt(y0)),comts(rpofrt(y0),rpofrt(ov"rt"(x0,y0))),t34):is(ts(rpofrt(y0),rpofrt(ov"rt"(x0,y0))),rpofrt(x0))
4547 -5155
4548 y0@satz155d:=satz153g(rpofrt(x0),rpofrt(y0),rpofrt(ov"rt"(x0,y0)),t35".5155"):is(rpofrt(ov"rt"(x0,y0)),ov(rpofrt(x0),rpofrt(y0)))
4549 @[x:nat][y:nat]
4550 satz155e:=tris(cut,rpofnt(pl"n"(x,y)),rpofrt(pl"rt"(rtofn(x),rtofn(y))),pl(rpofnt(x),rpofnt(y)),isrterp(rtofn(pl"n"(x,y)),pl"rt"(rtofn(x),rtofn(y)),symis(rat,pl"rt"(rtofn(x),rtofn(y)),rtofn(pl"n"(x,y)),satz112h(x,y))),satz155a(rtofn(x),rtofn(y))):is(rpofnt(pl"n"(x,y)),pl(rpofnt(x),rpofnt(y)))
4551 satz155f:=tris(cut,rpofnt(ts"n"(x,y)),rpofrt(ts"rt"(rtofn(x),rtofn(y))),ts(rpofnt(x),rpofnt(y)),isrterp(rtofn(ts"n"(x,y)),ts"rt"(rtofn(x),rtofn(y)),symis(rat,ts"rt"(rtofn(x),rtofn(y)),rtofn(ts"n"(x,y)),satz112j(x,y))),satz155c(rtofn(x),rtofn(y))):is(rpofnt(ts"n"(x,y)),ts(rpofnt(x),rpofnt(y)))
4552 +nt
4553 @natt:=ot(cut,[t:cut]natrp(t)):'type'
4554 [ksi:cut][nksi:natrp(ksi)]
4555 nttofrp:=out(cut,[t:cut]natrp(t),ksi,nksi):natt
4556 @[xt:natt][yt:natt]
4557 is:=is"e"(natt,xt,yt):'prop'
4558 nis:=not(is(xt,yt)):'prop'
4559 @[p:[x:natt]'prop']
4560 all:=all"l"(natt,p):'prop'
4561 some:=some"l"(natt,p):'prop'
4562 one:=one"e"(natt,p):'prop'
4563 xt@[st:set(natt)]
4564 in:=esti(natt,xt,st):'prop'
4565 xt@rpofntt:=in"e"(cut,[t:cut]natrp(t),xt):cut
4566 natrpi:=inp(cut,[t:cut]natrp(t),xt):natrp(rpofntt(xt))
4567 nksi@[eta:cut][neta:natrp(eta)][i:is"rp"(ksi,eta)]
4568 isrpentt:=isouti(cut,[t:cut]natrp(t),ksi,nksi,eta,neta,i):is(nttofrp(ksi,nksi),nttofrp(eta,neta))
4569 neta@[i:is(nttofrp(ksi,nksi),nttofrp(eta,neta))]
4570 isrpintt:=isoute(cut,[t:cut]natrp(t),ksi,nksi,eta,neta,i):is"rp"(ksi,eta)
4571 yt@[i:is(xt,yt)]
4572 isntterp:=isini(cut,[t:cut]natrp(t),xt,yt,i):is"rp"(rpofntt(xt),rpofntt(yt))
4573 yt@[i:is"rp"(rpofntt(xt),rpofntt(yt))]
4574 isnttirp:=isine(cut,[t:cut]natrp(t),xt,yt,i):is(xt,yt)
4575 nksi@isrpntt1:=isinout(cut,[t:cut]natrp(t),ksi,nksi):is"rp"(ksi,rpofntt(nttofrp(ksi,nksi)))
4576 xt@isnttrp1:=isoutin(cut,[t:cut]natrp(t),xt):is(xt,nttofrp(rpofntt(xt),natrpi(xt)))
4577 @[x:nat]
4578 nttofnt:=nttofrp(rpofnt(x),natrpi"rp"(x)):natt
4579 [y:nat][i:is"n"(x,y)]
4580 isntentt:=isrpentt(rpofnt(x),natrpi"rp"(x),rpofnt(y),natrpi"rp"(y),isnterp(x,y,i)):is(nttofnt(x),nttofnt(y))
4581 y@[i:is(nttofnt(x),nttofnt(y))]
4582 isntintt:=isntirp(x,y,isrpintt(rpofnt(x),natrpi"rp"(x),rpofnt(y),natrpi"rp"(y),i)):is"n"(x,y)
4583 xt@ntofntt:=ntofrp(rpofntt(xt),natrpi(xt)):nat
4584 yt@[i:is(xt,yt)]
4585 isnttent:=isrpent(rpofntt(xt),natrpi(xt),rpofntt(yt),natrpi(yt),isntterp(xt,yt,i)):is"n"(ntofntt(xt),ntofntt(yt))
4586 yt@[i:is"n"(ntofntt(xt),ntofntt(yt))]
4587 isnttint:=isnttirp(xt,yt,isrpint(rpofntt(xt),natrpi(xt),rpofntt(yt),natrpi(yt),i)):is(xt,yt)
4588 +iii5
4589 x@t5:=isrpntt1(rpofnt(x),natrpi"rp"(x)):is"rp"(rpofnt(x),rpofntt(nttofnt(x)))
4590 t6:=isrpent(rpofnt(x),natrpi"rp"(x),rpofntt(nttofnt(x)),natrpi(nttofnt(x)),t5):is"n"(ntofrp(rpofnt(x),natrpi"rp"(x)),ntofntt(nttofnt(x)))
4591 -iii5
4592 x@isntntt1:=tris(nat,x,ntofrp(rpofnt(x),natrpi"rp"(x)),ntofntt(nttofnt(x)),isntrp1(x),t6".iii5"):is"n"(x,ntofntt(nttofnt(x)))
4593 +*iii5
4594 xt@t7:=isrpnt1(rpofntt(xt),natrpi(xt)):is"rp"(rpofntt(xt),rpofnt(ntofntt(xt)))
4595 t8:=isrpentt(rpofntt(xt),natrpi(xt),rpofnt(ntofntt(xt)),natrpi"rp"(ntofntt(xt)),t7):is(nttofrp(rpofntt(xt),natrpi(xt)),nttofnt(ntofntt(xt)))
4596 -iii5
4597 xt@isnttnt1:=tris(natt,xt,nttofrp(rpofntt(xt),natrpi(xt)),nttofnt(ntofntt(xt)),isnttrp1(xt),t8".iii5"):is(xt,nttofnt(ntofntt(xt)))
4598 x@isntntt2:=symis(nat,x,ntofntt(nttofnt(x)),isntntt1):is"n"(ntofntt(nttofnt(x)),x)
4599 xt@isnttnt2:=symis(natt,xt,nttofnt(ntofntt(xt)),isnttnt1):is(nttofnt(ntofntt(xt)),xt)
4600 @1t:=nttofnt(1):natt
4601 suct:=[x:natt]nttofnt(<ntofntt(x)>suc):[x:natt]natt
4602 +5156
4603 xt@[j:is(<xt>suct,1t)]
4604 t1:=isntintt(<ntofntt(xt)>suc,1,j):is"n"(<ntofntt(xt)>suc,1)
4605 -5156
4606 xt@satz156a:=th3"l.imp"(is(<xt>suct,1t),is"n"(<ntofntt(xt)>suc,1),<ntofntt(xt)>ax3,[t:is(<xt>suct,1t)]t1".5156"(t)):nis(<xt>suct,1t)
4607 yt@[i:is(<xt>suct,<yt>suct)]
4608 +*5156
4609 i@t2:=isntintt(<ntofntt(xt)>suc,<ntofntt(yt)>suc,i):is"n"(<ntofntt(xt)>suc,<ntofntt(yt)>suc)
4610 -5156
4611 i@satz156b:=isnttint(xt,yt,<t2".5156"><ntofntt(yt)><ntofntt(xt)>ax4):is(xt,yt)
4612 @[st:set(natt)]
4613 cond1:=in(1t,st):'prop'
4614 cond2:=all([x:natt]imp(in(x,st),in(<x>suct,st))):'prop'
4615 [c1:cond1][c2:cond2]
4616 +*5156
4617 c2@[x:nat]
4618 prop1:=in(nttofnt(x),st):'prop'
4619 [p:prop1(x)]
4620 t3:=<p><nttofnt(x)>c2:in(<nttofnt(x)>suct,st)
4621 t4:=isp(nat,[t:nat]in(nttofnt(<t>suc),st),ntofntt(nttofnt(x)),x,t3,isntntt2(x)):prop1(<x>suc)
4622 -5156
4623 c2@[xt:natt]
4624 +*5156
4625 xt@t5:=induction([t:nat]prop1(t),c1,[t:nat][u:prop1(t)]t4(t,u),ntofntt(xt)):in(nttofnt(ntofntt(xt)),st)
4626 -5156
4627 xt@satz156c:=isp(natt,[t:natt]in(t,st),nttofnt(ntofntt(xt)),xt,t5".5156",isnttnt2(xt)):in(xt,st)
4628 @ax3t:=[x:natt]satz156a(x):[x:natt]nis(<x>suct,1t)
4629 ax4t:=[x:natt][y:natt][u:is(<x>suct,<y>suct)]satz156b(x,y,u):[x:natt][y:natt][u:is(<x>suct,<y>suct)]is(x,y)
4630 ax5t:=[s:set(natt)][u:cond1(s)][v:cond2(s)][x:natt]satz156c(s,u,v,x):[s:set(natt)][u:cond1(s)][v:cond2(s)][x:natt]in(x,s)
4631 -nt
4632 +rtt
4633 @ratt:=ot(cut,[t:cut]ratrp(t)):'type'
4634 [ksi:cut][rtksi:ratrp(ksi)]
4635 rttofrp:=out(cut,[t:cut]ratrp(t),ksi,rtksi):ratt
4636 @[x0t:ratt][y0t:ratt]
4637 is:=is"e"(ratt,x0t,y0t):'prop'
4638 nis:=not(is(x0t,y0t)):'prop'
4639 @[p:[x:ratt]'prop']
4640 all:=all"l"(ratt,p):'prop'
4641 some:=some"l"(ratt,p):'prop'
4642 one:=one"e"(ratt,p):'prop'
4643 x0t@rpofrtt:=in"e"(cut,[t:cut]ratrp(t),x0t):cut
4644 ratrpi:=inp(cut,[t:cut]ratrp(t),x0t):ratrp(rpofrtt(x0t))
4645 rtksi@[eta:cut][rteta:ratrp(eta)][i:is"rp"(ksi,eta)]
4646 isrpertt:=isouti(cut,[t:cut]ratrp(t),ksi,rtksi,eta,rteta,i):is(rttofrp(ksi,rtksi),rttofrp(eta,rteta))
4647 rteta@[i:is(rttofrp(ksi,rtksi),rttofrp(eta,rteta))]
4648 isrpirtt:=isoute(cut,[t:cut]ratrp(t),ksi,rtksi,eta,rteta,i):is"rp"(ksi,eta)
4649 y0t@[i:is(x0t,y0t)]
4650 isrtterp:=isini(cut,[t:cut]ratrp(t),x0t,y0t,i):is"rp"(rpofrtt(x0t),rpofrtt(y0t))
4651 y0t@[i:is"rp"(rpofrtt(x0t),rpofrtt(y0t))]
4652 isrttirp:=isine(cut,[t:cut]ratrp(t),x0t,y0t,i):is(x0t,y0t)
4653 rtksi@isrprtt1:=isinout(cut,[t:cut]ratrp(t),ksi,rtksi):is"rp"(ksi,rpofrtt(rttofrp(ksi,rtksi)))
4654 x0t@isrttrp1:=isoutin(cut,[t:cut]ratrp(t),x0t):is(x0t,rttofrp(rpofrtt(x0t),ratrpi(x0t)))
4655 @[x0:rat]
4656 rttofrt:=rttofrp(rpofrt(x0),ratrpi"rp"(x0)):ratt
4657 [y0:rat][i:is"rt"(x0,y0)]
4658 isrtertt:=isrpertt(rpofrt(x0),ratrpi"rp"(x0),rpofrt(y0),ratrpi"rp"(y0),isrterp(x0,y0,i)):is(rttofrt(x0),rttofrt(y0))
4659 y0@[i:is(rttofrt(x0),rttofrt(y0))]
4660 isrtirtt:=isrtirp(x0,y0,isrpirtt(rpofrt(x0),ratrpi"rp"(x0),rpofrt(y0),ratrpi"rp"(y0),i)):is"rt"(x0,y0)
4661 x0t@rtofrtt:=rtofrp(rpofrtt(x0t),ratrpi(x0t)):rat
4662 y0t@[i:is(x0t,y0t)]
4663 isrttert:=isrpert(rpofrtt(x0t),ratrpi(x0t),rpofrtt(y0t),ratrpi(y0t),isrtterp(x0t,y0t,i)):is"rt"(rtofrtt(x0t),rtofrtt(y0t))
4664 y0t@[i:is"rt"(rtofrtt(x0t),rtofrtt(y0t))]
4665 isrttirt:=isrttirp(x0t,y0t,isrpirt(rpofrtt(x0t),ratrpi(x0t),rpofrtt(y0t),ratrpi(y0t),i)):is(x0t,y0t)
4666 +iii5
4667 x0@t9:=isrprtt1(rpofrt(x0),ratrpi"rp"(x0)):is"rp"(rpofrt(x0),rpofrtt(rttofrt(x0)))
4668 t10:=isrpert(rpofrt(x0),ratrpi"rp"(x0),rpofrtt(rttofrt(x0)),ratrpi(rttofrt(x0)),t9):is"rt"(rtofrp(rpofrt(x0),ratrpi"rp"(x0)),rtofrtt(rttofrt(x0)))
4669 -iii5
4670 x0@isrtrtt1:=tris(rat,x0,rtofrp(rpofrt(x0),ratrpi"rp"(x0)),rtofrtt(rttofrt(x0)),isrtrp1(x0),t10".iii5"):is"rt"(x0,rtofrtt(rttofrt(x0)))
4671 +*iii5
4672 x0t@t11:=isrprt1(rpofrtt(x0t),ratrpi(x0t)):is"rp"(rpofrtt(x0t),rpofrt(rtofrtt(x0t)))
4673 t12:=isrpertt(rpofrtt(x0t),ratrpi(x0t),rpofrt(rtofrtt(x0t)),ratrpi"rp"(rtofrtt(x0t)),t11):is(rttofrp(rpofrtt(x0t),ratrpi(x0t)),rttofrt(rtofrtt(x0t)))
4674 -iii5
4675 x0t@isrttrt1:=tris(ratt,x0t,rttofrp(rpofrtt(x0t),ratrpi(x0t)),rttofrt(rtofrtt(x0t)),isrttrp1(x0t),t12".iii5"):is(x0t,rttofrt(rtofrtt(x0t)))
4676 -rtt
4677 @[ksi:cut]
4678 example2:=satz153c(1rp,ksi):is(ts(ksi,ov(1rp,ksi)),1rp)
4679 [rtksi:ratrp(ksi)]
4680 +5157
4681 x01:=rtofrp(ksi,rtksi):rat
4682 ksi@s1:=setof(rat,[x:rat]urt(ksi,x)):set(rat)
4683 rtksi@[y0:rat][i:in(y0,s1)]
4684 t1:=estie(rat,[x:rat]urt(ksi,x),y0,i):urt(ksi,y0)
4685 [m:more"rt"(x01,y0)]
4686 t2:=lrtrpofrt(x01,y0,satz82(x01,y0,m)):lrt(rpofrt(x01),y0)
4687 t3:=isp(cut,[x:cut]lrt(x,y0),rpofrt(x01),ksi,t2,isrprt2(ksi,rtksi)):lrt(ksi,y0)
4688 i@t4:=th3"l.imp"(more"rt"(x01,y0),lrt(ksi,y0),t1,[t:more"rt"(x01,y0)]t3(t)):not(more"rt"(x01,y0))
4689 t5:=satz81e(x01,y0,t4):lessis"rt"(x01,y0)
4690 rtksi@t6:=[x:rat][t:in(x,s1)]t5(x,t):lb(s1,x01)
4691 t7:=urtrpofrt(x01,x01,moreisi2"rt"(x01,x01,refis(rat,x01))):urt(rpofrt(x01),x01)
4692 t8:=isp(cut,[x:cut]urt(x,x01),rpofrt(x01),ksi,t7,isrprt2(ksi,rtksi)):urt(ksi,x01)
4693 t9:=estii(rat,[x:rat]urt(ksi,x),x01,t8):in(x01,s1)
4694 t10:=andi(lb(s1,x01),in(x01,s1),t6,t9):min(s1,x01)
4695 -5157
4696 satz157a:=t10".5157":min(setof(rat,[x:rat]urt(ksi,x)),rtofrp(ksi,rtksi))
4697 satz157b:=somei(rat,[x:rat]min(s1".5157",x),x01".5157",t10".5157"):some"rt"([x:rat]min(setof(rat,[y:rat]urt(ksi,y)),x))
4698 ksi@[x0:rat][m:min(setof(rat,[x:rat]urt(ksi,x)),x0)]
4699 +*5157
4700 m"rp"@t11:=ande1(lb(s1,x0),in(x0,s1),m):lb(s1,x0)
4701 t12:=ande2(lb(s1,x0),in(x0,s1),m):in(x0,s1)
4702 t13:=estie(rat,[x:rat]urt(ksi,x),x0,t12):urt(ksi,x0)
4703 [y0:rat][ly:lrt(ksi,y0)]
4704 t14:=cutapp2a(ksi,y0,ly,x0,t13):less"rt"(y0,x0)
4705 t15:=lrtrpofrt(x0,y0,t14):lrt(rpofrt(x0),y0)
4706 y0@[uy:urt(ksi,y0)]
4707 t17:=estii(rat,[x:rat]urt(ksi,x),y0,uy):in(y0,s1)
4708 t18:=satz85(x0,y0,<t17><y0>t11):moreis"rt"(y0,x0)
4709 t19:=urtrpofrt(x0,y0,t18):urt(rpofrt(x0),y0)
4710 y0@t20:=cp(lrt(rpofrt(x0),y0),lrt(ksi,y0),[t:urt(ksi,y0)]t19(t)):imp(lrt(rpofrt(x0),y0),lrt(ksi,y0))
4711 -5157
4712 m@satz157c:=isi1(ksi,rpofrt(x0),[x:rat][t:lrt(ksi,x)]t15".5157"(x,t),[x:rat]t20".5157"(x)):is(ksi,rpofrt(x0))
4713 ksi@[s:some"rt"([x:rat]min(setof(rat,[y:rat]urt(ksi,y)),x))]
4714 +*5157
4715 s@[x0:rat][m:min(s1,x0)]
4716 t21:=somei(rat,[x:rat]is(ksi,rpofrt(x)),x0,satz157c(x0,m)):ratrp(ksi)
4717 -5157
4718 s@satz157d:=someapp(rat,[x:rat]min(s1".5157",x),s,ratrp(ksi),[x:rat][t:min(s1".5157",x)]t21".5157"(x,t)):ratrp(ksi)
4719 ksi@[x0:rat][lx:lrt(ksi,x0)]
4720 +5158
4721 x0@xr:=rpofrt(x0):cut
4722 lx@t1:=urtrpofrt(x0,x0,moreisi2"rt"(x0,x0,refis(rat,x0))):urt(xr,x0)
4723 t2:=andi(urt(xr,x0),lrt(ksi,x0),t1,lx):and(urt(xr,x0),lrt(ksi,x0))
4724 -5158
4725 satz158a:=somei(rat,[x:rat]and(urt(rpofrt(x0),x),lrt(ksi,x)),x0,t2".5158"):less(rpofrt(x0),ksi)
4726 x0@[ux:urt(ksi,x0)]
4727 +*5158
4728 ux@s1:=setof(rat,[x:rat]urt(ksi,x)):set(rat)
4729 [m:min(s1,x0)]
4730 t3:=symis(cut,ksi,xr,satz157c(ksi,x0,m)):is(xr,ksi)
4731 t4:=moreisi2(xr,ksi,t3):moreis(xr,ksi)
4732 ux@[n:not(min(s1,x0))]
4733 t5:=estii(rat,[x:rat]urt(ksi,x),x0,ux):in(x0,s1)
4734 t6:=th4"l.and"(lb(s1,x0),in(x0,s1),n,t5):not(lb(s1,x0))
4735 t7:=th1"l.some"(rat,[x:rat]imp(in(x,s1),lessis"rt"(x0,x)),t6):some"rt"([x:rat]not(imp(in(x,s1),lessis"rt"(x0,x))))
4736 [y0:rat][o:not(imp(in(y0,s1),lessis"rt"(x0,y0)))]
4737 t8:=th5"l.imp"(in(y0,s1),lessis"rt"(x0,y0),o):in(y0,s1)
4738 t9:=estie(rat,[x:rat]urt(ksi,x),y0,t8):urt(ksi,y0)
4739 t10:=th6"l.imp"(in(y0,s1),lessis"rt"(x0,y0),o):not(lessis"rt"(x0,y0))
4740 t11:=satz82(x0,y0,satz81k(x0,y0,t10)):less"rt"(y0,x0)
4741 t12:=lrtrpofrt(x0,y0,t11):lrt(xr,y0)
4742 t13:=andi(lrt(xr,y0),urt(ksi,y0),t12,t9):and(lrt(xr,y0),urt(ksi,y0))
4743 t14:=somei(rat,[x:rat]and(lrt(xr,x),urt(ksi,x)),y0,t13):more(xr,ksi)
4744 n@t15:=someapp(rat,[x:rat]not(imp(in(x,s1),lessis"rt"(x0,x))),t7,more(xr,ksi),[x:rat][t:not(imp(in(x,s1),lessis"rt"(x0,x)))]t14(x,t)):more(xr,ksi)
4745 t16:=moreisi1(xr,ksi,t15):moreis(xr,ksi)
4746 -5158
4747 ux@satz158b:=th1"l.imp"(min(s1".5158",x0),moreis(rpofrt(x0),ksi),[t:min(s1".5158",x0)]t4".5158"(t),[t:not(min(s1".5158",x0))]t16".5158"(t)):moreis(rpofrt(x0),ksi)
4748 x0@[l:less(rpofrt(x0),ksi)]
4749 +*5158
4750 l@t17:=satz123h(xr,ksi,l):not(moreis(xr,ksi))
4751 t18:=th3"l.imp"(urt(ksi,x0),moreis(xr,ksi),t17,[t:urt(ksi,x0)]satz158b(t)):not(urt(ksi,x0))
4752 -5158
4753 l@satz158c:=et(lrt(ksi,x0),t18".5158"):lrt(ksi,x0)
4754 x0@[m:moreis(rpofrt(x0),ksi)]
4755 +*5158
4756 m"rp"@t19:=satz123c(xr,ksi,m):not(less(xr,ksi))
4757 -5158
4758 m@satz158d:=th3"l.imp"(lrt(ksi,x0),less(rpofrt(x0),ksi),t19".5158",[t:lrt(ksi,x0)]satz158a(t)):urt(ksi,x0)
4759 ksi@[eta:cut][l:less(ksi,eta)]
4760 +5159
4761 [x0:rat]
4762 xr:=rpofrt(x0):cut
4763 [ux:urt(ksi,x0)][lx:lrt(eta,x0)][z0:rat]
4764 zr:=rpofrt(z0):cut
4765 [lz:lrt(eta,z0)][k:less"rt"(x0,z0)]
4766 t1:=satz127a(ksi,xr,zr,satz124(xr,ksi,satz158b(ksi,x0,ux)),satz154c(x0,z0,k)):less(ksi,zr)
4767 t2:=andi(less(ksi,zr),less(zr,eta),t1,satz158a(eta,z0,lz)):and(less(ksi,zr),less(zr,eta))
4768 t3:=somei(rat,[x:rat]and(less(ksi,rpofrt(x)),less(rpofrt(x),eta)),z0,t2):some"rt"([x:rat]and(less(ksi,rpofrt(x)),less(rpofrt(x),eta)))
4769 lx@t4:=cutapp3(eta,x0,lx,some"rt"([x:rat]and(less(ksi,rpofrt(x)),less(rpofrt(x),eta))),[x:rat][t:lrt(eta,x)][u:less"rt"(x0,x)]t3(x,t,u)):some"rt"([x:rat]and(less(ksi,rpofrt(x)),less(rpofrt(x),eta)))
4770 -5159
4771 satz159:=lessapp(ksi,eta,l,some"rt"([x:rat]and(less(ksi,rpofrt(x)),less(rpofrt(x),eta))),[x:rat][t:urt(ksi,x)][u:lrt(eta,x)]t4".5159"(x,t,u)):some"rt"([x:rat]and(less(ksi,rpofrt(x)),less(rpofrt(x),eta)))
4772 +*5159
4773 x0@[a:and(less(ksi,xr),less(xr,eta))]
4774 t5:=andi(ratrp(xr),and(less(ksi,xr),less(xr,eta)),ratrpi(x0),a):and3(ratrp(xr),less(ksi,xr),less(xr,eta))
4775 t6:=somei(cut,[c:cut]and3(ratrp(c),less(ksi,c),less(c,eta)),xr,t5):some([c:cut]and3(ratrp(c),less(ksi,c),less(c,eta)))
4776 -5159
4777 l@satz159a:=someapp(rat,[x:rat]and(less(ksi,rpofrt(x)),less(rpofrt(x),eta)),satz159,some([a:cut]and3(ratrp(a),less(ksi,a),less(a,eta))),[x:rat][t:and(less(ksi,rpofrt(x)),less(rpofrt(x),eta))]t6".5159"(x,t)):some([a:cut]and3(ratrp(a),less(ksi,a),less(a,eta)))
4778 [p:'prop'][p1:[x:rat][t:less(ksi,rpofrt(x))][u:less(rpofrt(x),eta)]p]
4779 +*5159
4780 p1@[y0:rat]
4781 yr:=rpofrt(y0):cut
4782 [a:and(less(ksi,yr),less(yr,eta))]
4783 t7:=ande1(less(ksi,yr),less(yr,eta),a):less(ksi,yr)
4784 t8:=ande2(less(ksi,yr),less(yr,eta),a):less(yr,eta)
4785 t9:=<t8><t7><y0>p1:p
4786 -5159
4787 p1@satz159app:=someapp(rat,[x:rat]and(less(ksi,rpofrt(x)),less(rpofrt(x),eta)),satz159,p,[x:rat][t:and(less(ksi,rpofrt(x)),less(rpofrt(x),eta))]t9".5159"(x,t)):p
4788 eta@[z0:rat][m:more(rpofrt(z0),ts(ksi,eta))]
4789 +5160
4790 zr:=rpofrt(z0):cut
4791 nm:=mn(zr,ts(ksi,eta),m):cut
4792 dn:=pl(pl(ksi,eta),1rp):cut
4793 fr:=ov(nm,dn):cut
4794 zeta:=ite(less(fr,1rp),cut,fr,1rp):cut
4795 [l:less(fr,1rp)]
4796 t1:=itet(less(fr,1rp),cut,fr,1rp,l):is(zeta,fr)
4797 t2:=lessisi2(zeta,fr,t1):lessis(zeta,fr)
4798 t3:=lessisi1(zeta,1rp,satz127a(zeta,fr,1rp,t2,l)):lessis(zeta,1rp)
4799 m@[n:not(less(fr,1rp))]
4800 t4:=itef(less(fr,1rp),cut,fr,1rp,n):is(zeta,1rp)
4801 t5:=lessisi2(zeta,1rp,t4):lessis(zeta,1rp)
4802 t6:=trlessis(zeta,1rp,fr,t5,satz124(fr,1rp,satz123f(fr,1rp,n))):lessis(zeta,fr)
4803 m@t7:=th1"l.imp"(less(fr,1rp),lessis(zeta,1rp),[t:less(fr,1rp)]t3(t),[t:not(less(fr,1rp))]t5(t)):lessis(zeta,1rp)
4804 t8:=th1"l.imp"(less(fr,1rp),lessis(zeta,fr),[t:less(fr,1rp)]t2(t),[t:not(less(fr,1rp))]t6(t)):lessis(zeta,fr)
4805 [z1:rat]
4806 zr1:=rpofrt(z1):cut
4807 [l1:less(ksi,zr1)][l2:less(zr1,pl(ksi,zeta))][z2:rat]
4808 zr2:=rpofrt(z2):cut
4809 [l3:less(eta,zr2)][l4:less(zr2,pl(eta,zeta))]
4810 t9:=isless2(ts(pl(ksi,zeta),pl(eta,zeta)),pl(ts(pl(ksi,zeta),eta),ts(pl(ksi,zeta),zeta)),ts(zr1,zr2),disttp2(pl(ksi,zeta),eta,zeta),satz147a(zr1,pl(ksi,zeta),zr2,pl(eta,zeta),l2,l4)):less(ts(zr1,zr2),pl(ts(pl(ksi,zeta),eta),ts(pl(ksi,zeta),zeta)))
4811 t10:=lessisi2(ts(pl(ksi,zeta),eta),pl(ts(ksi,eta),ts(eta,zeta)),tris(cut,ts(pl(ksi,zeta),eta),pl(ts(ksi,eta),ts(zeta,eta)),pl(ts(ksi,eta),ts(eta,zeta)),disttp1(ksi,zeta,eta),ispl2(ts(zeta,eta),ts(eta,zeta),ts(ksi,eta),comts(zeta,eta)))):lessis(ts(pl(ksi,zeta),eta),pl(ts(ksi,eta),ts(eta,zeta)))
4812 t11:=satz149a(pl(ksi,zeta),pl(ksi,1rp),zeta,zeta,satz139a(ksi,ksi,zeta,1rp,lessisi2(ksi,ksi,refis(cut,ksi)),t7),lessisi2(zeta,zeta,refis(cut,zeta))):lessis(ts(pl(ksi,zeta),zeta),ts(pl(ksi,1rp),zeta))
4813 t12:=satz139a(ts(pl(ksi,zeta),eta),pl(ts(ksi,eta),ts(eta,zeta)),ts(pl(ksi,zeta),zeta),ts(pl(ksi,1rp),zeta),t10,t11):lessis(pl(ts(pl(ksi,zeta),eta),ts(pl(ksi,zeta),zeta)),pl(pl(ts(ksi,eta),ts(eta,zeta)),ts(pl(ksi,1rp),zeta)))
4814 t13:=satz127b(ts(zr1,zr2),pl(ts(pl(ksi,zeta),eta),ts(pl(ksi,zeta),zeta)),pl(pl(ts(ksi,eta),ts(eta,zeta)),ts(pl(ksi,1rp),zeta)),t9,t12):less(ts(zr1,zr2),pl(pl(ts(ksi,eta),ts(eta,zeta)),ts(pl(ksi,1rp),zeta)))
4815 t14:=tris(cut,pl(eta,pl(ksi,1rp)),pl(pl(eta,ksi),1rp),pl(pl(ksi,eta),1rp),asspl2(eta,ksi,1rp),ispl1(pl(eta,ksi),pl(ksi,eta),1rp,compl(eta,ksi))):is(pl(eta,pl(ksi,1rp)),dn)
4816 t15:=tris(cut,pl(ts(eta,zeta),ts(pl(ksi,1rp),zeta)),ts(pl(eta,pl(ksi,1rp)),zeta),ts(dn,zeta),distpt1(eta,pl(ksi,1rp),zeta),ists1(pl(eta,pl(ksi,1rp)),dn,zeta,t14)):is(pl(ts(eta,zeta),ts(pl(ksi,1rp),zeta)),ts(dn,zeta))
4817 t16:=tris(cut,pl(pl(ts(ksi,eta),ts(eta,zeta)),ts(pl(ksi,1rp),zeta)),pl(ts(ksi,eta),pl(ts(eta,zeta),ts(pl(ksi,1rp),zeta))),pl(ts(ksi,eta),ts(dn,zeta)),asspl1(ts(ksi,eta),ts(eta,zeta),ts(pl(ksi,1rp),zeta)),ispl2(pl(ts(eta,zeta),ts(pl(ksi,1rp),zeta)),ts(dn,zeta),ts(ksi,eta),t15)):is(pl(pl(ts(ksi,eta),ts(eta,zeta)),ts(pl(ksi,1rp),zeta)),pl(ts(ksi,eta),ts(dn,zeta)))
4818 t17:=isless2(pl(pl(ts(ksi,eta),ts(eta,zeta)),ts(pl(ksi,1rp),zeta)),pl(ts(ksi,eta),ts(dn,zeta)),ts(zr1,zr2),t16,t13):less(ts(zr1,zr2),pl(ts(ksi,eta),ts(dn,zeta)))
4819 t18:=islessis12(ts(zeta,dn),ts(dn,zeta),ts(fr,dn),nm,comts(zeta,dn),satz153e(nm,dn),satz149a(zeta,fr,dn,dn,t8,lessisi2(dn,dn,refis(cut,dn)))):lessis(ts(dn,zeta),nm)
4820 t19:=satz139a(ts(ksi,eta),ts(ksi,eta),ts(dn,zeta),nm,lessisi2(ts(ksi,eta),ts(ksi,eta),refis(cut,ts(ksi,eta))),t18):lessis(pl(ts(ksi,eta),ts(dn,zeta)),pl(ts(ksi,eta),nm))
4821 t20:=satz127b(ts(zr1,zr2),pl(ts(ksi,eta),ts(dn,zeta)),pl(ts(ksi,eta),nm),t17,t19):less(ts(zr1,zr2),pl(ts(ksi,eta),nm))
4822 t21:=isless2(pl(ts(ksi,eta),nm),zr,ts(zr1,zr2),satz140c(zr,ts(ksi,eta),m),t20):less(ts(zr1,zr2),zr)
4823 t22:=satz154f(ts"rt"(z1,z2),z0,isless1(ts(zr1,zr2),rpofrt(ts"rt"(z1,z2)),zr,symis(cut,rpofrt(ts"rt"(z1,z2)),ts(zr1,zr2),satz155c(z1,z2)),t21)):less"rt"(ts"rt"(z1,z2),z0)
4824 x0:=ov"rt"(z0,z2):rat
4825 xr:=rpofrt(x0):cut
4826 y0:=z2:rat
4827 yr:=rpofrt(y0):cut
4828 t23:=satz110e(z0,z2):is"rt"(ts"rt"(x0,y0),z0)
4829 t24:=ismore1"rt"(z0,ts"rt"(x0,z2),ts"rt"(z1,z2),symis(rat,ts"rt"(x0,z2),z0,t23),satz83(ts"rt"(z1,z2),z0,t22)):more"rt"(ts"rt"(x0,z2),ts"rt"(z1,z2))
4830 t25:=satz106a(x0,z1,z2,t24):more"rt"(x0,z1)
4831 t26:=trmore(xr,zr1,ksi,satz154a(x0,z1,t25),satz122(ksi,zr1,l1)):more(xr,ksi)
4832 t27:=satz122(eta,yr,l3):more(yr,eta)
4833 z0@[u0:rat]
4834 ur:=rpofrt(u0):cut
4835 [v0:rat]
4836 vr:=rpofrt(v0):cut
4837 prop1:=and3(more(ur,ksi),more(vr,eta),is"rt"(ts"rt"(u0,v0),z0)):'prop'
4838 z0@prop2:=some"rt"([x:rat]some"rt"([y:rat]prop1(x,y))):'prop'
4839 l4@t28:=and3i(more(xr,ksi),more(yr,eta),is"rt"(ts"rt"(x0,y0),z0),t26,t27,t23):prop1(x0,y0)
4840 t29:=somei(rat,[y:rat]prop1(x0,y),y0,t28):some"rt"([y:rat]prop1(x0,y))
4841 t30:=somei(rat,[x:rat]some"rt"([y:rat]prop1(x,y)),x0,t29):prop2
4842 l2@t31:=satz159app(eta,pl(eta,zeta),satz133a(eta,zeta),prop2,[x:rat][t:less(eta,rpofrt(x))][u:less(rpofrt(x),pl(eta,zeta))]t30(x,t,u)):prop2
4843 -5160
4844 satz160:=satz159app(ksi,pl(ksi,zeta".5160"),satz133a(ksi,zeta".5160"),prop2".5160",[x:rat][t:less(ksi,rpofrt(x))][u:less(rpofrt(x),pl(ksi,zeta".5160"))]t31".5160"(x,t,u)):some"rt"([x:rat]some"rt"([y:rat]and3(more(rpofrt(x),ksi),more(rpofrt(y),eta),is"rt"(ts"rt"(x,y),z0))))
4845 [p:'prop'][p1:[x:rat][t:more(rpofrt(x),ksi)][y:rat][u:more(rpofrt(y),eta)][v:is"rt"(ts"rt"(x,y),z0)]p]
4846 +*5160
4847 p1@[x1:rat]
4848 xr1:=rpofrt(x1):cut
4849 [px:some"rt"([y:rat]prop1(x1,y))][y1:rat]
4850 yr1:=rpofrt(y1):cut
4851 [py:prop1(x1,y1)]
4852 t32:=and3e1(more(xr1,ksi),more(yr1,eta),is"rt"(ts"rt"(x1,y1),z0),py):more(xr1,ksi)
4853 t33:=and3e2(more(xr1,ksi),more(yr1,eta),is"rt"(ts"rt"(x1,y1),z0),py):more(yr1,eta)
4854 t34:=and3e3(more(xr1,ksi),more(yr1,eta),is"rt"(ts"rt"(x1,y1),z0),py):is"rt"(ts"rt"(x1,y1),z0)
4855 t35:=<t34><t33><y1><t32><x1>p1:p
4856 px@t36:=someapp(rat,[y:rat]prop1(x1,y),px,p,[y:rat][v:prop1(x1,y)]t35(y,v)):p
4857 -5160
4858 p1@satz160app:=someapp(rat,[x:rat]some"rt"([y:rat]prop1".5160"(x,y)),satz160,p,[x:rat][t:some"rt"([y:rat]prop1".5160"(x,y))]t36".5160"(x,t)):p
4859 +5161
4860 @[ksi:cut][eta:cut]
4861 min:=ite(less(ksi,eta),cut,ksi,eta):cut
4862 max:=ite(more(ksi,eta),cut,ksi,eta):cut
4863 [u0:rat]
4864 ur:=rpofrt(u0):cut
4865 [lu:lrt(min,u0)]
4866 t1:=satz158a(min,u0,lu):less(ur,min)
4867 [l:less(ksi,eta)]
4868 t2:=isless2(min,ksi,ur,itet(less(ksi,eta),cut,ksi,eta,l),t1):less(ur,ksi)
4869 t3:=trless(ur,ksi,eta,t2,l):less(ur,eta)
4870 lu@[n:not(less(ksi,eta))]
4871 t4:=isless2(min,eta,ur,itef(less(ksi,eta),cut,ksi,eta,n),t1):less(ur,eta)
4872 t5:=satz127b(ur,eta,ksi,t4,satz124(ksi,eta,satz123f(ksi,eta,n))):less(ur,ksi)
4873 lu@t6:=th1"l.imp"(less(ksi,eta),less(ur,ksi),[t:less(ksi,eta)]t2(t),[t:not(less(ksi,eta))]t5(t)):less(ur,ksi)
4874 t7:=th1"l.imp"(less(ksi,eta),less(ur,eta),[t:less(ksi,eta)]t3(t),[t:not(less(ksi,eta))]t4(t)):less(ur,eta)
4875 u0@[uu:urt(max,u0)]
4876 t8:=satz158b(max,u0,uu):moreis(ur,max)
4877 [m:more(ksi,eta)]
4878 t9:=ismoreis2(max,ksi,ur,itet(more(ksi,eta),cut,ksi,eta,m),t8):moreis(ur,ksi)
4879 t10:=trmoreis(ur,ksi,eta,t9,moreisi1(ksi,eta,m)):moreis(ur,eta)
4880 uu@[n:not(more(ksi,eta))]
4881 t11:=ismoreis2(max,eta,ur,itef(more(ksi,eta),cut,ksi,eta,n),t8):moreis(ur,eta)
4882 t12:=trmoreis(ur,eta,ksi,t11,satz125(ksi,eta,satz123e(ksi,eta,n))):moreis(ur,ksi)
4883 uu@t13:=th1"l.imp"(more(ksi,eta),moreis(ur,ksi),[t:more(ksi,eta)]t9(t),[t:not(more(ksi,eta))]t12(t)):moreis(ur,ksi)
4884 t14:=th1"l.imp"(more(ksi,eta),moreis(ur,eta),[t:more(ksi,eta)]t10(t),[t:not(more(ksi,eta))]t11(t)):moreis(ur,eta)
4885 -5161
4886 @[zeta:cut]
4887 +*5161
4888 zeta@[ksi1:cut][ksi2:cut][m:more(ksi1,ksi2)]
4889 t15:=satz147(ksi1,ksi2,ksi1,ksi2,m,m):more(ts(ksi1,ksi1),ts(ksi2,ksi2))
4890 ksi2@sq1:=ts(ksi1,ksi1):cut
4891 sq2:=ts(ksi2,ksi2):cut
4892 m@t16:=ec3e21(is(sq1,sq2),more(sq1,sq2),less(sq1,sq2),satz123b(sq1,sq2),t15):nis(sq1,sq2)
4893 ksi2@[i:is(sq1,zeta)][j:is(sq2,zeta)]
4894 t17:=tris2(cut,sq1,sq2,zeta,i,j):is(sq1,sq2)
4895 t18:=[t:more(ksi1,ksi2)]<t17>t16(t):not(more(ksi1,ksi2))
4896 t19:=[t:less(ksi1,ksi2)]<symis(cut,sq1,sq2,t17)>t16(ksi2,ksi1,satz122(ksi1,ksi2,t)):not(less(ksi1,ksi2))
4897 t20:=or3e1(is(ksi1,ksi2),more(ksi1,ksi2),less(ksi1,ksi2),satz123a(ksi1,ksi2),t18,t19):is(ksi1,ksi2)
4898 zeta@t21:=[a:cut][b:cut][t:is(ts(a,a),zeta)][u:is(ts(b,b),zeta)]t20(a,b,t,u):amone(cut,[a:cut]is(ts(a,a),zeta))
4899 sqrtset:=setof(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta)):set(rat)
4900 [x0:rat]
4901 xr:=rpofrt(x0):cut
4902 [lx:lrt(min(1rp,zeta),x0)]
4903 t22:=t6(1rp,zeta,x0,lx):less(xr,1rp)
4904 t23:=t7(1rp,zeta,x0,lx):less(xr,zeta)
4905 t24:=isless1(xr,ts(xr,1rp),zeta,satz151a(xr),t23):less(ts(xr,1rp),zeta)
4906 t25:=trless(ts(xr,xr),ts(xr,1rp),zeta,satz148c(xr,xr,xr,1rp,lessisi2(xr,xr,refis(cut,xr)),t22),t24):less(ts(xr,xr),zeta)
4907 t26:=estii(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),x0,t25):in(x0,sqrtset)
4908 x0@[ux:urt(max(1rp,zeta),x0)]
4909 t27:=t13(1rp,zeta,x0,ux):moreis(xr,1rp)
4910 t28:=t14(1rp,zeta,x0,ux):moreis(xr,zeta)
4911 t29:=ismoreis1(xr,ts(xr,1rp),zeta,satz151a(xr),t28):moreis(ts(xr,1rp),zeta)
4912 t30:=trmoreis(ts(xr,xr),ts(xr,1rp),zeta,satz149(xr,xr,xr,1rp,moreisi2(xr,xr,refis(cut,xr)),t27),t29):moreis(ts(xr,xr),zeta)
4913 t31:=satz123c(ts(xr,xr),zeta,t30):not(less(ts(xr,xr),zeta))
4914 t32:=th3"l.imp"(in(x0,sqrtset),less(ts(xr,xr),zeta),t31,[t:in(x0,sqrtset)]estie(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),x0,t)):not(in(x0,sqrtset))
4915 x0@[i:in(x0,sqrtset)][y0:rat]
4916 yr:=rpofrt(y0):cut
4917 [l:less"rt"(y0,x0)]
4918 i@t33:=estie(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),x0,i):less(ts(xr,xr),zeta)
4919 l@t34:=satz154c(y0,x0,l):less(yr,xr)
4920 t35:=trless(ts(yr,yr),ts(xr,xr),zeta,satz147a(yr,xr,yr,xr,t34,t34),t33):less(ts(yr,yr),zeta)
4921 t36:=estii(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),y0,t35):in(y0,sqrtset)
4922 i@t37:=satz122(ts(xr,xr),zeta,t33):more(zeta,ts(xr,xr))
4923 nm:=mn(zeta,ts(xr,xr),t37):cut
4924 dn:=pl(xr,pl(xr,1rp)):cut
4925 fr:=ov(nm,dn):cut
4926 [z0:rat]
4927 zr:=rpofrt(z0):cut
4928 [lz:lrt(min(1rp,fr),z0)]
4929 t38:=t6(1rp,fr,z0,lz):less(zr,1rp)
4930 t39:=t7(1rp,fr,z0,lz):less(zr,fr)
4931 t40:=satz94(x0,z0):more"rt"(pl"rt"(x0,z0),x0)
4932 t41:=tris(cut,ts(rpofrt(pl"rt"(x0,z0)),rpofrt(pl"rt"(x0,z0))),ts(pl(xr,zr),pl(xr,zr)),pl(ts(pl(xr,zr),xr),ts(pl(xr,zr),zr)),ists12(rpofrt(pl"rt"(x0,z0)),pl(xr,zr),rpofrt(pl"rt"(x0,z0)),pl(xr,zr),satz155a(x0,z0),satz155a(x0,z0)),disttp2(pl(xr,zr),xr,zr)):is(ts(rpofrt(pl"rt"(x0,z0)),rpofrt(pl"rt"(x0,z0))),pl(ts(pl(xr,zr),xr),ts(pl(xr,zr),zr)))
4933 t42:=symis(cut,ts(rpofrt(pl"rt"(x0,z0)),rpofrt(pl"rt"(x0,z0))),pl(ts(pl(xr,zr),xr),ts(pl(xr,zr),zr)),t41):is(pl(ts(pl(xr,zr),xr),ts(pl(xr,zr),zr)),ts(rpofrt(pl"rt"(x0,z0)),rpofrt(pl"rt"(x0,z0))))
4934 t43:=lessisi2(ts(pl(xr,zr),xr),pl(ts(xr,xr),ts(xr,zr)),tris(cut,ts(pl(xr,zr),xr),pl(ts(xr,xr),ts(zr,xr)),pl(ts(xr,xr),ts(xr,zr)),disttp1(xr,zr,xr),ispl2(ts(zr,xr),ts(xr,zr),ts(xr,xr),comts(zr,xr)))):lessis(ts(pl(xr,zr),xr),pl(ts(xr,xr),ts(xr,zr)))
4935 t44:=satz145c(pl(xr,zr),pl(xr,1rp),zr,satz138c(xr,xr,zr,1rp,lessisi2(xr,xr,refis(cut,xr)),t38)):less(ts(pl(xr,zr),zr),ts(pl(xr,1rp),zr))
4936 t45:=satz138c(ts(pl(xr,zr),xr),pl(ts(xr,xr),ts(xr,zr)),ts(pl(xr,zr),zr),ts(pl(xr,1rp),zr),t43,t44):less(pl(ts(pl(xr,zr),xr),ts(pl(xr,zr),zr)),pl(pl(ts(xr,xr),ts(xr,zr)),ts(pl(xr,1rp),zr)))
4937 t46:=tris(cut,pl(pl(ts(xr,xr),ts(xr,zr)),ts(pl(xr,1rp),zr)),pl(ts(xr,xr),pl(ts(xr,zr),ts(pl(xr,1rp),zr))),pl(ts(xr,xr),ts(dn,zr)),asspl1(ts(xr,xr),ts(xr,zr),ts(pl(xr,1rp),zr)),ispl2(pl(ts(xr,zr),ts(pl(xr,1rp),zr)),ts(dn,zr),ts(xr,xr),distpt1(xr,pl(xr,1rp),zr))):is(pl(pl(ts(xr,xr),ts(xr,zr)),ts(pl(xr,1rp),zr)),pl(ts(xr,xr),ts(dn,zr)))
4938 t47:=isless12(pl(ts(pl(xr,zr),xr),ts(pl(xr,zr),zr)),ts(rpofrt(pl"rt"(x0,z0)),rpofrt(pl"rt"(x0,z0))),pl(pl(ts(xr,xr),ts(xr,zr)),ts(pl(xr,1rp),zr)),pl(ts(xr,xr),ts(dn,zr)),t42,t46,t45):less(ts(rpofrt(pl"rt"(x0,z0)),rpofrt(pl"rt"(x0,z0))),pl(ts(xr,xr),ts(dn,zr)))
4939 t48:=isless2(ts(dn,fr),nm,ts(dn,zr),satz153c(nm,dn),satz148c(dn,dn,zr,fr,lessisi2(dn,dn,refis(cut,dn)),t39)):less(ts(dn,zr),nm)
4940 t49:=satz138c(ts(xr,xr),ts(xr,xr),ts(dn,zr),nm,lessisi2(ts(xr,xr),ts(xr,xr),refis(cut,ts(xr,xr))),t48):less(pl(ts(xr,xr),ts(dn,zr)),pl(ts(xr,xr),nm))
4941 t50:=isless2(pl(ts(xr,xr),nm),zeta,pl(ts(xr,xr),ts(dn,zr)),satz140c(zeta,ts(xr,xr),t37),t49):less(pl(ts(xr,xr),ts(dn,zr)),zeta)
4942 t51:=trless(ts(rpofrt(pl"rt"(x0,z0)),rpofrt(pl"rt"(x0,z0))),pl(ts(xr,xr),ts(dn,zr)),zeta,t47,t50):less(ts(rpofrt(pl"rt"(x0,z0)),rpofrt(pl"rt"(x0,z0))),zeta)
4943 t52:=estii(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),pl"rt"(x0,z0),t51):in(pl"rt"(x0,z0),sqrtset)
4944 t53:=andi(in(pl"rt"(x0,z0),sqrtset),more"rt"(pl"rt"(x0,z0),x0),t52,t40):and(in(pl"rt"(x0,z0),sqrtset),more"rt"(pl"rt"(x0,z0),x0))
4945 t54:=somei(rat,[y:rat]and(in(y,sqrtset),more"rt"(y,x0)),pl"rt"(x0,z0),t53):some"rt"([y:rat]and(in(y,sqrtset),more"rt"(y,x0)))
4946 i@t55:=cutapp1a(min(1rp,fr),some"rt"([y:rat]and(in(y,sqrtset),more"rt"(y,x0))),[x:rat][t:lrt(min(1rp,fr),x)]t54(x,t)):some"rt"([y:rat]and(in(y,sqrtset),more"rt"(y,x0)))
4947 x0@[lx:lrt(min(1rp,zeta),x0)][y0:rat][uy:urt(max(1rp,zeta),y0)]
4948 t56:=cut2(sqrtset,x0,t26(lx),y0,t32(y0,uy),[x:rat][t:in(x,sqrtset)][y:rat][u:less"rt"(y,x)]t36(x,t,y,u),[x:rat][t:in(x,sqrtset)]t55(x,t)):cutprop(sqrtset)
4949 lx@t57:=cutapp1b(max(1rp,zeta),cutprop(sqrtset),[y:rat][t:urt(max(1rp,zeta),y)]t56(y,t)):cutprop(sqrtset)
4950 zeta@t58:=cutapp1a(min(1rp,zeta),cutprop(sqrtset),[x:rat][t:lrt(min(1rp,zeta),x)]t57(x,t)):cutprop(sqrtset)
4951 rtc:=cutof(sqrtset,t58):cut
4952 @[x0:rat][y0:rat][l:lessis"rt"(x0,y0)]
4953 t59:=th9"l.or"(less"rt"(x0,y0),is"rt"(x0,y0),less(rpofrt(x0),rpofrt(y0)),is(rpofrt(x0),rpofrt(y0)),l,[t:less"rt"(x0,y0)]satz154c(x0,y0,t),[t:is"rt"(x0,y0)]satz154b(x0,y0,t)):lessis(rpofrt(x0),rpofrt(y0))
4954 y0@[m:moreis"rt"(x0,y0)]
4955 t60:=satz125(rpofrt(y0),rpofrt(x0),t59(y0,x0,satz84(x0,y0,m))):moreis(rpofrt(x0),rpofrt(y0))
4956 zeta@[m:more(ts(rtc,rtc),zeta)]
4957 t61:=satz121(ts(rtc,rtc),zeta,m):less(zeta,ts(rtc,rtc))
4958 [z1:rat]
4959 zr1:=rpofrt(z1):cut
4960 [l1:less(zeta,zr1)][l2:less(zr1,ts(rtc,rtc))]
4961 t62:=satz158c(ts(rtc,rtc),z1,l2):lrt(ts(rtc,rtc),z1)
4962 [x1:rat]
4963 xr1:=rpofrt(x1):cut
4964 [lx1:lrt(rtc,x1)][x2:rat]
4965 xr2:=rpofrt(x2):cut
4966 [lx2:lrt(rtc,x2)][i:is"rt"(z1,ts"rt"(x1,x2))]
4967 xm:=ite(more"rt"(x1,x2),rat,x1,x2):rat
4968 xrm:=rpofrt(xm):cut
4969 [o:more"rt"(x1,x2)]
4970 t63:=symis(rat,xm,x1,itet(more"rt"(x1,x2),rat,x1,x2,o)):is"rt"(x1,xm)
4971 t64:=isp(rat,[x:rat]lrt(rtc,x),x1,xm,lx1,t63):lrt(rtc,xm)
4972 t65:=lessisi2"rt"(x1,xm,t63):lessis"rt"(x1,xm)
4973 t66:=lessisi1"rt"(x2,xm,satz87b(x2,x1,xm,satz82(x1,x2,o),t65)):lessis"rt"(x2,xm)
4974 i@[n:not(more"rt"(x1,x2))]
4975 t67:=symis(rat,xm,x2,itef(more"rt"(x1,x2),rat,x1,x2,n)):is"rt"(x2,xm)
4976 t68:=isp(rat,[x:rat]lrt(rtc,x),x2,xm,lx2,t67):lrt(rtc,xm)
4977 t69:=lessisi2"rt"(x2,xm,t67):lessis"rt"(x2,xm)
4978 t70:=satz88(x1,x2,xm,satz81e(x1,x2,n),t69):lessis"rt"(x1,xm)
4979 i@t71:=th1"l.imp"(more"rt"(x1,x2),lrt(rtc,xm),[t:more"rt"(x1,x2)]t64(t),[t:not(more"rt"(x1,x2))]t68(t)):lrt(rtc,xm)
4980 t72:=th1"l.imp"(more"rt"(x1,x2),lessis"rt"(x1,xm),[t:more"rt"(x1,x2)]t65(t),[t:not(more"rt"(x1,x2))]t70(t)):lessis"rt"(x1,xm)
4981 t73:=th1"l.imp"(more"rt"(x1,x2),lessis"rt"(x2,xm),[t:more"rt"(x1,x2)]t66(t),[t:not(more"rt"(x1,x2))]t69(t)):lessis"rt"(x2,xm)
4982 t74:=ini(sqrtset,t58,xm,t71):in(xm,sqrtset)
4983 t75:=t59(x1,xm,t72):lessis(xr1,xrm)
4984 t76:=t59(x2,xm,t73):lessis(xr2,xrm)
4985 t77:=tris(cut,zr1,rpofrt(ts"rt"(x1,x2)),ts(xr1,xr2),satz154b(z1,ts"rt"(x1,x2),i),satz155c(x1,x2)):is(zr1,ts(xr1,xr2))
4986 t78:=islessis1(ts(xr1,xr2),zr1,ts(xrm,xrm),symis(cut,zr1,ts(xr1,xr2),t77),satz149a(xr1,xrm,xr2,xrm,t75,t76)):lessis(zr1,ts(xrm,xrm))
4987 t79:=estie(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),xm,t74):less(ts(xrm,xrm),zeta)
4988 t80:=satz127a(zr1,ts(xrm,xrm),zeta,t78,t79):less(zr1,zeta)
4989 t81:=<t80>ec3e23(is(zr1,zeta),more(zr1,zeta),less(zr1,zeta),satz123b(zr1,zeta),satz122(zeta,zr1,l1)):con
4990 t82:=tsapp(rtc,rtc,z1,t62,con,[x:rat][t:lrt(rtc,x)][y:rat][u:lrt(rtc,y)][v:is"rt"(z1,ts"rt"(x,y))]t81(x,t,y,u,v)):con
4991 l2@t82a:=tsapp(rtc,rtc,z1,t62,con,[x:rat][t:lrt(rtc,x)][y:rat][u:lrt(rtc,y)][v:is"rt"(z1,ts"rt"(x,y))]t82(x,t,y,u,v)):con
4992 m@t83:=satz159app(zeta,ts(rtc,rtc),t61,con,[x:rat][t:less(zeta,rpofrt(x))][u:less(rpofrt(x),ts(rtc,rtc))]t82a(x,t,u)):con
4993 zeta@[l:less(ts(rtc,rtc),zeta)][z2:rat]
4994 zr2:=rpofrt(z2):cut
4995 [l3:less(ts(rtc,rtc),zr2)][l4:less(zr2,zeta)]
4996 t84:=satz122(ts(rtc,rtc),zr2,l3):more(zr2,ts(rtc,rtc))
4997 [y1:rat]
4998 yr1:=rpofrt(y1):cut
4999 [m1:more(yr1,rtc)][y2:rat]
5000 yr2:=rpofrt(y2):cut
5001 [m2:more(yr2,rtc)][i:is"rt"(ts"rt"(y1,y2),z2)]
5002 ym:=ite(less"rt"(y1,y2),rat,y1,y2):rat
5003 yrm:=rpofrt(ym):cut
5004 [k:less"rt"(y1,y2)]
5005 t85:=symis(rat,ym,y1,itet(less"rt"(y1,y2),rat,y1,y2,k)):is"rt"(y1,ym)
5006 t86:=satz154b(y1,ym,t85):is(yr1,yrm)
5007 t87:=ismore1(yr1,yrm,rtc,t86,m1):more(yrm,rtc)
5008 t88:=moreisi2(yr1,yrm,t86):moreis(yr1,yrm)
5009 t89:=moreisi1(yr2,yrm,satz127d(yr2,yr1,yrm,satz122(yr1,yr2,satz154c(y1,y2,k)),t88)):moreis(yr2,yrm)
5010 i@[n:not(less"rt"(y1,y2))]
5011 t90:=symis(rat,ym,y2,itef(less"rt"(y1,y2),rat,y1,y2,n)):is"rt"(y2,ym)
5012 t91:=satz154b(y2,ym,t90):is(yr2,yrm)
5013 t92:=ismore1(yr2,yrm,rtc,t91,m2):more(yrm,rtc)
5014 t93:=moreisi2(yr2,yrm,t91):moreis(yr2,yrm)
5015 t94:=trmoreis(yr1,yr2,yrm,t60(y1,y2,satz81f(y1,y2,n)),t93):moreis(yr1,yrm)
5016 i@t95:=th1"l.imp"(less"rt"(y1,y2),more(yrm,rtc),[t:less"rt"(y1,y2)]t87(t),[t:not(less"rt"(y1,y2))]t92(t)):more(yrm,rtc)
5017 t96:=th1"l.imp"(less"rt"(y1,y2),moreis(yr1,yrm),[t:less"rt"(y1,y2)]t88(t),[t:not(less"rt"(y1,y2))]t94(t)):moreis(yr1,yrm)
5018 t97:=th1"l.imp"(less"rt"(y1,y2),moreis(yr2,yrm),[t:less"rt"(y1,y2)]t89(t),[t:not(less"rt"(y1,y2))]t93(t)):moreis(yr2,yrm)
5019 t98:=satz158d(rtc,ym,moreisi1(yrm,rtc,t95)):urt(rtc,ym)
5020 t99:=th3"l.imp"(in(ym,sqrtset),lrt(rtc,ym),t98,[t:in(ym,sqrtset)]ine(sqrtset,t58,ym,t)):not(in(ym,sqrtset))
5021 t100:=th3"l.imp"(less(ts(yrm,yrm),zeta),in(ym,sqrtset),t99,[t:less(ts(yrm,yrm),zeta)]estii(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),ym,t)):not(less(ts(yrm,yrm),zeta))
5022 t101:=satz123f(ts(yrm,yrm),zeta,t100):moreis(ts(yrm,yrm),zeta)
5023 t101a:=satz149(yr1,yrm,yr2,yrm,t96,t97):moreis(ts(yr1,yr2),ts(yrm,yrm))
5024 t102:=ismoreis1(ts(yr1,yr2),zr2,ts(yrm,yrm),tris(cut,ts(yr1,yr2),rpofrt(ts"rt"(y1,y2)),zr2,symis(cut,rpofrt(ts"rt"(y1,y2)),ts(yr1,yr2),satz155c(y1,y2)),satz154b(ts"rt"(y1,y2),z2,i)),t101a):moreis(zr2,ts(yrm,yrm))
5025 t103:=trmoreis(zr2,ts(yrm,yrm),zeta,t102,t101):moreis(zr2,zeta)
5026 t104:=<l4>satz123c(zr2,zeta,t103):con
5027 l4@t105:=satz160app(rtc,rtc,z2,t84,con,[x:rat][t:more(rpofrt(x),rtc)][y:rat][u:more(rpofrt(y),rtc)][v:is"rt"(ts"rt"(x,y),z2)]t104(x,t,y,u,v)):con
5028 l@t106:=satz159app(ts(rtc,rtc),zeta,l,con,[x:rat][t:less(ts(rtc,rtc),rpofrt(x))][u:less(rpofrt(x),zeta)]t105(x,t,u)):con
5029 zeta@t107:=or3e1(is(ts(rtc,rtc),zeta),more(ts(rtc,rtc),zeta),less(ts(rtc,rtc),zeta),satz123a(ts(rtc,rtc),zeta),[t:more(ts(rtc,rtc),zeta)]t83(t),[t:less(ts(rtc,rtc),zeta)]t106(t)):is(ts(rtc,rtc),zeta)
5030 t108:=somei(cut,[a:cut]is(ts(a,a),zeta),rtc,t107):some([a:cut]is(ts(a,a),zeta))
5031 -5161
5032 zeta@satz161:=onei(cut,[a:cut]is(ts(a,a),zeta),t21".5161",t108".5161"):one([a:cut]is(ts(a,a),zeta))
5033 @[ksi:cut]
5034 irratrp:=not(ratrp(ksi)):'prop'
5035 -rp
5036 -rt
5037 +5162
5038 @[v:nat]
5039 t1:=tris(nat,pl(v,v),pl(ts(1,v),v),ts(<1>suc,v),ispl1(v,ts(1,v),v,satz28g(v)),satz28h(1,v)):is(pl(v,v),ts(<1>suc,v))
5040 t2:=isless2(pl(v,v),ts(<1>suc,v),v,t1,satz18a(v,v)):less(v,ts(<1>suc,v))
5041 [w:nat][l:less(ts(v,v),ts(w,w))]
5042 t3:=satz10j(v,w,th3"l.imp"(moreis(v,w),moreis(ts(v,v),ts(w,w)),satz10h(ts(v,v),ts(w,w),l),[t:moreis(v,w)]satz36(v,w,v,w,t,t))):less(v,w)
5043 w@t4:=tris(nat,ts(pl(v,w),v),pl(ts(v,v),ts(w,v)),pl(ts(v,v),ts(v,w)),disttp1(v,w,v),ispl2(ts(w,v),ts(v,w),ts(v,v),comts(w,v))):is(ts(pl(v,w),v),pl(ts(v,v),ts(v,w)))
5044 t5:=tr3is(nat,ts(pl(v,w),pl(v,w)),pl(ts(pl(v,w),v),ts(pl(v,w),w)),pl(pl(ts(v,v),ts(v,w)),pl(ts(v,w),ts(w,w))),pl(pl(pl(ts(v,v),ts(v,w)),ts(v,w)),ts(w,w)),disttp2(pl(v,w),v,w),ispl12(ts(pl(v,w),v),pl(ts(v,v),ts(v,w)),ts(pl(v,w),w),pl(ts(v,w),ts(w,w)),t4,disttp1(v,w,w)),asspl2(pl(ts(v,v),ts(v,w)),ts(v,w),ts(w,w))):is(ts(pl(v,w),pl(v,w)),pl(pl(pl(ts(v,v),ts(v,w)),ts(v,w)),ts(w,w)))
5045 t6:=tris(nat,pl(pl(ts(v,v),ts(v,w)),ts(v,w)),pl(ts(v,v),pl(ts(v,w),ts(v,w))),pl(ts(v,v),ts(<1>suc,ts(v,w))),asspl1(ts(v,v),ts(v,w),ts(v,w)),ispl2(pl(ts(v,w),ts(v,w)),ts(<1>suc,ts(v,w)),ts(v,v),t1(ts(v,w)))):is(pl(pl(ts(v,v),ts(v,w)),ts(v,w)),pl(ts(v,v),ts(<1>suc,ts(v,w))))
5046 nun:=tris(nat,ts(pl(v,w),pl(v,w)),pl(pl(pl(ts(v,v),ts(v,w)),ts(v,w)),ts(w,w)),pl(pl(ts(v,v),ts(<1>suc,ts(v,w))),ts(w,w)),t5,ispl1(pl(pl(ts(v,v),ts(v,w)),ts(v,w)),pl(ts(v,v),ts(<1>suc,ts(v,w))),ts(w,w),t6)):is(ts(pl(v,w),pl(v,w)),pl(pl(ts(v,v),ts(<1>suc,ts(v,w))),ts(w,w)))
5047 nun1:=symis(nat,ts(pl(v,w),pl(v,w)),pl(pl(ts(v,v),ts(<1>suc,ts(v,w))),ts(w,w)),nun):is(pl(pl(ts(v,v),ts(<1>suc,ts(v,w))),ts(w,w)),ts(pl(v,w),pl(v,w)))
5048 prop1:=eq(tf(fr(w,v),fr(w,v)),fr(<1>suc,1)):'prop'
5049 v@prop2:=some([t:nat]prop1(t)):'prop'
5050 @prop3:=some([u:nat]prop2(u)):'prop'
5051 [p:prop3]
5052 y:=ind(nat,[t:nat]min([u:nat]prop2(u),t),satz27a([u:nat]prop2(u),p)):nat
5053 t7:=oneax(nat,[t:nat]min([u:nat]prop2(u),t),satz27a([u:nat]prop2(u),p)):min([u:nat]prop2(u),y)
5054 t8:=ande1(lb([u:nat]prop2(u),y),prop2(y),t7):lb([u:nat]prop2(u),y)
5055 t9:=ande2(lb([u:nat]prop2(u),y),prop2(y),t7):prop2(y)
5056 [x:nat][q:prop1(y,x)]
5057 t10:=treq1(fr(<1>suc,1),fr(ts(x,x),ts(y,y)),tf(fr(x,y),fr(x,y)),q,tfeq12a(x,y,x,y)):eq(fr(<1>suc,1),fr(ts(x,x),ts(y,y)))
5058 t11:=tr4is(nat,ts(<1>suc,ts(y,y)),ts(num(fr(<1>suc,1)),den(fr(ts(x,x),ts(y,y)))),ts(num(fr(ts(x,x),ts(y,y))),den(fr(<1>suc,1))),ts(ts(x,x),1),ts(x,x),12isnd(<1>suc,1,ts(x,x),ts(y,y)),t10,ndis12(ts(x,x),ts(y,y),<1>suc,1),satz28a(ts(x,x))):is(ts(<1>suc,ts(y,y)),ts(x,x))
5059 t12:=isless2(ts(<1>suc,ts(y,y)),ts(x,x),ts(y,y),t11,t2(ts(y,y))):less(ts(y,y),ts(x,x))
5060 t13:=isless1(ts(ts(<1>suc,y),y),ts(x,x),ts(ts(<1>suc,y),ts(<1>suc,y)),tris(nat,ts(ts(<1>suc,y),y),ts(<1>suc,ts(y,y)),ts(x,x),assts1(<1>suc,y,y),t11),satz35c(ts(<1>suc,y),ts(<1>suc,y),y,ts(<1>suc,y),lessisi2(ts(<1>suc,y),ts(<1>suc,y),refis(nat,ts(<1>suc,y))),t2(y))):less(ts(x,x),ts(ts(<1>suc,y),ts(<1>suc,y)))
5061 t14:=t3(y,x,t12):less(y,x)
5062 t15:=t3(x,ts(<1>suc,y),t13):less(x,ts(<1>suc,y))
5063 [u:nat][i:is(x,pl(y,u))]
5064 t16:=isless12(x,pl(y,u),ts(<1>suc,y),pl(y,y),i,symis(nat,pl(y,y),ts(<1>suc,y),t1(y)),t15):less(pl(y,u),pl(y,y))
5065 t17:=satz20f(u,y,y,t16):less(u,y)
5066 [t:nat][j:is(y,pl(u,t))]
5067 t18:=symis(nat,y,pl(u,t),j):is(pl(u,t),y)
5068 t19:=tris(nat,ts(x,x),ts(pl(y,u),pl(y,u)),pl(pl(ts(y,y),ts(<1>suc,ts(y,u))),ts(u,u)),ists12(x,pl(y,u),x,pl(y,u),i,i),nun(y,u)):is(ts(x,x),pl(pl(ts(y,y),ts(<1>suc,ts(y,u))),ts(u,u)))
5069 t20:=tris(nat,pl(ts(x,x),ts(t,t)),pl(pl(pl(ts(y,y),ts(<1>suc,ts(y,u))),ts(u,u)),ts(t,t)),pl(pl(ts(y,y),ts(<1>suc,ts(y,u))),pl(ts(u,u),ts(t,t))),ispl1(ts(x,x),pl(pl(ts(y,y),ts(<1>suc,ts(y,u))),ts(u,u)),ts(t,t),t19),asspl1(pl(ts(y,y),ts(<1>suc,ts(y,u))),ts(u,u),ts(t,t))):is(pl(ts(x,x),ts(t,t)),pl(pl(ts(y,y),ts(<1>suc,ts(y,u))),pl(ts(u,u),ts(t,t))))
5070 t21:=tr3is(nat,ts(y,u),ts(u,y),ts(u,pl(u,t)),pl(ts(u,u),ts(u,t)),comts(y,u),ists2(y,pl(u,t),u,j),disttp2(u,u,t)):is(ts(y,u),pl(ts(u,u),ts(u,t)))
5071 t22:=tris(nat,ts(<1>suc,ts(y,u)),ts(<1>suc,pl(ts(u,u),ts(u,t))),pl(ts(<1>suc,ts(u,u)),ts(<1>suc,ts(u,t))),ists2(ts(y,u),pl(ts(u,u),ts(u,t)),<1>suc,t21),disttp2(<1>suc,ts(u,u),ts(u,t))):is(ts(<1>suc,ts(y,u)),pl(ts(<1>suc,ts(u,u)),ts(<1>suc,ts(u,t))))
5072 t23:=tris(nat,pl(ts(y,y),ts(<1>suc,ts(y,u))),pl(ts(y,y),pl(ts(<1>suc,ts(u,u)),ts(<1>suc,ts(u,t)))),pl(pl(ts(y,y),ts(<1>suc,ts(u,u))),ts(<1>suc,ts(u,t))),ispl2(ts(<1>suc,ts(y,u)),pl(ts(<1>suc,ts(u,u)),ts(<1>suc,ts(u,t))),ts(y,y),t22),asspl2(ts(y,y),ts(<1>suc,ts(u,u)),ts(<1>suc,ts(u,t)))):is(pl(ts(y,y),ts(<1>suc,ts(y,u))),pl(pl(ts(y,y),ts(<1>suc,ts(u,u))),ts(<1>suc,ts(u,t))))
5073 t24:=tr3is(nat,pl(ts(x,x),ts(t,t)),pl(pl(ts(y,y),ts(<1>suc,ts(y,u))),pl(ts(u,u),ts(t,t))),pl(pl(pl(ts(y,y),ts(<1>suc,ts(u,u))),ts(<1>suc,ts(u,t))),pl(ts(u,u),ts(t,t))),pl(pl(ts(y,y),ts(<1>suc,ts(u,u))),pl(ts(<1>suc,ts(u,t)),pl(ts(u,u),ts(t,t)))),t20,ispl1(pl(ts(y,y),ts(<1>suc,ts(y,u))),pl(pl(ts(y,y),ts(<1>suc,ts(u,u))),ts(<1>suc,ts(u,t))),pl(ts(u,u),ts(t,t)),t23),asspl1(pl(ts(y,y),ts(<1>suc,ts(u,u))),ts(<1>suc,ts(u,t)),pl(ts(u,u),ts(t,t)))):is(pl(ts(x,x),ts(t,t)),pl(pl(ts(y,y),ts(<1>suc,ts(u,u))),pl(ts(<1>suc,ts(u,t)),pl(ts(u,u),ts(t,t)))))
5074 t25:=tr4is(nat,pl(ts(<1>suc,ts(u,t)),pl(ts(u,u),ts(t,t))),pl(pl(ts(<1>suc,ts(u,t)),ts(u,u)),ts(t,t)),pl(pl(ts(u,u),ts(<1>suc,ts(u,t))),ts(t,t)),ts(pl(u,t),pl(u,t)),ts(y,y),asspl2(ts(<1>suc,ts(u,t)),ts(u,u),ts(t,t)),ispl1(pl(ts(<1>suc,ts(u,t)),ts(u,u)),pl(ts(u,u),ts(<1>suc,ts(u,t))),ts(t,t),compl(ts(<1>suc,ts(u,t)),ts(u,u))),nun1(u,t),ists12(pl(u,t),y,pl(u,t),y,t18,t18)):is(pl(ts(<1>suc,ts(u,t)),pl(ts(u,u),ts(t,t))),ts(y,y))
5075 t26:=tr4is(nat,pl(ts(x,x),ts(t,t)),pl(pl(ts(y,y),ts(<1>suc,ts(u,u))),pl(ts(<1>suc,ts(u,t)),pl(ts(u,u),ts(t,t)))),pl(pl(ts(y,y),ts(<1>suc,ts(u,u))),ts(y,y)),pl(ts(y,y),pl(ts(y,y),ts(<1>suc,ts(u,u)))),pl(pl(ts(y,y),ts(y,y)),ts(<1>suc,ts(u,u))),t24,ispl2(pl(ts(<1>suc,ts(u,t)),pl(ts(u,u),ts(t,t))),ts(y,y),pl(ts(y,y),ts(<1>suc,ts(u,u))),t25),compl(pl(ts(y,y),ts(<1>suc,ts(u,u))),ts(y,y)),asspl2(ts(y,y),ts(y,y),ts(<1>suc,ts(u,u)))):is(pl(ts(x,x),ts(t,t)),pl(pl(ts(y,y),ts(y,y)),ts(<1>suc,ts(u,u))))
5076 t27:=tris(nat,pl(ts(y,y),ts(y,y)),ts(<1>suc,ts(y,y)),ts(x,x),t1(ts(y,y)),t11):is(pl(ts(y,y),ts(y,y)),ts(x,x))
5077 t28:=tris(nat,pl(ts(x,x),ts(t,t)),pl(pl(ts(y,y),ts(y,y)),ts(<1>suc,ts(u,u))),pl(ts(x,x),ts(<1>suc,ts(u,u))),t26,ispl1(pl(ts(y,y),ts(y,y)),ts(x,x),ts(<1>suc,ts(u,u)),t27)):is(pl(ts(x,x),ts(t,t)),pl(ts(x,x),ts(<1>suc,ts(u,u))))
5078 t29:=satz20e(ts(t,t),ts(<1>suc,ts(u,u)),ts(x,x),t28):is(ts(t,t),ts(<1>suc,ts(u,u)))
5079 t30:=tr4is(nat,ts(num(fr(<1>suc,1)),den(fr(ts(t,t),ts(u,u)))),ts(<1>suc,ts(u,u)),ts(t,t),ts(ts(t,t),1),ts(num(fr(ts(t,t),ts(u,u))),den(fr(<1>suc,1))),ndis12(<1>suc,1,ts(t,t),ts(u,u)),symis(nat,ts(t,t),ts(<1>suc,ts(u,u)),t29),satz28e(ts(t,t)),12isnd(ts(t,t),ts(u,u),<1>suc,1)):eq(fr(<1>suc,1),fr(ts(t,t),ts(u,u)))
5080 t31:=treq2(tf(fr(t,u),fr(t,u)),fr(<1>suc,1),fr(ts(t,t),ts(u,u)),tfeq12a(t,u,t,u),t30):prop1(u,t)
5081 t32:=somei(nat,[v:nat]prop1(u,v),t,t31):prop2(u)
5082 t33:=<t32><u>t8:lessis(y,u)
5083 t34:=<t33>satz10g(y,u,satz12(u,y,t17)):con
5084 i@t35:=someapp(nat,[v:nat]diffprop(y,u,v),t17,con,[v:nat][w:diffprop(y,u,v)]t34(v,w)):con
5085 q@t36:=someapp(nat,[v:nat]diffprop(x,y,v),t14,con,[v:nat][w:diffprop(x,y,v)]t35(v,w)):con
5086 p@t37:=someapp(nat,[v:nat]prop1(y,v),t9,con,[v:nat][w:prop1(y,v)]t36(v,w)):con
5087 -5162
5088 +*rt
5089 +5162
5090 @[x0:rat][i:is(ts(x0,x0),rtofn(<1>suc))][x:frac][xix0:inf(x,class(x0))]
5091 t38:=ise(ts(x0,x0),rtofn(<1>suc),tf(x,x),fr(<1>suc,1),tict(x0,x0,x,x,xix0,xix0),inclass(fr(<1>suc,1)),i):eq"n"(tf(x,x),fr(<1>suc,1))
5092 t39:=refeq1(fr(num(x),den(x)),x,fris(x)):eq"n"(fr(num(x),den(x)),x)
5093 t40:=eqtf12(fr(num(x),den(x)),x,fr(num(x),den(x)),x,t39,t39):eq"n"(tf(fr(num(x),den(x)),fr(num(x),den(x))),tf(x,x))
5094 t41:=treq"n"(tf(fr(num(x),den(x)),fr(num(x),den(x))),tf(x,x),fr(<1>suc,1),t40,t38):prop1"n.5162"(den(x),num(x))
5095 t42:=somei(nat,[t:nat]prop1"n.5162"(den(x),t),num(x),t41):prop2"n.5162"(den(x))
5096 t43:=somei(nat,[t:nat]prop2"n.5162"(t),den(x),t42):prop3"n.5162"
5097 t44:=t37"n.5162"(t43):con
5098 i@t45:=ratapp1(x0,con,[x:frac][t:inf(x,class(x0))]t44(x,t)):con
5099 -5162
5100 +*rp
5101 +5162
5102 @ksi:=ind(cut,[a:cut]is(ts(a,a),rpofnt(<1>suc)),satz161(rpofnt(<1>suc))):cut
5103 t46:=oneax(cut,[a:cut]is(ts(a,a),rpofnt(<1>suc)),satz161(rpofnt(<1>suc))):is(ts(ksi,ksi),rpofnt(<1>suc))
5104 [r:ratrp(ksi)]
5105 x0:=rtofrp(ksi,r):rat
5106 t47:=tr3is(cut,rpofrt(ts"rt"(x0,x0)),ts(rpofrt(x0),rpofrt(x0)),ts(ksi,ksi),rpofnt(<1>suc),satz155c(x0,x0),ists12(rpofrt(x0),ksi,rpofrt(x0),ksi,isrprt2(ksi,r),isrprt2(ksi,r)),t46):is(rpofrt(ts"rt"(x0,x0)),rpofnt(<1>suc))
5107 t48:=isrtirp(ts"rt"(x0,x0),rtofn(<1>suc),t47):is"rt"(ts"rt"(x0,x0),rtofn(<1>suc))
5108 t49:=t45"rt.5162"(x0,t48):con
5109 -5162
5110 @satz162:=somei(cut,[a:cut]irratrp(a),ksi".5162",[t:ratrp(ksi".5162")]t49".5162"(t)):some([a:cut]irratrp(a))
5111 [zeta:cut]
5112 sqrt:=ind(cut,[a:cut]is(ts(a,a),zeta),satz161(zeta)):cut
5113 thsqrt1:=oneax(cut,[a:cut]is(ts(a,a),zeta),satz161(zeta)):is(ts(sqrt(zeta),sqrt(zeta)),zeta)
5114 [ksi:cut][i:is(ts(ksi,ksi),zeta)]
5115 thsqrt2:=t20".5161"(zeta,ksi,sqrt,i,thsqrt1):is(ksi,sqrt)
5116 @[ksi:cut][eta:cut][i:is(ksi,eta)]
5117 issqrt:=isf(cut,cut,[t:cut]sqrt(t),ksi,eta,i):is(sqrt(ksi),sqrt(eta))
5118 @[ksi:cut][nx:natrp(ksi)][eta:cut][ny:natrp(eta)]
5119 +iiia
5120 x:=ntofrp(ksi,nx):nat
5121 y:=ntofrp(eta,ny):nat
5122 t1:=isrpnt1(ksi,nx):is(ksi,rpofnt(x))
5123 t2:=isrpnt1(eta,ny):is(eta,rpofnt(y))
5124 t3:=ispl12(ksi,rpofnt(x),eta,rpofnt(y),t1,t2):is(pl(ksi,eta),pl(rpofnt(x),rpofnt(y)))
5125 x0:=rtofn(x):rat
5126 y0:=rtofn(y):rat
5127 t4:=natrti(x):natrt(x0)
5128 t5:=natrti(y):natrt(y0)
5129 t6:=symis(cut,rpofrt(pl"rt"(x0,y0)),pl(rpofnt(x),rpofnt(y)),satz155a(x0,y0)):is(pl(rpofnt(x),rpofnt(y)),rpofrt(pl"rt"(x0,y0)))
5130 t7:=satz112d(x0,t4,y0,t5):natrt(pl"rt"(x0,y0))
5131 xpy:=nofrt(pl"rt"(x0,y0),t7):nat
5132 t8:=isrtn1(pl"rt"(x0,y0),t7):is"rt"(pl"rt"(x0,y0),rtofn(xpy))
5133 t9:=isrterp(pl"rt"(x0,y0),rtofn(xpy),t8):is(rpofrt(pl"rt"(x0,y0)),rpofnt(xpy))
5134 t10:=tr3is(cut,pl(ksi,eta),pl(rpofnt(x),rpofnt(y)),rpofrt(pl"rt"(x0,y0)),rpofnt(xpy),t3,t6,t9):is(pl(ksi,eta),rpofnt(xpy))
5135 -iiia
5136 natpl:=somei(nat,[t:nat]is(pl(ksi,eta),rpofnt(t)),xpy".iiia",t10".iiia"):natrp(pl(ksi,eta))
5137 +*iiia
5138 ny@t11:=ists12(ksi,rpofnt(x),eta,rpofnt(y),t1,t2):is(ts(ksi,eta),ts(rpofnt(x),rpofnt(y)))
5139 t12:=symis(cut,rpofrt(ts"rt"(x0,y0)),ts(rpofnt(x),rpofnt(y)),satz155c(x0,y0)):is(ts(rpofnt(x),rpofnt(y)),rpofrt(ts"rt"(x0,y0)))
5140 t13:=satz112f(x0,t4,y0,t5):natrt(ts"rt"(x0,y0))
5141 xty:=nofrt(ts"rt"(x0,y0),t13):nat
5142 t14:=isrtn1(ts"rt"(x0,y0),t13):is"rt"(ts"rt"(x0,y0),rtofn(xty))
5143 t15:=isrterp(ts"rt"(x0,y0),rtofn(xty),t14):is(rpofrt(ts"rt"(x0,y0)),rpofnt(xty))
5144 t16:=tr3is(cut,ts(ksi,eta),ts(rpofnt(x),rpofnt(y)),rpofrt(ts"rt"(x0,y0)),rpofnt(xty),t11,t12,t15):is(ts(ksi,eta),rpofnt(xty))
5145 -iiia
5146 ny@natts:=somei(nat,[t:nat]is(ts(ksi,eta),rpofnt(t)),xty".iiia",t16".iiia"):natrp(ts(ksi,eta))
5147 [m:more(ksi,eta)]
5148 +*iiia
5149 m@t17:=ismore12(ksi,rpofnt(x),eta,rpofnt(y),t1,t2,m):more(rpofnt(x),rpofnt(y))
5150 t18:=satz154d(x0,y0,t17):more"rt"(x0,y0)
5151 t20:=ismn12(ksi,rpofnt(x),eta,rpofnt(y),m,satz154a(x0,y0,t18),t1,t2):is(mn(ksi,eta,m),mn(rpofnt(x),rpofnt(y),satz154a(x0,y0,t18)))
5152 t21:=symis(cut,rpofrt(mn"rt"(x0,y0,t18)),mn(rpofnt(x),rpofnt(y),satz154a(x0,y0,t18)),satz155b(x0,y0,t18)):is(mn(rpofnt(x),rpofnt(y),satz154a(x0,y0,t18)),rpofrt(mn"rt"(x0,y0,t18)))
5153 t22:=satz112g(x0,t4,y0,t5,t18):natrt(mn"rt"(x0,y0,t18))
5154 xmy:=nofrt(mn"rt"(x0,y0,t18),t22):nat
5155 t23:=isrtn1(mn"rt"(x0,y0,t18),t22):is"rt"(mn"rt"(x0,y0,t18),rtofn(xmy))
5156 t24:=isrterp(mn"rt"(x0,y0,t18),rtofn(xmy),t23):is(rpofrt(mn"rt"(x0,y0,t18)),rpofnt(xmy))
5157 t25:=tr3is(cut,mn(ksi,eta,m),mn(rpofnt(x),rpofnt(y),satz154a(x0,y0,t18)),rpofrt(mn"rt"(x0,y0,t18)),rpofnt(xmy),t20,t21,t24):is(mn(ksi,eta,m),rpofnt(xmy))
5158 -iiia
5159 m@natmn:=somei(nat,[t:nat]is(mn(ksi,eta,m),rpofnt(t)),xmy".iiia",t25".iiia"):natrp(mn(ksi,eta,m))
5160 @[p:cut][q:cut][r:cut]
5161 3pl13:=tr3is(cut,pl(p,pl(q,r)),pl(pl(q,r),p),pl(pl(r,q),p),pl(r,pl(q,p)),compl(p,pl(q,r)),ispl1(pl(q,r),pl(r,q),p,compl(q,r)),asspl1(r,q,p)):is(pl(p,pl(q,r)),pl(r,pl(q,p)))
5162 [s:cut]
5163 4pl24:=tr3is(cut,pl(pl(p,q),pl(r,s)),pl(p,pl(q,pl(r,s))),pl(p,pl(s,pl(r,q))),pl(pl(p,s),pl(r,q)),asspl1(p,q,pl(r,s)),ispl2(pl(q,pl(r,s)),pl(s,pl(r,q)),p,3pl13(q,r,s)),asspl2(p,s,pl(r,q))):is(pl(pl(p,q),pl(r,s)),pl(pl(p,s),pl(r,q)))
5164 r@3pl12:=tr3is(cut,pl(p,pl(q,r)),pl(pl(p,q),r),pl(pl(q,p),r),pl(q,pl(p,r)),asspl2(p,q,r),ispl1(pl(p,q),pl(q,p),r,compl(p,q)),asspl1(q,p,r)):is(pl(p,pl(q,r)),pl(q,pl(p,r)))
5165 s@4pl23:=tr3is(cut,pl(pl(p,q),pl(r,s)),pl(p,pl(q,pl(r,s))),pl(p,pl(r,pl(q,s))),pl(pl(p,r),pl(q,s)),asspl1(p,q,pl(r,s)),ispl2(pl(q,pl(r,s)),pl(r,pl(q,s)),p,3pl12(q,r,s)),asspl2(p,r,pl(q,s))):is(pl(pl(p,q),pl(r,s)),pl(pl(p,r),pl(q,s)))
5166 r@3pl23:=tr3is(cut,pl(pl(p,q),r),pl(p,pl(q,r)),pl(p,pl(r,q)),pl(pl(p,r),q),asspl1(p,q,r),ispl2(pl(q,r),pl(r,q),p,compl(q,r)),asspl2(p,r,q)):is(pl(pl(p,q),r),pl(pl(p,r),q))
5167 p@a2isapa:=tris(cut,ts(p,pl(1rp,1rp)),pl(ts(p,1rp),ts(p,1rp)),pl(p,p),disttp2(p,1rp,1rp),ispl12(ts(p,1rp),p,ts(p,1rp),p,satz151(p),satz151(p))):is(ts(p,pl(1rp,1rp)),pl(p,p))
5168 @dif:=pair1type(cut):'type'
5169 [a1:cut][a2:cut]
5170 df:=pair1(cut,a1,a2):dif
5171 @[a:dif]
5172 stm:=first1(cut,a):cut
5173 std:=second1(cut,a):cut
5174 a2@stmis:=first1is1(cut,a1,a2):is(stm(df(a1,a2)),a1)
5175 isstm:=first1is2(cut,a1,a2):is(a1,stm(df(a1,a2)))
5176 stdis:=second1is1(cut,a1,a2):is(std(df(a1,a2)),a2)
5177 isstd:=second1is2(cut,a1,a2):is(a2,std(df(a1,a2)))
5178 a@1a:=stm(a):cut
5179 2a:=std(a):cut
5180 dfis:=pair1is1(cut,a):is"e"(dif,df(1a,2a),a)
5181 isdf:=pair1is2(cut,a):is"e"(dif,a,df(1a,2a))
5182 a2@[b1:cut][b2:cut]
5183 12issmsd:=ispl12(a1,stm(df(a1,a2)),b2,std(df(b1,b2)),isstm(a1,a2),isstd(b1,b2)):is(pl(a1,b2),pl(stm(df(a1,a2)),std(df(b1,b2))))
5184 smsdis12:=symis(cut,pl(a1,b2),pl(stm(df(a1,a2)),std(df(b1,b2))),12issmsd):is(pl(stm(df(a1,a2)),std(df(b1,b2))),pl(a1,b2))
5185 a@[r1:cut][r2:cut]
5186 1sdissmsd:=ispl1(r1,stm(df(r1,r2)),2a,isstm(r1,r2)):is(pl(r1,2a),pl(stm(df(r1,r2)),2a))
5187 smsdis1sd:=symis(cut,pl(r1,2a),pl(stm(df(r1,r2)),2a),1sdissmsd):is(pl(stm(df(r1,r2)),2a),pl(r1,2a))
5188 sm2issmsd:=ispl2(r2,std(df(r1,r2)),1a,isstd(r1,r2)):is(pl(1a,r2),pl(1a,std(df(r1,r2))))
5189 smsdissm2:=symis(cut,pl(1a,r2),pl(1a,std(df(r1,r2))),sm2issmsd):is(pl(1a,std(df(r1,r2))),pl(1a,r2))
5190 a2@[r:cut][i:is(a1,r)]
5191 issm:=isf(cut,dif,[t:cut]df(t,a2),a1,r,i):is"e"(dif,df(a1,a2),df(r,a2))
5192 r@[i:is(a2,r)]
5193 issd:=isf(cut,dif,[t:cut]df(a1,t),a2,r,i):is"e"(dif,df(a1,a2),df(a1,r))
5194 b2@[i:is(a1,b1)][j:is(a2,b2)]
5195 issmsd:=tris(dif,df(a1,a2),df(b1,a2),df(b1,b2),issm(a1,a2,b1,i),issd(b1,a2,b2,j)):is"e"(dif,df(a1,a2),df(b1,b2))
5196 a@[b:dif]
5197 1b:=stm(b):cut
5198 2b:=std(b):cut
5199 eq:=is(pl(1a,2b),pl(1b,2a)):'prop'
5200 b2@[i:is(pl(a1,b2),pl(b1,a2))]
5201 eqi12:=tr3is(cut,pl(stm(df(a1,a2)),std(df(b1,b2))),pl(a1,b2),pl(b1,a2),pl(stm(df(b1,b2)),std(df(a1,a2))),smsdis12(a1,a2,b1,b2),i,12issmsd(b1,b2,a1,a2)):eq(df(a1,a2),df(b1,b2))
5202 r2@[i:is(pl(1a,r2),pl(r1,2a))]
5203 eqi1:=isp(dif,[x:dif]eq(x,df(r1,r2)),df(1a,2a),a,eqi12(1a,2a,r1,r2,i),dfis):eq(a,df(r1,r2))
5204 r2@[i:is(pl(r1,2a),pl(1a,r2))]
5205 eqi2:=isp(dif,[x:dif]eq(df(r1,r2),x),df(1a,2a),a,eqi12(r1,r2,1a,2a,i),dfis):eq(df(r1,r2),a)
5206 b2@[e:eq(df(a1,a2),df(b1,b2))]
5207 eqe12:=tr3is(cut,pl(a1,b2),pl(stm(df(a1,a2)),std(df(b1,b2))),pl(stm(df(b1,b2)),std(df(a1,a2))),pl(b1,a2),12issmsd(a1,a2,b1,b2),e,smsdis12(b1,b2,a1,a2)):is(pl(a1,b2),pl(b1,a2))
5208 a@satzd163:=refis(cut,pl(1a,2a)):eq(a,a)
5209 refeq:=satzd163:eq(a,a)
5210 b@[i:is"e"(dif,a,b)]
5211 refeq1:=isp(dif,[x:dif]eq(a,x),a,b,refeq,i):eq(a,b)
5212 refeq2:=isp(dif,[x:dif]eq(x,a),a,b,refeq,i):eq(b,a)
5213 b2@[i:is(a1,b1)][j:is(a2,b2)]
5214 eqsmsd:=refeq1(df(a1,a2),df(b1,b2),issmsd(i,j)):eq(df(a1,a2),df(b1,b2))
5215 r@[i:is(a1,r)]
5216 eqsm:=refeq1(df(a1,a2),df(r,a2),issm(i)):eq(df(a1,a2),df(r,a2))
5217 r@[i:is(a2,r)]
5218 eqsd:=refeq1(df(a1,a2),df(a1,r),issd(i)):eq(df(a1,a2),df(a1,r))
5219 b@[e:eq(a,b)]
5220 satzd164:=symis(cut,pl(1a,2b),pl(1b,2a),e):eq(b,a)
5221 symeq:=satzd164:eq(b,a)
5222 b@[c:dif]
5223 1c:=stm(c):cut
5224 2c:=std(c):cut
5225 [e:eq(a,b)][f:eq(b,c)]
5226 +1d165
5227 t1:=ispl12(pl(1a,2b),pl(1b,2a),pl(1b,2c),pl(1c,2b),e,f):is(pl(pl(1a,2b),pl(1b,2c)),pl(pl(1b,2a),pl(1c,2b)))
5228 t2:=tr4is(cut,pl(pl(1a,2c),pl(1b,2b)),pl(pl(1a,2b),pl(1b,2c)),pl(pl(1b,2a),pl(1c,2b)),pl(pl(1c,2b),pl(1b,2a)),pl(pl(1c,2a),pl(1b,2b)),4pl24(1a,2c,1b,2b),t1,compl(pl(1b,2a),pl(1c,2b)),4pl24(1c,2b,1b,2a)):is(pl(pl(1a,2c),pl(1b,2b)),pl(pl(1c,2a),pl(1b,2b)))
5229 -1d165
5230 satzd165:=satz136b(pl(1a,2c),pl(1c,2a),pl(1b,2b),t2".1d165"):eq(a,c)
5231 treq:=satzd165:eq(a,c)
5232 c@[e:eq(c,a)][f:eq(c,b)]
5233 treq1:=treq(a,c,b,symeq(c,a,e),f):eq(a,b)
5234 c@[e:eq(a,c)][f:eq(b,c)]
5235 treq2:=treq(a,c,b,e,symeq(b,c,f)):eq(a,b)
5236 c@[d:dif][e1:eq(a,b)][e2:eq(b,c)][e3:eq(c,d)]
5237 tr3eq:=treq(a,b,d,e1,treq(b,c,d,e2,e3)):eq(a,d)
5238 d@[e:dif][e1:eq(a,b)][e2:eq(b,c)][e3:eq(c,d)][e4:eq(d,e)]
5239 tr4eq:=tr3eq(a,b,c,e,e1,e2,treq(c,d,e,e3,e4)):eq(a,e)
5240 a@posd:=more(1a,2a):'prop'
5241 zero:=is(1a,2a):'prop'
5242 negd:=less(1a,2a):'prop'
5243 a2@[m:more(a1,a2)]
5244 posdi:=ismore12(a1,stm(df(a1,a2)),a2,std(df(a1,a2)),isstm(a1,a2),isstd(a1,a2),m):posd(df(a1,a2))
5245 a2@[i:is(a1,a2)]
5246 zeroi:=tr3is(cut,stm(df(a1,a2)),a1,a2,std(df(a1,a2)),stmis(a1,a2),i,isstd(a1,a2)):zero(df(a1,a2))
5247 a2@[l:less(a1,a2)]
5248 negdi:=isless12(a1,stm(df(a1,a2)),a2,std(df(a1,a2)),isstm(a1,a2),isstd(a1,a2),l):negd(df(a1,a2))
5249 a@axrde:=satz123b(1a,2a):ec3(zero(a),posd(a),negd(a))
5250 axrdo:=satz123a(1a,2a):or3(zero(a),posd(a),negd(a))
5251 axrd:=orec3i(zero(a),posd(a),negd(a),axrdo,axrde):orec3(zero(a),posd(a),negd(a))
5252 [p:'prop'][p1:[t:posd(a)]p][p2:[t:zero(a)]p][p3:[t:negd(a)]p]
5253 rappd:=or3app(zero(a),posd(a),negd(a),p,axrdo,p2,p1,p3):p
5254 a@[p:posd(a)]
5255 pnot0d:=ec3e21(zero(a),posd(a),negd(a),axrde,p):not(zero(a))
5256 pnotnd:=ec3e23(zero(a),posd(a),negd(a),axrde,p):not(negd(a))
5257 a@[z:zero(a)]
5258 0notpd:=ec3e12(zero(a),posd(a),negd(a),axrde,z):not(posd(a))
5259 0notnd:=ec3e13(zero(a),posd(a),negd(a),axrde,z):not(negd(a))
5260 a@[n:negd(a)]
5261 nnotpd:=ec3e32(zero(a),posd(a),negd(a),axrde,n):not(posd(a))
5262 nnot0d:=ec3e31(zero(a),posd(a),negd(a),axrde,n):not(zero(a))
5263 b@[e:eq(a,b)][p:posd(a)]
5264 +iv1d
5265 t1:=ismore12(pl(1a,2b),pl(1b,2a),pl(2a,2b),pl(2b,2a),e,compl(2a,2b),satz135a(1a,2a,2b,p)):more(pl(1b,2a),pl(2b,2a))
5266 -iv1d
5267 eqposd:=satz136a(1b,2b,2a,t1".iv1d"):posd(b)
5268 e@[z:zero(a)]
5269 +*iv1d
5270 z@t2:=tr3is(cut,pl(1b,2a),pl(1a,2b),pl(2a,2b),pl(2b,2a),symeq(a,b,e),ispl1(1a,2a,2b,z),compl(2a,2b)):is(pl(1b,2a),pl(2b,2a))
5271 -iv1d
5272 z@eqzero:=satz136b(1b,2b,2a,t2".iv1d"):zero(b)
5273 e@[n:negd(a)]
5274 +*iv1d
5275 n@t3:=isless12(pl(1a,2b),pl(1b,2a),pl(2a,2b),pl(2b,2a),e,compl(2a,2b),satz135c(1a,2a,2b,n)):less(pl(1b,2a),pl(2b,2a))
5276 -iv1d
5277 n@eqnegd:=satz136c(1b,2b,2a,t3".iv1d"):negd(b)
5278 b@[z:zero(a)][y:zero(b)]
5279 zeroeq:=tris(cut,pl(1a,2b),pl(2a,1b),pl(1b,2a),ispl12(1a,2a,2b,1b,z,symis(cut,1b,2b,y)),compl(2a,1b)):eq(a,b)
5280 @[r:cut]
5281 pdofrp:=df(pl(r,1rp),1rp):dif
5282 ndofrp:=df(1rp,pl(r,1rp)):dif
5283 [s:cut][i:is(r,s)]
5284 isrpepd:=refeq1(pdofrp(r),pdofrp(s),isf(cut,dif,[x:cut]pdofrp(x),r,s,i)):eq(pdofrp(r),pdofrp(s))
5285 isrpend:=refeq1(ndofrp(r),ndofrp(s),isf(cut,dif,[x:cut]ndofrp(x),r,s,i)):eq(ndofrp(r),ndofrp(s))
5286 s@[e:eq(pdofrp(r),pdofrp(s))]
5287 +*iv1d
5288 e@t4:=satz136b(pl(r,1rp),pl(s,1rp),1rp,eqe12(pl(r,1rp),1rp,pl(s,1rp),1rp,e)):is(pl(r,1rp),pl(s,1rp))
5289 -iv1d
5290 e@isrpipd:=satz136b(r,s,1rp,t4".iv1d"):is(r,s)
5291 s@[e:eq(ndofrp(r),ndofrp(s))]
5292 +*iv1d
5293 e@t5:=satz136e(pl(s,1rp),pl(r,1rp),1rp,eqe12(1rp,pl(r,1rp),1rp,pl(s,1rp),e)):is(pl(s,1rp),pl(r,1rp))
5294 -iv1d
5295 e@isrpind:=symis(cut,s,r,satz136b(s,r,1rp,t5".iv1d")):is(r,s)
5296 r@posdirp:=posdi(pl(r,1rp),1rp,ismore1(pl(1rp,r),pl(r,1rp),1rp,compl(1rp,r),satz133(1rp,r))):posd(pdofrp(r))
5297 negdirp:=negdi(1rp,pl(r,1rp),isless2(pl(1rp,r),pl(r,1rp),1rp,compl(1rp,r),satz133a(1rp,r))):negd(ndofrp(r))
5298 a@[p:posd(a)]
5299 rpofpd:=mn(1a,2a,p):cut
5300 +*iv1d
5301 p@t6:=tr4is(cut,pl(1a,1rp),pl(pl(rpofpd,2a),1rp),pl(rpofpd,pl(2a,1rp)),pl(rpofpd,pl(1rp,2a)),pl(pl(rpofpd,1rp),2a),ispl1(1a,pl(rpofpd,2a),1rp,satz140f(1a,2a,p)),asspl1(rpofpd,2a,1rp),ispl2(pl(2a,1rp),pl(1rp,2a),rpofpd,compl(2a,1rp)),asspl2(rpofpd,1rp,2a)):is(pl(1a,1rp),pl(pl(rpofpd,1rp),2a))
5302 -iv1d
5303 p@eqpdrp1:=eqi1(a,pl(rpofpd,1rp),1rp,t6".iv1d"):eq(a,pdofrp(rpofpd(a,p)))
5304 eqpdrp2:=symeq(a,pdofrp(rpofpd(a,p)),eqpdrp1):eq(pdofrp(rpofpd(a,p)),a)
5305 a@[n:negd(a)]
5306 rpofnd:=mn(2a,1a,satz122(1a,2a,n)):cut
5307 +*iv1d
5308 n@t7:=tr3is(cut,pl(1a,pl(rpofnd,1rp)),pl(pl(1a,rpofnd),1rp),pl(2a,1rp),pl(1rp,2a),asspl2(1a,rpofnd,1rp),ispl1(pl(1a,rpofnd),2a,1rp,satz140c(2a,1a,satz122(1a,2a,n))),compl(2a,1rp)):is(pl(1a,pl(rpofnd,1rp)),pl(1rp,2a))
5309 -iv1d
5310 n@eqndrp1:=eqi1(a,1rp,pl(rpofnd,1rp),t7".iv1d"):eq(a,ndofrp(rpofnd(a,n)))
5311 eqndrp2:=symeq(a,ndofrp(rpofnd(a,n)),eqndrp1):eq(ndofrp(rpofnd(a,n)),a)
5312 @[h:dif][p:posd(h)][k:dif][q:posd(k)][e:eq(h,k)]
5313 +*iv1d
5314 e@t8:=tr3eq(pdofrp(rpofpd(h,p)),h,k,pdofrp(rpofpd(k,q)),eqpdrp2(h,p),e,eqpdrp1(k,q)):eq(pdofrp(rpofpd(h,p)),pdofrp(rpofpd(k,q)))
5315 -iv1d
5316 e@eqpderp:=isrpipd(rpofpd(h,p),rpofpd(k,q),t8".iv1d"):is(rpofpd(h,p),rpofpd(k,q))
5317 q@[i:is(rpofpd(h,p),rpofpd(k,q))]
5318 +*iv1d
5319 i@t9:=isrpepd(rpofpd(h,p),rpofpd(k,q),i):eq(pdofrp(rpofpd(h,p)),pdofrp(rpofpd(k,q)))
5320 -iv1d
5321 i@eqpdirp:=tr3eq(h,pdofrp(rpofpd(h,p)),pdofrp(rpofpd(k,q)),k,eqpdrp1(h,p),t9".iv1d",eqpdrp2(k,q)):eq(h,k)
5322 h@[n:negd(h)][k:dif][o:negd(k)][e:eq(h,k)]
5323 +*iv1d
5324 e@t10:=tr3eq(ndofrp(rpofnd(h,n)),h,k,ndofrp(rpofnd(k,o)),eqndrp2(h,n),e,eqndrp1(k,o)):eq(ndofrp(rpofnd(h,n)),ndofrp(rpofnd(k,o)))
5325 -iv1d
5326 e@eqnderp:=isrpind(rpofnd(h,n),rpofnd(k,o),t10".iv1d"):is(rpofnd(h,n),rpofnd(k,o))
5327 o@[i:is(rpofnd(h,n),rpofnd(k,o))]
5328 +*iv1d
5329 i@t11:=isrpend(rpofnd(h,n),rpofnd(k,o),i):eq(ndofrp(rpofnd(h,n)),ndofrp(rpofnd(k,o)))
5330 -iv1d
5331 i@eqndirp:=tr3eq(h,ndofrp(rpofnd(h,n)),ndofrp(rpofnd(k,o)),k,eqndrp1(h,n),t11".iv1d",eqndrp2(k,o)):eq(h,k)
5332 @[r:cut]
5333 +*iv1d
5334 r@t12:=eqpdrp1(pdofrp(r),posdirp(r)):eq(pdofrp(r),pdofrp(rpofpd(pdofrp(r),posdirp(r))))
5335 -iv1d
5336 r@isrppd1:=isrpipd(r,rpofpd(pdofrp(r),posdirp(r)),t12".iv1d"):is(r,rpofpd(pdofrp(r),posdirp(r)))
5337 isrppd2:=symis(cut,r,rpofpd(pdofrp(r),posdirp(r)),isrppd1):is(rpofpd(pdofrp(r),posdirp(r)),r)
5338 +*iv1d
5339 r@t13:=eqndrp1(ndofrp(r),negdirp(r)):eq(ndofrp(r),ndofrp(rpofnd(ndofrp(r),negdirp(r))))
5340 -iv1d
5341 r@isrpnd1:=isrpind(r,rpofnd(ndofrp(r),negdirp(r)),t13".iv1d"):is(r,rpofnd(ndofrp(r),negdirp(r)))
5342 isrpnd2:=symis(cut,r,rpofnd(ndofrp(r),negdirp(r)),isrpnd1):is(rpofnd(ndofrp(r),negdirp(r)),r)
5343 a2@[r:cut]
5344 lemmad1:=eqi12(a1,a2,pl(a1,r),pl(a2,r),tris(cut,pl(a1,pl(a2,r)),pl(a1,pl(r,a2)),pl(pl(a1,r),a2),ispl2(pl(a2,r),pl(r,a2),a1,compl(a2,r)),asspl2(a1,r,a2))):eq(df(a1,a2),df(pl(a1,r),pl(a2,r)))
5345 lemmad2:=symeq(df(a1,a2),df(pl(a1,r),pl(a2,r)),lemmad1):eq(df(pl(a1,r),pl(a2,r)),df(a1,a2))
5346 a@[r:cut]
5347 lemmad3:=treq(a,df(1a,2a),df(pl(1a,r),pl(2a,r)),refeq1(a,df(1a,2a),isdf),lemmad1(1a,2a,r)):eq(a,df(pl(1a,r),pl(2a,r)))
5348 lemmad4:=symeq(a,df(pl(1a,r),pl(2a,r)),lemmad3):eq(df(pl(1a,r),pl(2a,r)),a)
5349 a@absd:=ite(negd(a),dif,df(2a,1a),a):dif
5350 [n:negd(a)]
5351 absnd:=refeq1(absd(a),df(2a,1a),itet(negd(a),dif,df(2a,1a),a,n)):eq(absd(a),df(2a,1a))
5352 a@[n:not(negd(a))]
5353 absnnd:=refeq1(absd(a),a,itef(negd(a),dif,df(2a,1a),a,n)):eq(absd(a),a)
5354 a2@[l:less(a1,a2)]
5355 absdeql:=treq(absd(df(a1,a2)),df(std(df(a1,a2)),stm(df(a1,a2))),df(a2,a1),absnd(df(a1,a2),negdi(a1,a2,l)),eqsmsd(std(df(a1,a2)),stm(df(a1,a2)),a2,a1,stdis(a1,a2),stmis(a1,a2))):eq(absd(df(a1,a2)),df(a2,a1))
5356 a2@[m:moreis(a1,a2)]
5357 absdeqm:=absnnd(df(a1,a2),th3"l.imp"(negd(df(a1,a2)),less(a1,a2),satz123c(a1,a2,m),[t:negd(df(a1,a2))]isless12(stm(df(a1,a2)),a1,std(df(a1,a2)),a2,stmis(a1,a2),stdis(a1,a2),t))):eq(absd(df(a1,a2)),df(a1,a2))
5358 b@[e:eq(a,b)]
5359 +iv2d
5360 [n:negd(a)]
5361 t1:=tr3is(cut,pl(2a,1b),pl(1b,2a),pl(1a,2b),pl(2b,1a),compl(2a,1b),symeq(a,b,e),compl(1a,2b)):is(pl(2a,1b),pl(2b,1a))
5362 t2:=tr3eq(absd(a),df(2a,1a),df(2b,1b),absd(b),absnd(a,n),eqi12(2a,1a,2b,1b,t1),symeq(absd(b),df(2b,1b),absnd(b,eqnegd(a,b,e,n)))):eq(absd(a),absd(b))
5363 e@[n:not(negd(a))]
5364 t3:=tr3eq(absd(a),a,b,absd(b),absnnd(a,n),e,symeq(absd(b),b,absnnd(b,th3"l.imp"(negd(b),negd(a),n,[t:negd(b)]eqnegd(b,a,symeq(a,b,e),t))))):eq(absd(a),absd(b))
5365 -iv2d
5366 eqabsd:=th1"l.imp"(negd(a),eq(absd(a),absd(b)),[t:negd(a)]t2".iv2d"(t),[t:not(negd(a))]t3".iv2d"(t)):eq(absd(a),absd(b))
5367 a@[p:posd(a)]
5368 satzd166a:=eqposd(a,absd(a),symeq(absd(a),a,absnnd(a,pnotnd(a,p))),p):posd(absd(a))
5369 a@[n:negd(a)]
5370 +2d166
5371 t1:=posdi(2a,1a,satz122(1a,2a,n)):posd(df(2a,1a))
5372 -2d166
5373 satzd166b:=eqposd(df(2a,1a),absd(a),symeq(absd(a),df(2a,1a),absnd(a,n)),t1".2d166"):posd(absd(a))
5374 b@[p:posd(a)][q:posd(b)][e:eq(absd(a),absd(b))]
5375 satzd166c:=tr3eq(a,absd(a),absd(b),b,symeq(absd(a),a,absnnd(a,pnotnd(a,p))),e,absnnd(b,pnotnd(b,q))):eq(a,b)
5376 b@[n:negd(a)][o:negd(b)][e:eq(absd(a),absd(b))]
5377 +*2d166
5378 e@t2:=tr3eq(df(2a,1a),absd(a),absd(b),df(2b,1b),symeq(absd(a),df(2a,1a),absnd(a,n)),e,absnd(b,o)):eq(df(2a,1a),df(2b,1b))
5379 -2d166
5380 e@satzd166d:=tr3is(cut,pl(1a,2b),pl(2b,1a),pl(2a,1b),pl(1b,2a),compl(1a,2b),symis(cut,pl(2a,1b),pl(2b,1a),eqe12(2a,1a,2b,1b,t2".2d166")),compl(2a,1b)):eq(a,b)
5381 a@[n:not(zero(a))]
5382 satzd166e:=rappd(a,posd(absd(a)),[t:posd(a)]satzd166a(a,t),th2"l.imp"(zero(a),posd(absd(a)),n),[t:negd(a)]satzd166b(a,t)):posd(absd(a))
5383 a@[z:zero(a)]
5384 satzd166f:=eqzero(a,absd(a),symeq(absd(a),a,absnnd(a,0notnd(a,z))),z):zero(absd(a))
5385 b@mored:=more(pl(1a,2b),pl(1b,2a)):'prop'
5386 b2@[m:more(pl(a1,b2),pl(b1,a2))]
5387 moredi12:=ismore12(pl(a1,b2),pl(stm(df(a1,a2)),std(df(b1,b2))),pl(b1,a2),pl(stm(df(b1,b2)),std(df(a1,a2))),12issmsd(a1,a2,b1,b2),12issmsd(b1,b2,a1,a2),m):mored(df(a1,a2),df(b1,b2))
5388 r2@[m:more(pl(1a,r2),pl(r1,2a))]
5389 moredi1:=ismore12(pl(1a,r2),pl(1a,std(df(r1,r2))),pl(r1,2a),pl(stm(df(r1,r2)),2a),sm2issmsd(a,r1,r2),1sdissmsd(a,r1,r2),m):mored(a,df(r1,r2))
5390 r2@[m:more(pl(r1,2a),pl(1a,r2))]
5391 moredi2:=ismore12(pl(r1,2a),pl(stm(df(r1,r2)),2a),pl(1a,r2),pl(1a,std(df(r1,r2))),1sdissmsd(a,r1,r2),sm2issmsd(a,r1,r2),m):mored(df(r1,r2),a)
5392 b2@[m:mored(df(a1,a2),df(b1,b2))]
5393 morede12:=ismore12(pl(stm(df(a1,a2)),std(df(b1,b2))),pl(a1,b2),pl(stm(df(b1,b2)),std(df(a1,a2))),pl(b1,a2),smsdis12(a1,a2,b1,b2),smsdis12(b1,b2,a1,a2),m):more(pl(a1,b2),pl(b1,a2))
5394 b@lessd:=less(pl(1a,2b),pl(1b,2a)):'prop'
5395 [m:mored(a,b)]
5396 lemmad5:=satz121(pl(1a,2b),pl(1b,2a),m):lessd(b,a)
5397 b@[l:lessd(a,b)]
5398 lemmad6:=satz122(pl(1a,2b),pl(1b,2a),l):mored(b,a)
5399 b2@[l:less(pl(a1,b2),pl(b1,a2))]
5400 lessdi12:=lemmad5(df(b1,b2),df(a1,a2),moredi12(b1,b2,a1,a2,satz122(pl(a1,b2),pl(b1,a2),l))):lessd(df(a1,a2),df(b1,b2))
5401 r2@[l:less(pl(1a,r2),pl(r1,2a))]
5402 lessdi1:=lemmad5(df(r1,r2),a,moredi2(a,r1,r2,satz122(pl(1a,r2),pl(r1,2a),l))):lessd(a,df(r1,r2))
5403 r2@[l:less(pl(r1,2a),pl(1a,r2))]
5404 lessdi2:=lemmad5(a,df(r1,r2),moredi1(a,r1,r2,satz122(pl(r1,2a),pl(1a,r2),l))):lessd(df(r1,r2),a)
5405 b2@[l:lessd(df(a1,a2),df(b1,b2))]
5406 lessde12:=satz121(pl(b1,a2),pl(a1,b2),morede12(b1,b2,a1,a2,lemmad6(df(a1,a2),df(b1,b2),l))):less(pl(a1,b2),pl(b1,a2))
5407 b@satzd167:=satz123(pl(1a,2b),pl(1b,2a)):orec3(eq(a,b),mored(a,b),lessd(a,b))
5408 satzd167a:=satz123a(pl(1a,2b),pl(1b,2a)):or3(eq(a,b),mored(a,b),lessd(a,b))
5409 satzd167b:=satz123b(pl(1a,2b),pl(1b,2a)):ec3(eq(a,b),mored(a,b),lessd(a,b))
5410 d@1d:=stm(d):cut
5411 2d:=std(d):cut
5412 [e:eq(a,b)][f:eq(c,d)][m:mored(a,c)]
5413 +*iv2d
5414 m@t4:=tr4is(cut,pl(pl(1b,2d),pl(1c,2a)),pl(pl(1b,2a),pl(1c,2d)),pl(pl(1a,2b),pl(1d,2c)),pl(pl(1a,2c),pl(1d,2b)),pl(pl(1d,2b),pl(1a,2c)),4pl24(1b,2d,1c,2a),ispl12(pl(1b,2a),pl(1a,2b),pl(1c,2d),pl(1d,2c),symeq(a,b,e),f),4pl24(1a,2b,1d,2c),compl(pl(1a,2c),pl(1d,2b))):is(pl(pl(1b,2d),pl(1c,2a)),pl(pl(1d,2b),pl(1a,2c)))
5415 t5:=ismore2(pl(pl(1b,2d),pl(1c,2a)),pl(pl(1d,2b),pl(1a,2c)),pl(pl(1b,2d),pl(1a,2c)),t4,satz135d(pl(1a,2c),pl(1c,2a),pl(1b,2d),m)):more(pl(pl(1b,2d),pl(1a,2c)),pl(pl(1d,2b),pl(1a,2c)))
5416 -iv2d
5417 m@eqmored12:=satz136a(pl(1b,2d),pl(1d,2b),pl(1a,2c),t5".iv2d"):mored(b,d)
5418 f@[l:lessd(a,c)]
5419 eqlessd12:=lemmad5(d,b,eqmored12(c,d,a,b,f,e,lemmad6(a,c,l))):lessd(b,d)
5420 c@[e:eq(a,b)][m:mored(a,c)]
5421 eqmored1:=eqmored12(a,b,c,c,e,refeq(c),m):mored(b,c)
5422 e@[m:mored(c,a)]
5423 eqmored2:=eqmored12(c,c,a,b,refeq(c),e,m):mored(c,b)
5424 e@[l:lessd(a,c)]
5425 eqlessd1:=eqlessd12(a,b,c,c,e,refeq(c),l):lessd(b,c)
5426 e@[l:lessd(c,a)]
5427 eqlessd2:=eqlessd12(c,c,a,b,refeq(c),e,l):lessd(c,b)
5428 b@moreq:=or(mored(a,b),eq(a,b)):'prop'
5429 lesseq:=or(lessd(a,b),eq(a,b)):'prop'
5430 [m:moreq(a,b)]
5431 satzd168a:=th9"l.or"(mored(a,b),eq(a,b),lessd(b,a),eq(b,a),m,[t:mored(a,b)]lemmad5(a,b,t),[t:eq(a,b)]symeq(a,b,t)):lesseq(b,a)
5432 b@[l:lesseq(a,b)]
5433 satzd168b:=th9"l.or"(lessd(a,b),eq(a,b),mored(b,a),eq(b,a),l,[t:lessd(a,b)]lemmad6(a,b,t),[t:eq(a,b)]symeq(a,b,t)):moreq(b,a)
5434 c@[e:eq(a,b)][m:moreq(a,c)]
5435 eqmoreq1:=th9"l.or"(mored(a,c),eq(a,c),mored(b,c),eq(b,c),m,[t:mored(a,c)]eqmored1(a,b,c,e,t),[t:eq(a,c)]treq1(b,c,a,e,t)):moreq(b,c)
5436 e@[m:moreq(c,a)]
5437 eqmoreq2:=th9"l.or"(mored(c,a),eq(c,a),mored(c,b),eq(c,b),m,[t:mored(c,a)]eqmored2(a,b,c,e,t),[t:eq(c,a)]treq(c,a,b,t,e)):moreq(c,b)
5438 e@[l:lesseq(a,c)]
5439 eqlesseq1:=satzd168a(c,b,eqmoreq2(a,b,c,e,satzd168b(a,c,l))):lesseq(b,c)
5440 e@[l:lesseq(c,a)]
5441 eqlesseq2:=satzd168a(b,c,eqmoreq1(a,b,c,e,satzd168b(c,a,l))):lesseq(c,b)
5442 d@[e:eq(a,b)][f:eq(c,d)][m:moreq(a,c)]
5443 eqmoreq12:=eqmoreq1(a,b,d,e,eqmoreq2(c,d,a,f,m)):moreq(b,d)
5444 f@[l:lesseq(a,c)]
5445 eqlesseq12:=eqlesseq1(a,b,d,e,eqlesseq2(c,d,a,f,l)):lesseq(b,d)
5446 b@[m:mored(a,b)]
5447 moreqi1:=ori1(mored(a,b),eq(a,b),m):moreq(a,b)
5448 b@[l:lessd(a,b)]
5449 lesseqi1:=ori1(lessd(a,b),eq(a,b),l):lesseq(a,b)
5450 b@[e:eq(a,b)]
5451 moreqi2:=ori2(mored(a,b),eq(a,b),e):moreq(a,b)
5452 lesseqi2:=ori2(lessd(a,b),eq(a,b),e):lesseq(a,b)
5453 b@[m:moreq(a,b)]
5454 satzd167c:=th7"l.ec3"(eq(a,b),mored(a,b),lessd(a,b),satzd167b,comor(mored(a,b),eq(a,b),m)):not(lessd(a,b))
5455 b@[l:lesseq(a,b)]
5456 satzd167d:=th9"l.ec3"(eq(a,b),mored(a,b),lessd(a,b),satzd167b,l):not(mored(a,b))
5457 b@[n:not(mored(a,b))]
5458 satzd167e:=th2"l.or3"(eq(a,b),mored(a,b),lessd(a,b),satzd167a,n):lesseq(a,b)
5459 b@[n:not(lessd(a,b))]
5460 satzd167f:=comor(eq(a,b),mored(a,b),th3"l.or3"(eq(a,b),mored(a,b),lessd(a,b),satzd167a,n)):moreq(a,b)
5461 b@[m:mored(a,b)]
5462 satzd167g:=th3"l.imp"(lesseq(a,b),not(mored(a,b)),weli(mored(a,b),m),[t:lesseq(a,b)]satzd167d(t)):not(lesseq(a,b))
5463 b@[l:lessd(a,b)]
5464 satzd167h:=th3"l.imp"(moreq(a,b),not(lessd(a,b)),weli(lessd(a,b),l),[t:moreq(a,b)]satzd167c(t)):not(moreq(a,b))
5465 b@[n:not(moreq(a,b))]
5466 satzd167j:=or3e3(eq(a,b),mored(a,b),lessd(a,b),satzd167a,th5"l.or"(mored(a,b),eq(a,b),n),th4"l.or"(mored(a,b),eq(a,b),n)):lessd(a,b)
5467 b@[n:not(lesseq(a,b))]
5468 satzd167k:=or3e2(eq(a,b),mored(a,b),lessd(a,b),satzd167a,th4"l.or"(lessd(a,b),eq(a,b),n),th5"l.or"(lessd(a,b),eq(a,b),n)):mored(a,b)
5469 b@[z:zero(b)][p:posd(a)]
5470 satzd169a:=ismore12(pl(1a,1b),pl(1a,2b),pl(2a,1b),pl(1b,2a),ispl2(1b,2b,1a,z),compl(2a,1b),satz135a(1a,2a,1b,p)):mored(a,b)
5471 z@[m:mored(a,b)]
5472 satzd169b:=satz136d(1a,2a,2b,ismore12(pl(1a,2b),pl(2b,1a),pl(1b,2a),pl(2b,2a),compl(1a,2b),ispl1(1b,2b,2a,z),m)):posd(a)
5473 z@[n:negd(a)]
5474 satzd169c:=isless12(pl(1a,1b),pl(1a,2b),pl(2a,1b),pl(1b,2a),ispl2(1b,2b,1a,z),compl(2a,1b),satz135c(1a,2a,1b,n)):lessd(a,b)
5475 z@[l:lessd(a,b)]
5476 satzd169d:=satz136f(1a,2a,2b,isless12(pl(1a,2b),pl(2b,1a),pl(1b,2a),pl(2b,2a),compl(1a,2b),ispl1(1b,2b,2a,z),l)):negd(a)
5477 +2d170
5478 z@[p:posd(a)]
5479 t1:=moreqi1(absd(a),b,satzd169a(absd(a),b,z,satzd166a(a,p))):moreq(absd(a),b)
5480 z@[y:zero(a)]
5481 t2:=moreqi2(absd(a),b,treq(absd(a),a,b,absnnd(a,0notnd(a,y)),zeroeq(a,b,y,z))):moreq(absd(a),b)
5482 z@[n:negd(a)]
5483 t3:=moreqi1(absd(a),b,satzd169a(absd(a),b,z,satzd166b(a,n))):moreq(absd(a),b)
5484 -2d170
5485 z@satzd170:=rappd(a,moreq(absd(a),b),[t:posd(a)]t1".2d170"(t),[t:zero(a)]t2".2d170"(t),[t:negd(a)]t3".2d170"(t)):moreq(absd(a),b)
5486 c@[l:lessd(a,b)][k:lessd(b,c)]
5487 +2d171
5488 t1:=satz137a(pl(1a,2b),pl(1b,2a),pl(1b,2c),pl(1c,2b),l,k):less(pl(pl(1a,2b),pl(1b,2c)),pl(pl(1b,2a),pl(1c,2b)))
5489 t2:=isless12(pl(pl(1a,2b),pl(1b,2c)),pl(pl(1a,2c),pl(1b,2b)),pl(pl(1b,2a),pl(1c,2b)),pl(pl(1c,2a),pl(1b,2b)),4pl24(1a,2b,1b,2c),tris(cut,pl(pl(1b,2a),pl(1c,2b)),pl(pl(1b,2b),pl(1c,2a)),pl(pl(1c,2a),pl(1b,2b)),4pl24(1b,2a,1c,2b),compl(pl(1b,2b),pl(1c,2a))),t1):less(pl(pl(1a,2c),pl(1b,2b)),pl(pl(1c,2a),pl(1b,2b)))
5490 -2d171
5491 satzd171:=satz136c(pl(1a,2c),pl(1c,2a),pl(1b,2b),t2".2d171"):lessd(a,c)
5492 trlessd:=satzd171:lessd(a,c)
5493 c@[m:mored(a,b)][n:mored(b,c)]
5494 trmored:=lemmad6(c,a,trlessd(c,b,a,lemmad5(b,c,n),lemmad5(a,b,m))):mored(a,c)
5495 c@[l:lesseq(a,b)][k:lessd(b,c)]
5496 satzd172a:=orapp(lessd(a,b),eq(a,b),lessd(a,c),l,[t:lessd(a,b)]trlessd(t,k),[t:eq(a,b)]eqlessd1(b,a,c,symeq(a,b,t),k)):lessd(a,c)
5497 c@[l:lessd(a,b)][k:lesseq(b,c)]
5498 satzd172b:=orapp(lessd(b,c),eq(b,c),lessd(a,c),k,[t:lessd(b,c)]trlessd(l,t),[t:eq(b,c)]eqlessd2(b,c,a,t,l)):lessd(a,c)
5499 c@[m:moreq(a,b)][n:mored(b,c)]
5500 satzd172c:=lemmad6(c,a,satzd172b(c,b,a,lemmad5(b,c,n),satzd168a(a,b,m))):mored(a,c)
5501 c@[m:mored(a,b)][n:moreq(b,c)]
5502 satzd172d:=lemmad6(c,a,satzd172a(c,b,a,satzd168a(b,c,n),lemmad5(a,b,m))):mored(a,c)
5503 c@[l:lesseq(a,b)][k:lesseq(b,c)]
5504 +2d173
5505 [j:lessd(a,b)]
5506 t1:=lesseqi1(a,c,satzd172b(j,k)):lesseq(a,c)
5507 k@[e:eq(a,b)]
5508 t2:=eqlesseq1(b,a,c,symeq(a,b,e),k):lesseq(a,c)
5509 -2d173
5510 satzd173:=orapp(lessd(a,b),eq(a,b),lesseq(a,c),l,[t:lessd(a,b)]t1".2d173"(t),[t:eq(a,b)]t2".2d173"(t)):lesseq(a,c)
5511 trlesseq:=satzd173:lesseq(a,c)
5512 c@[m:moreq(a,b)][n:moreq(b,c)]
5513 trmoreq:=satzd168b(c,a,trlesseq(c,b,a,satzd168a(b,c,n),satzd168a(a,b,m))):moreq(a,c)
5514 a@ratd:=[t:not(zero(a))]ratrp(rpofpd(absd(a),satzd166e(a,t))):'prop'
5515 irratd:=not(ratd(a)):'prop'
5516 b@[e:eq(a,b)][r:ratd(a)]
5517 +*iv2d
5518 r@[n:not(zero(b))]
5519 t6:=th3"l.imp"(zero(a),zero(b),n,[t:zero(a)]eqzero(a,b,e,t)):not(zero(a))
5520 t7:=eqpderp(absd(a),satzd166e(a,t6),absd(b),satzd166e(b,n),eqabsd(a,b,e)):is(rpofpd(absd(a),satzd166e(a,t6)),rpofpd(absd(b),satzd166e(b,n)))
5521 t8:=isp(cut,[t:cut]ratrp(t),rpofpd(absd(a),satzd166e(a,t6)),rpofpd(absd(b),satzd166e(b,n)),<t6>r,t7):ratrp(rpofpd(absd(b),satzd166e(b,n)))
5522 -iv2d
5523 r@eqratd:=[t:not(zero(b))]t8".iv2d"(t):ratd(b)
5524 e@[i:irratd(a)]
5525 eqirratd:=th3"l.imp"(ratd(b),ratd(a),i,[t:ratd(b)]eqratd(b,a,symeq(a,b,e),t)):irratd(b)
5526 a@[z:zero(a)]
5527 ratdi0:=th2"l.r.imp"(not(zero(a)),[t:not(zero(a))]ratrp(rpofpd(absd(a),satzd166e(a,t))),weli(zero(a),z)):ratd(a)
5528 @[r:cut][i:irratrp(r)][x0:rat]
5529 +*iv2d
5530 x0@[s:ratrp(pl(r,rpofrt(x0)))][y0:rat][j:is(pl(r,rpofrt(x0)),rpofrt(y0))]
5531 t9:=tris(cut,pl(rpofrt(x0),r),pl(r,rpofrt(x0)),rpofrt(y0),compl(rpofrt(x0),r),j):is(pl(rpofrt(x0),r),rpofrt(y0))
5532 t10:=ismore1(pl(rpofrt(x0),r),rpofrt(y0),rpofrt(x0),t9,satz133(rpofrt(x0),r)):more(rpofrt(y0),rpofrt(x0))
5533 t11:=satz154d(y0,x0,t10):more"rt"(y0,x0)
5534 t12:=satz155b(y0,x0,t11):is(rpofrt(mn"rt"(y0,x0,t11)),mn(rpofrt(y0),rpofrt(x0),satz154a(y0,x0,t11)))
5535 t13:=satz140g(rpofrt(y0),rpofrt(x0),r,satz154a(y0,x0,t11),t9):is(r,mn(rpofrt(y0),rpofrt(x0),satz154a(y0,x0,t11)))
5536 t14:=tris2(cut,r,rpofrt(mn"rt"(y0,x0,t11)),mn(rpofrt(y0),rpofrt(x0),satz154a(y0,x0,t11)),t13,t12):is(r,rpofrt(mn"rt"(y0,x0,t11)))
5537 t15:=somei(rat,[x:rat]is(r,rpofrt(x)),mn"rt"(y0,x0,t11),t14):ratrp(r)
5538 s@t16:=someapp(rat,[x:rat]is(pl(r,rpofrt(x0)),rpofrt(x)),s,con,[x:rat][t:is(pl(r,rpofrt(x0)),rpofrt(x))]<t15(x,t)>i):con
5539 -iv2d
5540 x0@remark1:=[t:ratrp(pl(r,rpofrt(x0)))]t16".iv2d"(t):irratrp(pl(r,rpofrt(x0)))
5541 +*iv2d
5542 r@rp:=pdofrp(r):dif
5543 rn:=ndofrp(r):dif
5544 t17:=posdirp(r):posd(rp)
5545 t18:=pnot0d(rp,t17):not(zero(rp))
5546 t19:=nnot0d(rn,negdirp(r)):not(zero(rn))
5547 [n:not(zero(rp))]
5548 t20:=tris2(cut,r,rpofpd(absd(rp),satzd166e(rp,n)),rpofpd(rp,t17),isrppd1(r),eqpderp(absd(rp),satzd166e(rp,n),rp,t17,absnnd(rp,pnotnd(rp,t17)))):is(r,rpofpd(absd(rp),satzd166e(rp,n)))
5549 r@t21:=treq(absd(rn),df(std(rn),stm(rn)),rp,absnd(rn,negdirp(r)),eqsmsd(std(rn),stm(rn),pl(r,1rp),1rp,stdis(1rp,pl(r,1rp)),stmis(1rp,pl(r,1rp)))):eq(absd(rn),rp)
5550 [n:not(zero(rn))]
5551 t22:=tris2(cut,r,rpofpd(absd(rn),satzd166e(rn,n)),rpofpd(rp,t17),isrppd1(r),eqpderp(absd(rn),satzd166e(rn,n),rp,t17,t21)):is(r,rpofpd(absd(rn),satzd166e(rn,n)))
5552 r@[s:cut][i:is(r,s)][rr:ratrp(r)]
5553 t23:=isp(cut,[x:cut]ratrp(x),r,s,rr,i):ratrp(s)
5554 i@[rs:ratrp(s)]
5555 t24:=isp1(cut,[x:cut]ratrp(x),s,r,rs,i):ratrp(r)
5556 -iv2d
5557 r@[rr:ratrp(r)]
5558 remark2a:=[t:not(zero(pdofrp(r)))]t23".iv2d"(rpofpd(absd(pdofrp(r)),satzd166e(pdofrp(r),t)),t20".iv2d"(t),rr):ratd(pdofrp(r))
5559 remark2b:=t17".iv2d":posd(pdofrp(r))
5560 remark3a:=[t:not(zero(ndofrp(r)))]t23".iv2d"(rpofpd(absd(ndofrp(r)),satzd166e(ndofrp(r),t)),t22".iv2d"(t),rr):ratd(ndofrp(r))
5561 remark3b:=negdirp(r):negd(ndofrp(r))
5562 r@[i:irratrp(r)]
5563 remark4a:=th3"l.imp"(ratd(pdofrp(r)),ratrp(r),i,[t:ratd(pdofrp(r))]t24".iv2d"(rpofpd(absd(pdofrp(r)),satzd166e(pdofrp(r),t18".iv2d")),t20".iv2d"(t18".iv2d"),<t18".iv2d">t)):irratd(pdofrp(r))
5564 remark4b:=t17".iv2d":posd(pdofrp(r))
5565 remark5a:=th3"l.imp"(ratd(ndofrp(r)),ratrp(r),i,[t:ratd(ndofrp(r))]t24".iv2d"(rpofpd(absd(ndofrp(r)),satzd166e(ndofrp(r),t19".iv2d")),t22".iv2d"(t19".iv2d"),<t19".iv2d">t)):irratd(ndofrp(r))
5566 remark5b:=negdirp(r):negd(ndofrp(r))
5567 a@natd:=and(posd(a),[t:posd(a)]natrp(rpofpd(a,t))):'prop'
5568 [n:natd(a)]
5569 natposd:=ande1(posd(a),[t:posd(a)]natrp(rpofpd(a,t)),n):posd(a)
5570 natderp:=ande2"l.r"(posd(a),[t:posd(a)]natrp(rpofpd(a,t)),n):natrp(rpofpd(a,natposd(a,n)))
5571 b@[e:eq(a,b)][n:natd(a)]
5572 +*iv2d
5573 n"rp"@t25:=eqposd(a,b,e,natposd(a,n)):posd(b)
5574 [p:posd(b)]
5575 t26:=eqpderp(a,natposd(a,n),b,p,e):is(rpofpd(a,natposd(a,n)),rpofpd(b,p))
5576 t27:=isp(cut,[t:cut]natrp(t),rpofpd(a,natposd(a,n)),rpofpd(b,p),natderp(a,n),t26):natrp(rpofpd(b,p))
5577 -iv2d
5578 n@eqnatd:=andi(posd(b),[t:posd(b)]natrp(rpofpd(b,t)),t25".iv2d",[t:posd(b)]t27".iv2d"(t)):natd(b)
5579 @[x:nat]
5580 pdofnt:=pdofrp(rpofnt(x)):dif
5581 +*iv2d
5582 x@t28:=posdirp(rpofnt(x)):posd(pdofnt(x))
5583 [p:posd(pdofnt(x))]
5584 t29:=isrppd1(rpofnt(x)):is(rpofnt(x),rpofpd(pdofnt(x),t28))
5585 t30:=eqpderp(pdofnt(x),t28,pdofnt(x),p,refeq(pdofnt(x))):is(rpofpd(pdofnt(x),t28),rpofpd(pdofnt(x),p))
5586 t31:=tris(cut,rpofnt(x),rpofpd(pdofnt(x),t28),rpofpd(pdofnt(x),p),t29,t30):is(rpofnt(x),rpofpd(pdofnt(x),p))
5587 t32:=isp(cut,[t:cut]natrp(t),rpofnt(x),rpofpd(pdofnt(x),p),natrpi(x),t31):natrp(rpofpd(pdofnt(x),p))
5588 -iv2d
5589 x@natdi:=andi(posd(pdofnt(x)),[t:posd(pdofnt(x))]natrp(rpofpd(pdofnt(x),t)),t28".iv2d",[t:posd(pdofnt(x))]t32".iv2d"(t)):natd(pdofnt(x))
5590 a@intd:=or(zero(a),natd(absd(a))):'prop'
5591 b@[e:eq(a,b)][i:intd(a)]
5592 +*iv2d
5593 i"rp"@[z:zero(a)]
5594 t33:=eqzero(a,b,e,z):zero(b)
5595 i"rp"@[n:natd(absd(a))]
5596 t34:=eqnatd(absd(a),absd(b),eqabsd(a,b,e),n):natd(absd(b))
5597 -iv2d
5598 i@eqintd:=th9"l.or"(zero(a),natd(absd(a)),zero(b),natd(absd(b)),i,[t:zero(a)]t33".iv2d"(t),[t:natd(absd(a))]t34".iv2d"(t)):intd(b)
5599 a@[n:natd(a)]
5600 +*iv2d
5601 n"rp"@t34a:=symeq(absd(a),a,absnnd(a,pnotnd(a,natposd(a,n)))):eq(a,absd(a))
5602 t35:=eqnatd(a,absd(a),t34a,n):natd(absd(a))
5603 -iv2d
5604 n@natintd:=ori2(zero(a),natd(absd(a)),t35".iv2d"):intd(a)
5605 a@[p:posd(a)][i:intd(a)]
5606 +*iv2d
5607 i"rp"@t36:=ore2(zero(a),natd(absd(a)),i,pnot0d(a,p)):natd(absd(a))
5608 -iv2d
5609 i@posintnatd:=eqnatd(absd(a),a,absnnd(a,pnotnd(a,p)),t36".iv2d"):natd(a)
5610 a@[z:zero(a)]
5611 intdi0:=ori1(zero(a),natd(absd(a)),z):intd(a)
5612 r@[n:natrp(r)]
5613 +*iv2d
5614 n"rp"@t37:=posdirp(r):posd(pdofrp(r))
5615 [p:posd(pdofrp(r))]
5616 t38:=tris(cut,r,rpofpd(pdofrp(r),t37),rpofpd(pdofrp(r),p),isrppd1(r),eqpderp(pdofrp(r),t37,pdofrp(r),p,refeq(pdofrp(r)))):is(r,rpofpd(pdofrp(r),p))
5617 t39:=isp(cut,[t:cut]natrp(t),r,rpofpd(pdofrp(r),p),n,t38):natrp(rpofpd(pdofrp(r),p))
5618 -iv2d
5619 n@remark6a:=andi(posd(pdofrp(r)),[t:posd(pdofrp(r))]natrp(rpofpd(pdofrp(r),t)),t37".iv2d",[t:posd(pdofrp(r))]t39".iv2d"(t)):natd(pdofrp(r))
5620 remark6:=natintd(pdofrp(r),remark6a):intd(pdofrp(r))
5621 +*iv2d
5622 n"rp"@t40:=absdeql(1rp,pl(r,1rp),isless2(pl(1rp,r),pl(r,1rp),1rp,compl(1rp,r),satz133a(1rp,r))):eq(absd(ndofrp(r)),pdofrp(r))
5623 t41:=eqnatd(pdofrp(r),absd(ndofrp(r)),symeq(absd(ndofrp(r)),pdofrp(r),t40),remark6a):natd(absd(ndofrp(r)))
5624 -iv2d
5625 n@remark7:=ori2(zero(ndofrp(r)),natd(absd(ndofrp(r))),t41".iv2d"):intd(ndofrp(r))
5626 a@[i:intd(a)]
5627 +2d174
5628 [n:not(zero(a))]
5629 t1:=ore2(zero(a),natd(absd(a)),i,n):natd(absd(a))
5630 t2:=ande2(posd(absd(a)),[t:posd(absd(a))]natrp(rpofpd(absd(a),t)),t1):[t:posd(absd(a))]natrp(rpofpd(absd(a),t))
5631 t3:=lemmaiii5(rpofpd(absd(a),satzd166e(a,n)),<satzd166e(a,n)>t2):ratrp(rpofpd(absd(a),satzd166e(a,n)))
5632 -2d174
5633 satzd174:=[t:not(zero(a))]t3".2d174"(t):ratd(a)
5634 b@pd:=df(pl(1a,1b),pl(2a,2b)):dif
5635 b2@pd12:=issmsd(pl(stm(df(a1,a2)),stm(df(b1,b2))),pl(std(df(a1,a2)),std(df(b1,b2))),pl(a1,b1),pl(a2,b2),ispl12(stm(df(a1,a2)),a1,stm(df(b1,b2)),b1,stmis(a1,a2),stmis(b1,b2)),ispl12(std(df(a1,a2)),a2,std(df(b1,b2)),b2,stdis(a1,a2),stdis(b1,b2))):is"e"(dif,pd(df(a1,a2),df(b1,b2)),df(pl(a1,b1),pl(a2,b2)))
5636 r2@pd1:=issmsd(pl(1a,stm(df(r1,r2))),pl(2a,std(df(r1,r2))),pl(1a,r1),pl(2a,r2),ispl2(stm(df(r1,r2)),r1,1a,stmis(r1,r2)),ispl2(std(df(r1,r2)),r2,2a,stdis(r1,r2))):is"e"(dif,pd(a,df(r1,r2)),df(pl(1a,r1),pl(2a,r2)))
5637 pd2:=issmsd(pl(stm(df(r1,r2)),1a),pl(std(df(r1,r2)),2a),pl(r1,1a),pl(r2,2a),ispl1(stm(df(r1,r2)),r1,1a,stmis(r1,r2)),ispl1(std(df(r1,r2)),r2,2a,stdis(r1,r2))):is"e"(dif,pd(df(r1,r2),a),df(pl(r1,1a),pl(r2,2a)))
5638 b2@pdeq12a:=refeq1(pd(df(a1,a2),df(b1,b2)),df(pl(a1,b1),pl(a2,b2)),pd12):eq(pd(df(a1,a2),df(b1,b2)),df(pl(a1,b1),pl(a2,b2)))
5639 pdeq12b:=refeq2(pd(df(a1,a2),df(b1,b2)),df(pl(a1,b1),pl(a2,b2)),pd12):eq(df(pl(a1,b1),pl(a2,b2)),pd(df(a1,a2),df(b1,b2)))
5640 r2@pdeq1a:=refeq1(pd(a,df(r1,r2)),df(pl(1a,r1),pl(2a,r2)),pd1):eq(pd(a,df(r1,r2)),df(pl(1a,r1),pl(2a,r2)))
5641 pdeq1b:=refeq2(pd(a,df(r1,r2)),df(pl(1a,r1),pl(2a,r2)),pd1):eq(df(pl(1a,r1),pl(2a,r2)),pd(a,df(r1,r2)))
5642 pdeq2a:=refeq1(pd(df(r1,r2),a),df(pl(r1,1a),pl(r2,2a)),pd2):eq(pd(df(r1,r2),a),df(pl(r1,1a),pl(r2,2a)))
5643 pdeq2b:=refeq2(pd(df(r1,r2),a),df(pl(r1,1a),pl(r2,2a)),pd2):eq(df(pl(r1,1a),pl(r2,2a)),pd(df(r1,r2),a))
5644 b@satzd175:=eqsmsd(pl(1a,1b),pl(2a,2b),pl(1b,1a),pl(2b,2a),compl(1a,1b),compl(2a,2b)):eq(pd(a,b),pd(b,a))
5645 compd:=satzd175:eq(pd(a,b),pd(b,a))
5646 c@[e:eq(a,b)]
5647 +iv3d
5648 t1:=tr3is(cut,pl(pl(1a,1c),pl(2b,2c)),pl(pl(1a,2b),pl(1c,2c)),pl(pl(1b,2a),pl(1c,2c)),pl(pl(1b,1c),pl(2a,2c)),4pl23(1a,1c,2b,2c),ispl1(pl(1a,2b),pl(1b,2a),pl(1c,2c),e),4pl23(1b,2a,1c,2c)):is(pl(pl(1a,1c),pl(2b,2c)),pl(pl(1b,1c),pl(2a,2c)))
5649 -iv3d
5650 eqpd1:=eqi12(pl(1a,1c),pl(2a,2c),pl(1b,1c),pl(2b,2c),t1".iv3d"):eq(pd(a,c),pd(b,c))
5651 eqpd2:=tr3eq(pd(c,a),pd(a,c),pd(b,c),pd(c,b),compd(c,a),eqpd1,compd(b,c)):eq(pd(c,a),pd(c,b))
5652 d@[e:eq(a,b)][f:eq(c,d)]
5653 eqpd12:=treq(pd(a,c),pd(b,c),pd(b,d),eqpd1(a,b,c,e),eqpd2(c,d,b,f)):eq(pd(a,c),pd(b,d))
5654 b@[z:zero(a)]
5655 +*iv3d
5656 z@t2:=tr4is(cut,pl(pl(1a,1b),2b),pl(1a,pl(1b,2b)),pl(2a,pl(2b,1b)),pl(pl(2a,2b),1b),pl(1b,pl(2a,2b)),asspl1(1a,1b,2b),ispl12(1a,2a,pl(1b,2b),pl(2b,1b),z,compl(1b,2b)),asspl2(2a,2b,1b),compl(pl(2a,2b),1b)):is(pl(pl(1a,1b),2b),pl(1b,pl(2a,2b)))
5657 -iv3d
5658 z@pd01:=eqi2(b,pl(1a,1b),pl(2a,2b),t2".iv3d"):eq(pd(a,b),b)
5659 b@[z:zero(b)]
5660 pd02:=treq(pd(a,b),pd(b,a),a,compd(a,b),pd01(b,a,z)):eq(pd(a,b),a)
5661 b@[p:posd(a)][q:posd(b)]
5662 ppd:=posdi(pl(1a,1b),pl(2a,2b),satz137(1a,2a,1b,2b,p,q)):posd(pd(a,b))
5663 b@[n:negd(a)][o:negd(b)]
5664 npd:=negdi(pl(1a,1b),pl(2a,2b),satz137a(1a,2a,1b,2b,n,o)):negd(pd(a,b))
5665 a@m0d:=df(2a,1a):dif
5666 a2@m0deqa:=eqsmsd(std(df(a1,a2)),stm(df(a1,a2)),a2,a1,stdis(a1,a2),stmis(a1,a2)):eq(m0d(df(a1,a2)),df(a2,a1))
5667 m0deqb:=symeq(m0d(df(a1,a2)),df(a2,a1),m0deqa):eq(df(a2,a1),m0d(df(a1,a2)))
5668 b@[e:eq(a,b)]
5669 +*iv3d
5670 e@t3:=tr3is(cut,pl(2a,1b),pl(1b,2a),pl(1a,2b),pl(2b,1a),compl(2a,1b),symeq(a,b,e),compl(1a,2b)):is(pl(2a,1b),pl(2b,1a))
5671 -iv3d
5672 e@eqm0d:=eqi12(2a,1a,2b,1b,t3".iv3d"):eq(m0d(a),m0d(b))
5673 a@[p:posd(a)]
5674 satzd176a:=negdi(2a,1a,satz121(1a,2a,p)):negd(m0d(a))
5675 a@[z:zero(a)]
5676 satzd176b:=zeroi(2a,1a,symis(cut,1a,2a,z)):zero(m0d(a))
5677 a@[n:negd(a)]
5678 satzd176c:=posdi(2a,1a,satz122(1a,2a,n)):posd(m0d(a))
5679 a@[n:negd(m0d(a))]
5680 satzd176d:=satz122(2a,1a,isless12(stm(m0d(a)),2a,std(m0d(a)),1a,stmis(2a,1a),stdis(2a,1a),n)):posd(a)
5681 a@[z:zero(m0d(a))]
5682 satzd176e:=symis(cut,2a,1a,tr3is(cut,2a,stm(df(2a,1a)),std(df(2a,1a)),1a,isstm(2a,1a),z,stdis(2a,1a))):zero(a)
5683 a@[p:posd(m0d(a))]
5684 satzd176f:=satz121(2a,1a,ismore12(stm(m0d(a)),2a,std(m0d(a)),1a,stmis(2a,1a),stdis(2a,1a),p)):negd(a)
5685 a@[z:zero(a)]
5686 m0d0:=zeroeq(m0d(a),a,satzd176b(a,z),z):eq(m0d(a),a)
5687 +3d177
5688 a@t1:=tris(dif,m0d(m0d(a)),df(1a,2a),a,issmsd(std(m0d(a)),stm(m0d(a)),1a,2a,stdis(2a,1a),stmis(2a,1a)),dfis(a)):is"e"(dif,m0d(m0d(a)),a)
5689 -3d177
5690 a@satzd177:=refeq1(m0d(m0d(a)),a,t1".3d177"):eq(m0d(m0d(a)),a)
5691 satzd177a:=symeq(m0d(m0d(a)),a,satzd177):eq(a,m0d(m0d(a)))
5692 b@[e:eq(a,m0d(b))]
5693 satzd177b:=treq(m0d(a),m0d(m0d(b)),b,eqm0d(a,m0d(b),e),satzd177(b)):eq(m0d(a),b)
5694 satzd177c:=symeq(m0d(a),b,satzd177b):eq(b,m0d(a))
5695 b@[e:eq(m0d(a),b)]
5696 satzd177d:=satzd177c(b,a,symeq(m0d(a),b,e)):eq(a,m0d(b))
5697 satzd177e:=symeq(a,m0d(b),satzd177d):eq(m0d(b),a)
5698 +3d178
5699 a@[p:posd(a)]
5700 t1:=tr3eq(absd(m0d(a)),m0d(m0d(a)),a,absd(a),absnd(m0d(a),satzd176a(a,p)),satzd177(a),symeq(absd(a),a,absnnd(a,pnotnd(a,p)))):eq(absd(m0d(a)),absd(a))
5701 a@[z:zero(a)]
5702 t2:=tr3eq(absd(m0d(a)),m0d(a),a,absd(a),absnnd(m0d(a),0notnd(m0d(a),satzd176b(a,z))),m0d0(a,z),symeq(absd(a),a,absnnd(a,0notnd(a,z)))):eq(absd(m0d(a)),absd(a))
5703 a@[n:negd(a)]
5704 t3:=treq(absd(m0d(a)),m0d(a),absd(a),absnnd(m0d(a),pnotnd(m0d(a),satzd176c(a,n))),symeq(absd(a),m0d(a),absnd(a,n))):eq(absd(m0d(a)),absd(a))
5705 -3d178
5706 a@satzd178:=rappd(a,eq(absd(m0d(a)),absd(a)),[t:posd(a)]t1".3d178"(t),[t:zero(a)]t2".3d178"(t),[t:negd(a)]t3".3d178"(t)):eq(absd(m0d(a)),absd(a))
5707 satzd178a:=symeq(absd(m0d(a)),absd(a),satzd178):eq(absd(a),absd(m0d(a)))
5708 +3d179
5709 t1:=pdeq1b(a,2a,1a):eq(df(pl(1a,2a),pl(2a,1a)),pd(a,m0d(a)))
5710 t2:=zeroi(pl(1a,2a),pl(2a,1a),compl(1a,2a)):zero(df(pl(1a,2a),pl(2a,1a)))
5711 -3d179
5712 satzd179:=eqzero(df(pl(1a,2a),pl(2a,1a)),pd(a,m0d(a)),t1".3d179",t2".3d179"):zero(pd(a,m0d(a)))
5713 satzd179a:=eqzero(pd(a,m0d(a)),pd(m0d(a),a),compd(a,m0d(a)),satzd179):zero(pd(m0d(a),a))
5714 b@satzd180:=treq(m0d(pd(a,b)),df(pl(2a,2b),pl(1a,1b)),pd(m0d(a),m0d(b)),m0deqa(pl(1a,1b),pl(2a,2b)),pdeq12b(2a,1a,2b,1b)):eq(m0d(pd(a,b)),pd(m0d(a),m0d(b)))
5715 satzd180a:=symeq(m0d(pd(a,b)),pd(m0d(a),m0d(b)),satzd180):eq(pd(m0d(a),m0d(b)),m0d(pd(a,b)))
5716 md:=pd(a,m0d(b)):dif
5717 b2@mdeq12a:=treq(md(df(a1,a2),df(b1,b2)),pd(df(a1,a2),df(b2,b1)),df(pl(a1,b2),pl(a2,b1)),eqpd2(m0d(df(b1,b2)),df(b2,b1),df(a1,a2),m0deqa(b1,b2)),pdeq12a(a1,a2,b2,b1)):eq(md(df(a1,a2),df(b1,b2)),df(pl(a1,b2),pl(a2,b1)))
5718 mdeq12b:=symeq(md(df(a1,a2),df(b1,b2)),df(pl(a1,b2),pl(a2,b1)),mdeq12a):eq(df(pl(a1,b2),pl(a2,b1)),md(df(a1,a2),df(b1,b2)))
5719 r2@mdeq1a:=treq(md(a,df(r1,r2)),pd(a,df(r2,r1)),df(pl(1a,r2),pl(2a,r1)),eqpd2(m0d(df(r1,r2)),df(r2,r1),a,m0deqa(r1,r2)),pdeq1a(a,r2,r1)):eq(md(a,df(r1,r2)),df(pl(1a,r2),pl(2a,r1)))
5720 mdeq1b:=symeq(md(a,df(r1,r2)),df(pl(1a,r2),pl(2a,r1)),mdeq1a):eq(df(pl(1a,r2),pl(2a,r1)),md(a,df(r1,r2)))
5721 mdeq2a:=pdeq12a(r1,r2,2a,1a):eq(md(df(r1,r2),a),df(pl(r1,2a),pl(r2,1a)))
5722 mdeq2b:=pdeq12b(r1,r2,2a,1a):eq(df(pl(r1,2a),pl(r2,1a)),md(df(r1,r2),a))
5723 c@[e:eq(a,b)]
5724 eqmd1:=eqpd1(a,b,m0d(c),e):eq(md(a,c),md(b,c))
5725 eqmd2:=eqpd2(m0d(a),m0d(b),c,eqm0d(a,b,e)):eq(md(c,a),md(c,b))
5726 d@[e:eq(a,b)][f:eq(c,d)]
5727 eqmd12:=treq(md(a,c),md(b,c),md(b,d),eqmd1(a,b,c,e),eqmd2(c,d,b,f)):eq(md(a,c),md(b,d))
5728 b@satzd181:=tr3eq(m0d(md(a,b)),pd(m0d(a),m0d(m0d(b))),pd(m0d(a),b),md(b,a),satzd180(a,m0d(b)),eqpd2(m0d(m0d(b)),b,m0d(a),satzd177(b)),compd(m0d(a),b)):eq(m0d(md(a,b)),md(b,a))
5729 satzd181a:=symeq(m0d(md(b,a)),md(a,b),satzd181(b,a)):eq(md(a,b),m0d(md(b,a)))
5730 +3d182
5731 t1:=treq(md(a,b),df(pl(1a,2b),pl(2a,1b)),df(pl(1a,2b),pl(1b,2a)),pdeq1a(a,2b,1b),eqsd(pl(1a,2b),pl(2a,1b),pl(1b,2a),compl(2a,1b))):eq(md(a,b),df(pl(1a,2b),pl(1b,2a)))
5732 t2:=symeq(md(a,b),df(pl(1a,2b),pl(1b,2a)),t1):eq(df(pl(1a,2b),pl(1b,2a)),md(a,b))
5733 t3:=stmis(pl(1a,2b),pl(1b,2a)):is(stm(df(pl(1a,2b),pl(1b,2a))),pl(1a,2b))
5734 t4:=stdis(pl(1a,2b),pl(1b,2a)):is(std(df(pl(1a,2b),pl(1b,2a))),pl(1b,2a))
5735 -3d182
5736 [p:posd(md(a,b))]
5737 +*3d182
5738 p@t5:=eqposd(md(a,b),df(pl(1a,2b),pl(1b,2a)),t1,p):posd(df(pl(1a,2b),pl(1b,2a)))
5739 -3d182
5740 p@satzd182a:=ismore12(stm(df(pl(1a,2b),pl(1b,2a))),pl(1a,2b),std(df(pl(1a,2b),pl(1b,2a))),pl(1b,2a),t3".3d182",t4".3d182",t5".3d182"):mored(a,b)
5741 b@[z:zero(md(a,b))]
5742 +*3d182
5743 z@t6:=eqzero(md(a,b),df(pl(1a,2b),pl(1b,2a)),t1,z):zero(df(pl(1a,2b),pl(1b,2a)))
5744 -3d182
5745 z@satzd182b:=tr3is(cut,pl(1a,2b),stm(df(pl(1a,2b),pl(1b,2a))),std(df(pl(1a,2b),pl(1b,2a))),pl(1b,2a),isstm(pl(1a,2b),pl(1b,2a)),t6".3d182",t4".3d182"):eq(a,b)
5746 b@[n:negd(md(a,b))]
5747 +*3d182
5748 n@t7:=eqnegd(md(a,b),df(pl(1a,2b),pl(1b,2a)),t1,n):negd(df(pl(1a,2b),pl(1b,2a)))
5749 -3d182
5750 n@satzd182c:=isless12(stm(df(pl(1a,2b),pl(1b,2a))),pl(1a,2b),std(df(pl(1a,2b),pl(1b,2a))),pl(1b,2a),t3".3d182",t4".3d182",t7".3d182"):lessd(a,b)
5751 b@[m:mored(a,b)]
5752 +*3d182
5753 m@t8:=posdi(pl(1a,2b),pl(1b,2a),m):posd(df(pl(1a,2b),pl(1b,2a)))
5754 -3d182
5755 m@satzd182d:=eqposd(df(pl(1a,2b),pl(1b,2a)),md(a,b),t2".3d182",t8".3d182"):posd(md(a,b))
5756 b@[e:eq(a,b)]
5757 +*3d182
5758 e@t9:=zeroi(pl(1a,2b),pl(1b,2a),e):zero(df(pl(1a,2b),pl(1b,2a)))
5759 -3d182
5760 e@satzd182e:=eqzero(df(pl(1a,2b),pl(1b,2a)),md(a,b),t2".3d182",t9".3d182"):zero(md(a,b))
5761 b@[l:lessd(a,b)]
5762 +*3d182
5763 l@t10:=negdi(pl(1a,2b),pl(1b,2a),l):negd(df(pl(1a,2b),pl(1b,2a)))
5764 -3d182
5765 l@satzd182f:=eqnegd(df(pl(1a,2b),pl(1b,2a)),md(a,b),t2".3d182",t10".3d182"):negd(md(a,b))
5766 +3d183
5767 b@t1:=tris(cut,pl(1a,2b),pl(2b,1a),pl(stm(m0d(b)),std(m0d(a))),compl(1a,2b),12issmsd(2b,1b,2a,1a)):is(pl(1a,2b),pl(stm(m0d(b)),std(m0d(a))))
5768 t2:=t1(b,a):is(pl(1b,2a),pl(stm(m0d(a)),std(m0d(b))))
5769 -3d183
5770 b@[m:mored(a,b)]
5771 satzd183a:=isless12(pl(1b,2a),pl(stm(m0d(a)),std(m0d(b))),pl(1a,2b),pl(stm(m0d(b)),std(m0d(a))),t2".3d183",t1".3d183",lemmad5(a,b,m)):lessd(m0d(a),m0d(b))
5772 b@[e:eq(a,b)]
5773 staz183b:=eqm0d(a,b,e):eq(m0d(a),m0d(b))
5774 b@[l:lessd(a,b)]
5775 satzd183c:=ismore12(pl(1b,2a),pl(stm(m0d(a)),std(m0d(b))),pl(1a,2b),pl(stm(m0d(b)),std(m0d(a))),t2".3d183",t1".3d183",lemmad6(a,b,l)):mored(m0d(a),m0d(b))
5776 b@[l:lessd(m0d(a),m0d(b))]
5777 satzd183d:=eqmored12(m0d(m0d(a)),a,m0d(m0d(b)),b,satzd177(a),satzd177(b),satzd183c(m0d(a),m0d(b),l)):mored(a,b)
5778 b@[e:eq(m0d(a),m0d(b))]
5779 satzd183e:=tr3eq(a,m0d(m0d(a)),m0d(m0d(b)),b,satzd177a(a),eqm0d(m0d(a),m0d(b),e),satzd177(b)):eq(a,b)
5780 b@[m:mored(m0d(a),m0d(b))]
5781 satzd183f:=eqlessd12(m0d(m0d(a)),a,m0d(m0d(b)),b,satzd177(a),satzd177(b),satzd183a(m0d(a),m0d(b),m)):lessd(a,b)
5782 +3d184
5783 a@t1:=tr3eq(a,df(pl(1a,pl(1rp,1rp)),pl(2a,pl(1rp,1rp))),df(pl(pl(1a,1rp),1rp),pl(1rp,pl(2a,1rp))),md(pdofrp(1a),pdofrp(2a)),lemmad3(a,pl(1rp,1rp)),eqsmsd(pl(1a,pl(1rp,1rp)),pl(2a,pl(1rp,1rp)),pl(pl(1a,1rp),1rp),pl(1rp,pl(2a,1rp)),asspl2(1a,1rp,1rp),3pl12(2a,1rp,1rp)),mdeq12b(pl(1a,1rp),1rp,pl(2a,1rp),1rp)):eq(a,md(pdofrp(1a),pdofrp(2a)))
5784 t2:=and3i(posd(pdofrp(1a)),posd(pdofrp(2a)),eq(a,md(pdofrp(1a),pdofrp(2a))),posdirp(1a),posdirp(2a),t1):and3(posd(pdofrp(1a)),posd(pdofrp(2a)),eq(a,md(pdofrp(1a),pdofrp(2a))))
5785 t3:=somei(dif,[x:dif]and3(posd(pdofrp(1a)),posd(x),eq(a,md(pdofrp(1a),x))),pdofrp(2a),t2):some"l"(dif,[x:dif]and3(posd(pdofrp(1a)),posd(x),eq(a,md(pdofrp(1a),x))))
5786 -3d184
5787 a@satzd184:=somei(dif,[x:dif]some"l"(dif,[y:dif]and3(posd(x),posd(y),eq(a,md(x,y)))),pdofrp(1a),t3".3d184"):some"l"(dif,[x:dif]some"l"(dif,[y:dif]and3(posd(x),posd(y),eq(a,md(x,y)))))
5788 c@asspd1:=tr3eq(pd(pd(a,b),c),df(pl(pl(1a,1b),1c),pl(pl(2a,2b),2c)),df(pl(1a,pl(1b,1c)),pl(2a,pl(2b,2c))),pd(a,pd(b,c)),pdeq2a(c,pl(1a,1b),pl(2a,2b)),eqsmsd(pl(pl(1a,1b),1c),pl(pl(2a,2b),2c),pl(1a,pl(1b,1c)),pl(2a,pl(2b,2c)),asspl1(1a,1b,1c),asspl1(2a,2b,2c)),pdeq1b(a,pl(1b,1c),pl(2b,2c))):eq(pd(pd(a,b),c),pd(a,pd(b,c)))
5789 asspd2:=symeq(pd(pd(a,b),c),pd(a,pd(b,c)),asspd1):eq(pd(a,pd(b,c)),pd(pd(a,b),c))
5790 3pd23:=tr3eq(pd(pd(a,b),c),pd(a,pd(b,c)),pd(a,pd(c,b)),pd(pd(a,c),b),asspd1(a,b,c),eqpd2(pd(b,c),pd(c,b),a,compd(b,c)),asspd2(a,c,b)):eq(pd(pd(a,b),c),pd(pd(a,c),b))
5791 d@4pd23:=tr3eq(pd(pd(a,b),pd(c,d)),pd(pd(pd(a,b),c),d),pd(pd(pd(a,c),b),d),pd(pd(a,c),pd(b,d)),asspd2(pd(a,b),c,d),eqpd1(pd(pd(a,b),c),pd(pd(a,c),b),d,3pd23),asspd1(pd(a,c),b,d)):eq(pd(pd(a,b),pd(c,d)),pd(pd(a,c),pd(b,d)))
5792 b@pdmd:=treq(pd(md(a,b),b),pd(a,pd(m0d(b),b)),a,asspd1(a,m0d(b),b),pd02(a,pd(m0d(b),b),satzd179a(b))):eq(pd(md(a,b),b),a)
5793 mdpd:=treq(md(pd(a,b),b),pd(a,pd(b,m0d(b))),a,asspd1(a,b,m0d(b)),pd02(a,pd(b,m0d(b)),satzd179(b))):eq(md(pd(a,b),b),a)
5794 d@satzd185:=treq(pd(md(a,b),md(c,d)),pd(pd(a,c),pd(m0d(b),m0d(d))),md(pd(a,c),pd(b,d)),4pd23(a,m0d(b),c,m0d(d)),eqpd2(pd(m0d(b),m0d(d)),m0d(pd(b,d)),pd(a,c),satzd180a(b,d))):eq(pd(md(a,b),md(c,d)),md(pd(a,c),pd(b,d)))
5795 c@satzd186:=asspd1:eq(pd(pd(a,b),c),pd(a,pd(b,c)))
5796 b@satzd187a:=treq(pd(b,md(a,b)),pd(md(a,b),b),a,compd(b,md(a,b)),pdmd):eq(pd(b,md(a,b)),a)
5797 [x:dif][e:eq(pd(b,x),a)]
5798 satzd187c:=treq(md(a,b),md(pd(x,b),b),x,eqmd1(a,pd(x,b),b,treq1(a,pd(x,b),pd(b,x),e,compd(b,x))),mdpd(x,b)):eq(md(a,b),x)
5799 satzd187d:=symeq(md(a,b),x,satzd187c):eq(x,md(a,b))
5800 x@[e:eq(pd(x,b),a)]
5801 satzd187e:=satzd187c(treq(pd(b,x),pd(x,b),a,compd(b,x),e)):eq(md(a,b),x)
5802 satzd187f:=symeq(md(a,b),x,satzd187e):eq(x,md(a,b))
5803 +3d188
5804 c@t1:=tr3eq(md(pd(a,c),pd(b,c)),pd(pd(a,c),pd(m0d(b),m0d(c))),pd(md(a,b),md(c,c)),md(a,b),eqpd2(m0d(pd(b,c)),pd(m0d(b),m0d(c)),pd(a,c),satzd180(b,c)),4pd23(a,c,m0d(b),m0d(c)),pd02(md(a,b),md(c,c),satzd179(c))):eq(md(pd(a,c),pd(b,c)),md(a,b))
5805 t2:=symeq(md(pd(a,c),pd(b,c)),md(a,b),t1):eq(md(a,b),md(pd(a,c),pd(b,c)))
5806 -3d188
5807 c@[m:mored(pd(a,c),pd(b,c))]
5808 +*3d188
5809 m@t3:=eqposd(md(pd(a,c),pd(b,c)),md(a,b),t1,satzd182d(pd(a,c),pd(b,c),m)):posd(md(a,b))
5810 -3d188
5811 m@satzd188a:=satzd182a(a,b,t3".3d188"):mored(a,b)
5812 c@[e:eq(pd(a,c),pd(b,c))]
5813 +*3d188
5814 e@t4:=eqzero(md(pd(a,c),pd(b,c)),md(a,b),t1,satzd182e(pd(a,c),pd(b,c),e)):zero(md(a,b))
5815 -3d188
5816 e@satzd188b:=satzd182b(a,b,t4".3d188"):eq(a,b)
5817 c@[l:lessd(pd(a,c),pd(b,c))]
5818 +*3d188
5819 l@t5:=eqnegd(md(pd(a,c),pd(b,c)),md(a,b),t1,satzd182f(pd(a,c),pd(b,c),l)):negd(md(a,b))
5820 -3d188
5821 l@satzd188c:=satzd182c(a,b,t5".3d188"):lessd(a,b)
5822 c@[m:mored(a,b)]
5823 +*3d188
5824 m@t6:=eqposd(md(a,b),md(pd(a,c),pd(b,c)),t2,satzd182d(a,b,m)):posd(md(pd(a,c),pd(b,c)))
5825 -3d188
5826 m@satzd188d:=satzd182a(pd(a,c),pd(b,c),t6".3d188"):mored(pd(a,c),pd(b,c))
5827 c@[e:eq(a,b)]
5828 satzd188e:=eqpd1(a,b,c,e):eq(pd(a,c),pd(b,c))
5829 c@[l:lessd(a,b)]
5830 +*3d188
5831 l@t7:=eqnegd(md(a,b),md(pd(a,c),pd(b,c)),t2,satzd182f(a,b,l)):negd(md(pd(a,c),pd(b,c)))
5832 -3d188
5833 l@satzd188f:=satzd182c(pd(a,c),pd(b,c),t7".3d188"):lessd(pd(a,c),pd(b,c))
5834 c@[m:mored(pd(c,a),pd(c,b))]
5835 satzd188g:=satzd188a(eqmored12(pd(c,a),pd(a,c),pd(c,b),pd(b,c),compd(c,a),compd(c,b),m)):mored(a,b)
5836 c@[e:eq(pd(c,a),pd(c,b))]
5837 satzd188h:=satzd188b(tr3eq(pd(a,c),pd(c,a),pd(c,b),pd(b,c),compd(a,c),e,compd(c,b))):eq(a,b)
5838 [l:lessd(pd(c,a),pd(c,b))]
5839 satzd188j:=satzd188c(eqlessd12(pd(c,a),pd(a,c),pd(c,b),pd(b,c),compd(c,a),compd(c,b),l)):lessd(a,b)
5840 c@[m:mored(a,b)]
5841 satzd188k:=eqmored12(pd(a,c),pd(c,a),pd(b,c),pd(c,b),compd(a,c),compd(b,c),satzd188d(m)):mored(pd(c,a),pd(c,b))
5842 c@[e:eq(a,b)]
5843 satzd188l:=eqpd2(a,b,c,e):eq(pd(c,a),pd(c,b))
5844 c@[l:lessd(a,b)]
5845 satzd188m:=eqlessd12(pd(a,c),pd(c,a),pd(b,c),pd(c,b),compd(a,c),compd(b,c),satzd188f(l)):lessd(pd(c,a),pd(c,b))
5846 d@[e:eq(a,b)][m:mored(c,d)]
5847 satzd188n:=eqmored2(pd(a,d),pd(b,d),pd(a,c),eqpd1(a,b,d,e),satzd188k(c,d,a,m)):mored(pd(a,c),pd(b,d))
5848 satzd188o:=eqmored12(pd(a,c),pd(c,a),pd(b,d),pd(d,b),compd(a,c),compd(b,d),satzd188n):mored(pd(c,a),pd(d,b))
5849 e@[l:lessd(c,d)]
5850 satzd188p:=eqlessd2(pd(a,d),pd(b,d),pd(a,c),eqpd1(a,b,d,e),satzd188m(c,d,a,l)):lessd(pd(a,c),pd(b,d))
5851 satzd188q:=eqlessd12(pd(a,c),pd(c,a),pd(b,d),pd(d,b),compd(a,c),compd(b,d),satzd188p):lessd(pd(c,a),pd(d,b))
5852 d@[m:mored(a,b)][n:mored(c,d)]
5853 satzd189:=trmored(pd(a,c),pd(b,c),pd(b,d),satzd188d(a,b,c,m),satzd188k(c,d,b,n)):mored(pd(a,c),pd(b,d))
5854 d@[l:lessd(a,b)][k:lessd(c,d)]
5855 satzd189a:=lemmad5(pd(b,d),pd(a,c),satzd189(b,a,d,c,lemmad6(a,b,l),lemmad6(c,d,k))):lessd(pd(a,c),pd(b,d))
5856 d@[m:moreq(a,b)][n:mored(c,d)]
5857 satzd190a:=orapp(mored(a,b),eq(a,b),mored(pd(a,c),pd(b,d)),m,[t:mored(a,b)]satzd189(t,n),[t:eq(a,b)]satzd188n(t,n)):mored(pd(a,c),pd(b,d))
5858 d@[m:mored(a,b)][n:moreq(c,d)]
5859 satzd190b:=eqmored12(pd(c,a),pd(a,c),pd(d,b),pd(b,d),compd(c,a),compd(d,b),satzd190a(c,d,a,b,n,m)):mored(pd(a,c),pd(b,d))
5860 d@[l:lesseq(a,b)][k:lessd(c,d)]
5861 satzd190c:=lemmad5(pd(b,d),pd(a,c),satzd190a(b,a,d,c,satzd168b(a,b,l),lemmad6(c,d,k))):lessd(pd(a,c),pd(b,d))
5862 d@[l:lessd(a,b)][k:lesseq(c,d)]
5863 satzd190d:=lemmad5(pd(b,d),pd(a,c),satzd190b(b,a,d,c,lemmad6(a,b,l),satzd168b(c,d,k))):lessd(pd(a,c),pd(b,d))
5864 d@[m:moreq(a,b)][n:moreq(c,d)]
5865 +3d191
5866 [e:eq(a,b)][f:eq(c,d)]
5867 t1:=moreqi2(pd(a,c),pd(b,d),eqpd12(a,b,c,d,e,f)):moreq(pd(a,c),pd(b,d))
5868 e@[o:mored(c,d)]
5869 t2:=moreqi1(pd(a,c),pd(b,d),satzd190a(m,o)):moreq(pd(a,c),pd(b,d))
5870 e@t3:=orapp(mored(c,d),eq(c,d),moreq(pd(a,c),pd(b,d)),n,[t:mored(c,d)]t2(t),[t:eq(c,d)]t1(t)):moreq(pd(a,c),pd(b,d))
5871 n@[o:mored(a,b)]
5872 t4:=moreqi1(pd(a,c),pd(b,d),satzd190b(o,n)):moreq(pd(a,c),pd(b,d))
5873 -3d191
5874 satzd191:=orapp(mored(a,b),eq(a,b),moreq(pd(a,c),pd(b,d)),m,[t:mored(a,b)]t4".3d191"(t),[t:eq(a,b)]t3".3d191"(t)):moreq(pd(a,c),pd(b,d))
5875 d@[l:lesseq(a,b)][k:lesseq(c,d)]
5876 satzd191a:=satzd168a(pd(b,d),pd(a,c),satzd191(b,a,d,c,satzd168b(a,b,l),satzd168b(c,d,k))):lesseq(pd(a,c),pd(b,d))
5877 b@td:=df(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b))):dif
5878 +iv4d
5879 a2@[r:cut]
5880 t1:=ists1(stm(df(a1,a2)),a1,r,stmis(a1,a2)):is(ts(stm(df(a1,a2)),r),ts(a1,r))
5881 t2:=ists2(stm(df(a1,a2)),a1,r,stmis(a1,a2)):is(ts(r,stm(df(a1,a2))),ts(r,a1))
5882 t3:=ists1(std(df(a1,a2)),a2,r,stdis(a1,a2)):is(ts(std(df(a1,a2)),r),ts(a2,r))
5883 t4:=ists2(std(df(a1,a2)),a2,r,stdis(a1,a2)):is(ts(r,std(df(a1,a2))),ts(r,a2))
5884 [s:cut]
5885 t5:=ispl12(ts(stm(df(a1,a2)),r),ts(a1,r),ts(std(df(a1,a2)),s),ts(a2,s),t1(r),t3(s)):is(pl(ts(stm(df(a1,a2)),r),ts(std(df(a1,a2)),s)),pl(ts(a1,r),ts(a2,s)))
5886 t6:=ispl12(ts(r,stm(df(a1,a2))),ts(r,a1),ts(s,std(df(a1,a2))),ts(s,a2),t2(r),t4(s)):is(pl(ts(r,stm(df(a1,a2))),ts(s,std(df(a1,a2)))),pl(ts(r,a1),ts(s,a2)))
5887 t7:=ispl12(ts(std(df(a1,a2)),r),ts(a2,r),ts(stm(df(a1,a2)),s),ts(a1,s),t3(r),t1(s)):is(pl(ts(std(df(a1,a2)),r),ts(stm(df(a1,a2)),s)),pl(ts(a2,r),ts(a1,s)))
5888 t8:=ispl12(ts(r,std(df(a1,a2))),ts(r,a2),ts(s,stm(df(a1,a2))),ts(s,a1),t4(r),t2(s)):is(pl(ts(r,std(df(a1,a2))),ts(s,stm(df(a1,a2)))),pl(ts(r,a2),ts(s,a1)))
5889 b2@t9:=tris(cut,pl(ts(stm(df(a1,a2)),stm(df(b1,b2))),ts(std(df(a1,a2)),std(df(b1,b2)))),pl(ts(a1,stm(df(b1,b2))),ts(a2,std(df(b1,b2)))),pl(ts(a1,b1),ts(a2,b2)),t5(a1,a2,stm(df(b1,b2)),std(df(b1,b2))),t6(b1,b2,a1,a2)):is(pl(ts(stm(df(a1,a2)),stm(df(b1,b2))),ts(std(df(a1,a2)),std(df(b1,b2)))),pl(ts(a1,b1),ts(a2,b2)))
5890 t10:=tris(cut,pl(ts(stm(df(a1,a2)),std(df(b1,b2))),ts(std(df(a1,a2)),stm(df(b1,b2)))),pl(ts(a1,std(df(b1,b2))),ts(a2,stm(df(b1,b2)))),pl(ts(a1,b2),ts(a2,b1)),t5(a1,a2,std(df(b1,b2)),stm(df(b1,b2))),t8(b1,b2,a1,a2)):is(pl(ts(stm(df(a1,a2)),std(df(b1,b2))),ts(std(df(a1,a2)),stm(df(b1,b2)))),pl(ts(a1,b2),ts(a2,b1)))
5891 -iv4d
5892 b2@td12:=issmsd(pl(ts(stm(df(a1,a2)),stm(df(b1,b2))),ts(std(df(a1,a2)),std(df(b1,b2)))),pl(ts(stm(df(a1,a2)),std(df(b1,b2))),ts(std(df(a1,a2)),stm(df(b1,b2)))),pl(ts(a1,b1),ts(a2,b2)),pl(ts(a1,b2),ts(a2,b1)),t9".iv4d",t10".iv4d"):is"e"(dif,td(df(a1,a2),df(b1,b2)),df(pl(ts(a1,b1),ts(a2,b2)),pl(ts(a1,b2),ts(a2,b1))))
5893 r2@td1:=issmsd(pl(ts(1a,stm(df(r1,r2))),ts(2a,std(df(r1,r2)))),pl(ts(1a,std(df(r1,r2))),ts(2a,stm(df(r1,r2)))),pl(ts(1a,r1),ts(2a,r2)),pl(ts(1a,r2),ts(2a,r1)),t6".iv4d"(r1,r2,1a,2a),t8".iv4d"(r1,r2,1a,2a)):is"e"(dif,td(a,df(r1,r2)),df(pl(ts(1a,r1),ts(2a,r2)),pl(ts(1a,r2),ts(2a,r1))))
5894 td2:=issmsd(pl(ts(stm(df(r1,r2)),1a),ts(std(df(r1,r2)),2a)),pl(ts(stm(df(r1,r2)),2a),ts(std(df(r1,r2)),1a)),pl(ts(r1,1a),ts(r2,2a)),pl(ts(r1,2a),ts(r2,1a)),t5".iv4d"(r1,r2,1a,2a),t5".iv4d"(r1,r2,2a,1a)):is"e"(dif,td(df(r1,r2),a),df(pl(ts(r1,1a),ts(r2,2a)),pl(ts(r1,2a),ts(r2,1a))))
5895 b2@tdeq12a:=refeq1(td(df(a1,a2),df(b1,b2)),df(pl(ts(a1,b1),ts(a2,b2)),pl(ts(a1,b2),ts(a2,b1))),td12):eq(td(df(a1,a2),df(b1,b2)),df(pl(ts(a1,b1),ts(a2,b2)),pl(ts(a1,b2),ts(a2,b1))))
5896 tdeq12b:=refeq2(td(df(a1,a2),df(b1,b2)),df(pl(ts(a1,b1),ts(a2,b2)),pl(ts(a1,b2),ts(a2,b1))),td12):eq(df(pl(ts(a1,b1),ts(a2,b2)),pl(ts(a1,b2),ts(a2,b1))),td(df(a1,a2),df(b1,b2)))
5897 r2@tdeq1a:=refeq1(td(a,df(r1,r2)),df(pl(ts(1a,r1),ts(2a,r2)),pl(ts(1a,r2),ts(2a,r1))),td1):eq(td(a,df(r1,r2)),df(pl(ts(1a,r1),ts(2a,r2)),pl(ts(1a,r2),ts(2a,r1))))
5898 tdeq1b:=refeq2(td(a,df(r1,r2)),df(pl(ts(1a,r1),ts(2a,r2)),pl(ts(1a,r2),ts(2a,r1))),td1):eq(df(pl(ts(1a,r1),ts(2a,r2)),pl(ts(1a,r2),ts(2a,r1))),td(a,df(r1,r2)))
5899 tdeq2a:=refeq1(td(df(r1,r2),a),df(pl(ts(r1,1a),ts(r2,2a)),pl(ts(r1,2a),ts(r2,1a))),td2):eq(td(df(r1,r2),a),df(pl(ts(r1,1a),ts(r2,2a)),pl(ts(r1,2a),ts(r2,1a))))
5900 tdeq2b:=refeq2(td(df(r1,r2),a),df(pl(ts(r1,1a),ts(r2,2a)),pl(ts(r1,2a),ts(r2,1a))),td2):eq(df(pl(ts(r1,1a),ts(r2,2a)),pl(ts(r1,2a),ts(r2,1a))),td(df(r1,r2),a))
5901 +4d194
5902 b@t1:=ispl12(ts(1a,1b),ts(1b,1a),ts(2a,2b),ts(2b,2a),comts(1a,1b),comts(2a,2b)):is(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1b,1a),ts(2b,2a)))
5903 t2:=tris(cut,pl(ts(1a,2b),ts(2a,1b)),pl(ts(2a,1b),ts(1a,2b)),pl(ts(1b,2a),ts(2b,1a)),compl(ts(1a,2b),ts(2a,1b)),ispl12(ts(2a,1b),ts(1b,2a),ts(1a,2b),ts(2b,1a),comts(2a,1b),comts(1a,2b))):is(pl(ts(1a,2b),ts(2a,1b)),pl(ts(1b,2a),ts(2b,1a)))
5904 -4d194
5905 b@satzd194:=eqsmsd(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b)),pl(ts(1b,1a),ts(2b,2a)),pl(ts(1b,2a),ts(2b,1a)),t1".4d194",t2".4d194"):eq(td(a,b),td(b,a))
5906 comtd:=satzd194:eq(td(a,b),td(b,a))
5907 c@[e:eq(a,b)]
5908 +*iv4d
5909 e@[r:cut]
5910 t11:=tr3is(cut,pl(ts(1a,r),ts(2b,r)),ts(pl(1a,2b),r),ts(pl(1b,2a),r),pl(ts(1b,r),ts(2a,r)),distpt1(1a,2b,r),ists1(pl(1a,2b),pl(1b,2a),r,e),disttp1(1b,2a,r)):is(pl(ts(1a,r),ts(2b,r)),pl(ts(1b,r),ts(2a,r)))
5911 e@t12:=tr3is(cut,pl(pl(ts(1a,1c),ts(2a,2c)),pl(ts(1b,2c),ts(2b,1c))),pl(pl(ts(1a,1c),ts(2b,1c)),pl(ts(1b,2c),ts(2a,2c))),pl(pl(ts(1b,1c),ts(2a,1c)),pl(ts(1a,2c),ts(2b,2c))),pl(pl(ts(1b,1c),ts(2b,2c)),pl(ts(1a,2c),ts(2a,1c))),4pl24(ts(1a,1c),ts(2a,2c),ts(1b,2c),ts(2b,1c)),ispl12(pl(ts(1a,1c),ts(2b,1c)),pl(ts(1b,1c),ts(2a,1c)),pl(ts(1b,2c),ts(2a,2c)),pl(ts(1a,2c),ts(2b,2c)),t11(1c),t11(b,a,c,symeq(a,b,e),2c)),4pl24(ts(1b,1c),ts(2a,1c),ts(1a,2c),ts(2b,2c))):is(pl(pl(ts(1a,1c),ts(2a,2c)),pl(ts(1b,2c),ts(2b,1c))),pl(pl(ts(1b,1c),ts(2b,2c)),pl(ts(1a,2c),ts(2a,1c))))
5912 -iv4d
5913 e@eqtd1:=eqi12(pl(ts(1a,1c),ts(2a,2c)),pl(ts(1a,2c),ts(2a,1c)),pl(ts(1b,1c),ts(2b,2c)),pl(ts(1b,2c),ts(2b,1c)),t12".iv4d"):eq(td(a,c),td(b,c))
5914 eqtd2:=tr3eq(td(c,a),td(a,c),td(b,c),td(c,b),comtd(c,a),eqtd1,comtd(b,c)):eq(td(c,a),td(c,b))
5915 d@[e:eq(a,b)][f:eq(c,d)]
5916 eqtd12:=treq(td(a,c),td(b,c),td(b,d),eqtd1(a,b,c,e),eqtd2(c,d,b,f)):eq(td(a,c),td(b,d))
5917 b@[z:zero(a)]
5918 +4d192
5919 t1:=tris(cut,pl(ts(1a,1b),ts(2a,2b)),pl(ts(2a,1b),ts(1a,2b)),pl(ts(1a,2b),ts(2a,1b)),ispl12(ts(1a,1b),ts(2a,1b),ts(2a,2b),ts(1a,2b),ists1(1a,2a,1b,z),ists1(2a,1a,2b,symis(cut,1a,2a,z))),compl(ts(2a,1b),ts(1a,2b))):is(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b)))
5920 -4d192
5921 satzd192a:=zeroi(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b)),t1".4d192"):zero(td(a,b))
5922 b@[z:zero(b)]
5923 satzd192b:=eqzero(td(b,a),td(a,b),comtd(b,a),satzd192a(b,a,z)):zero(td(a,b))
5924 b@[z:zero(a)]
5925 td01:=satzd192a(z):zero(td(a,b))
5926 b@[z:zero(b)]
5927 td02:=satzd192b(z):zero(td(a,b))
5928 b@satzd197a:=tr3eq(td(m0d(a),b),df(pl(ts(2a,1b),ts(1a,2b)),pl(ts(2a,2b),ts(1a,1b))),df(pl(ts(1a,2b),ts(2a,1b)),pl(ts(1a,1b),ts(2a,2b))),m0d(td(a,b)),tdeq2a(b,2a,1a),eqsmsd(pl(ts(2a,1b),ts(1a,2b)),pl(ts(2a,2b),ts(1a,1b)),pl(ts(1a,2b),ts(2a,1b)),pl(ts(1a,1b),ts(2a,2b)),compl(ts(2a,1b),ts(1a,2b)),compl(ts(2a,2b),ts(1a,1b))),m0deqb(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b)))):eq(td(m0d(a),b),m0d(td(a,b)))
5929 satzd197b:=tr3eq(td(a,m0d(b)),td(m0d(b),a),m0d(td(b,a)),m0d(td(a,b)),comtd(a,m0d(b)),satzd197a(b,a),eqm0d(td(b,a),td(a,b),comtd(b,a))):eq(td(a,m0d(b)),m0d(td(a,b)))
5930 satzd197c:=treq2(td(m0d(a),b),td(a,m0d(b)),m0d(td(a,b)),satzd197a,satzd197b):eq(td(m0d(a),b),td(a,m0d(b)))
5931 satzd197d:=symeq(td(m0d(a),b),td(a,m0d(b)),satzd197c):eq(td(a,m0d(b)),td(m0d(a),b))
5932 satzd197e:=symeq(td(m0d(a),b),m0d(td(a,b)),satzd197a):eq(m0d(td(a,b)),td(m0d(a),b))
5933 satzd197f:=symeq(td(a,m0d(b)),m0d(td(a,b)),satzd197b):eq(m0d(td(a,b)),td(a,m0d(b)))
5934 satzd198:=treq(td(m0d(a),m0d(b)),td(a,m0d(m0d(b))),td(a,b),satzd197c(a,m0d(b)),eqtd2(m0d(m0d(b)),b,a,satzd177(b))):eq(td(m0d(a),m0d(b)),td(a,b))
5935 satzd198a:=symeq(td(m0d(a),m0d(b)),td(a,b),satzd198):eq(td(a,b),td(m0d(a),m0d(b)))
5936 [p:posd(a)][q:posd(b)]
5937 +*iv4d
5938 q@[r:cut]
5939 t13:=tris(cut,pl(ts(r,mn(1b,2b,q)),ts(r,2b)),ts(r,pl(mn(1b,2b,q),2b)),ts(r,1b),distpt2(r,mn(1b,2b,q),2b),ists2(pl(mn(1b,2b,q),2b),1b,r,satz140e(1b,2b,q))):is(pl(ts(r,mn(1b,2b,q)),ts(r,2b)),ts(r,1b))
5940 [s:cut]
5941 t14:=tris(cut,pl(ts(r,mn(1b,2b,q)),pl(ts(r,2b),s)),pl(pl(ts(r,mn(1b,2b,q)),ts(r,2b)),s),pl(ts(r,1b),s),asspl2(ts(r,mn(1b,2b,q)),ts(r,2b),s),ispl1(pl(ts(r,mn(1b,2b,q)),ts(r,2b)),ts(r,1b),s,t13)):is(pl(ts(r,mn(1b,2b,q)),pl(ts(r,2b),s)),pl(ts(r,1b),s))
5942 t15:=tris(cut,pl(ts(r,mn(1b,2b,q)),pl(ts(r,2b),s)),pl(ts(r,1b),s),pl(s,ts(r,1b)),t14,compl(ts(r,1b),s)):is(pl(ts(r,mn(1b,2b,q)),pl(ts(r,2b),s)),pl(s,ts(r,1b)))
5943 q@t16:=satz135h(pl(ts(1a,2b),ts(2a,2b)),pl(ts(2a,2b),ts(1a,2b)),ts(1a,mn(1b,2b,q)),ts(2a,mn(1b,2b,q)),compl(ts(1a,2b),ts(2a,2b)),satz145a(1a,2a,mn(1b,2b,q),p)):more(pl(ts(1a,mn(1b,2b,q)),pl(ts(1a,2b),ts(2a,2b))),pl(ts(2a,mn(1b,2b,q)),pl(ts(2a,2b),ts(1a,2b))))
5944 t17:=ismore12(pl(ts(1a,mn(1b,2b,q)),pl(ts(1a,2b),ts(2a,2b))),pl(ts(1a,1b),ts(2a,2b)),pl(ts(2a,mn(1b,2b,q)),pl(ts(2a,2b),ts(1a,2b))),pl(ts(1a,2b),ts(2a,1b)),t14(1a,ts(2a,2b)),t15(2a,ts(1a,2b)),t16):more(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b)))
5945 -iv4d
5946 q@ptdpp:=posdi(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b)),t17".iv4d"):posd(td(a,b))
5947 b@[p:posd(a)][n:negd(b)]
5948 +*iv4d
5949 n@t18:=eqposd(td(a,m0d(b)),m0d(td(a,b)),satzd197b(a,b),ptdpp(a,m0d(b),p,satzd176c(b,n))):posd(m0d(td(a,b)))
5950 -iv4d
5951 n@ntdpn:=satzd176f(td(a,b),t18".iv4d"):negd(td(a,b))
5952 b@[n:negd(a)][p:posd(b)]
5953 ntdnp:=eqnegd(td(b,a),td(a,b),comtd(b,a),ntdpn(b,a,p,n)):negd(td(a,b))
5954 b@[n:negd(a)][o:negd(b)]
5955 ptdnn:=eqposd(td(m0d(a),m0d(b)),td(a,b),satzd198(a,b),ptdpp(m0d(a),m0d(b),satzd176c(a,n),satzd176c(b,o))):posd(td(a,b))
5956 b@[n:not(zero(a))][o:not(zero(b))]
5957 +*4d192
5958 o@[p:posd(a)][q:posd(b)]
5959 t2:=pnot0d(td(a,b),ptdpp(a,b,p,q)):not(zero(td(a,b)))
5960 p@[m:negd(b)]
5961 t3:=nnot0d(td(a,b),ntdpn(a,b,p,m)):not(zero(td(a,b)))
5962 p@t4:=rappd(b,not(zero(td(a,b))),[t:posd(b)]t2(t),th2"l.imp"(zero(b),not(zero(td(a,b))),o),[t:negd(b)]t3(t)):not(zero(td(a,b)))
5963 o@[m:negd(a)][p:posd(b)]
5964 t5:=nnot0d(td(a,b),ntdnp(a,b,m,p)):not(zero(td(a,b)))
5965 m@[l:negd(b)]
5966 t6:=pnot0d(td(a,b),ptdnn(a,b,m,l)):not(zero(td(a,b)))
5967 m@t7:=rappd(b,not(zero(td(a,b))),[t:posd(b)]t5(t),th2"l.imp"(zero(b),not(zero(td(a,b))),o),[t:negd(b)]t6(t)):not(zero(td(a,b)))
5968 -4d192
5969 o@satzd192d:=rappd(a,not(zero(td(a,b))),[t:posd(a)]t4".4d192"(t),th2"l.imp"(zero(a),not(zero(td(a,b))),n),[t:negd(a)]t7".4d192"(t)):not(zero(td(a,b)))
5970 b@[z:zero(td(a,b))]
5971 +*4d192
5972 z@[n:not(zero(a))]
5973 t8:=et(zero(b),th3"l.imp"(not(zero(b)),not(zero(td(a,b))),weli(zero(td(a,b)),z),[t:not(zero(b))]satzd192d(n,t))):zero(b)
5974 -4d192
5975 z@satzd192c:=[t:not(zero(a))]t8".4d192"(t):or(zero(a),zero(b))
5976 +4d193
5977 b@[p:posd(a)][q:posd(b)]
5978 t1:=treq2(absd(td(a,b)),td(absd(a),absd(b)),td(a,b),absnnd(td(a,b),pnotnd(td(a,b),ptdpp(a,b,p,q))),eqtd12(absd(a),a,absd(b),b,absnnd(a,pnotnd(a,p)),absnnd(b,pnotnd(b,q)))):eq(absd(td(a,b)),td(absd(a),absd(b)))
5979 p@[n:negd(b)]
5980 t2:=treq(absd(td(a,b)),m0d(td(a,b)),td(a,m0d(b)),absnd(td(a,b),ntdpn(a,b,p,n)),satzd197f(a,b)):eq(absd(td(a,b)),td(a,m0d(b)))
5981 t3:=treq2(absd(td(a,b)),td(absd(a),absd(b)),td(a,m0d(b)),t2,eqtd12(absd(a),a,absd(b),m0d(b),absnnd(a,pnotnd(a,p)),absnd(b,n))):eq(absd(td(a,b)),td(absd(a),absd(b)))
5982 b@[z:zero(a)]
5983 t4:=zeroeq(absd(td(a,b)),td(absd(a),absd(b)),satzd166f(td(a,b),td01(a,b,z)),td01(absd(a),absd(b),satzd166f(a,z))):eq(absd(td(a,b)),td(absd(a),absd(b)))
5984 b@[z:zero(b)]
5985 t5:=zeroeq(absd(td(a,b)),td(absd(a),absd(b)),satzd166f(td(a,b),td02(a,b,z)),td02(absd(a),absd(b),satzd166f(b,z))):eq(absd(td(a,b)),td(absd(a),absd(b)))
5986 b@[n:negd(a)][p:posd(b)]
5987 t6:=tr3eq(absd(td(a,b)),absd(td(b,a)),td(absd(b),absd(a)),td(absd(a),absd(b)),eqabsd(td(a,b),td(b,a),comtd(a,b)),t3(b,a,p,n),comtd(absd(b),absd(a))):eq(absd(td(a,b)),td(absd(a),absd(b)))
5988 n@[o:negd(b)]
5989 t7:=treq(td(absd(a),absd(b)),td(m0d(a),m0d(b)),td(a,b),eqtd12(absd(a),m0d(a),absd(b),m0d(b),absnd(a,n),absnd(b,o)),satzd198(a,b)):eq(td(absd(a),absd(b)),td(a,b))
5990 t8:=treq2(absd(td(a,b)),td(absd(a),absd(b)),td(a,b),absnnd(td(a,b),pnotnd(td(a,b),ptdnn(a,b,n,o))),t7):eq(absd(td(a,b)),td(absd(a),absd(b)))
5991 b@[p:posd(a)]
5992 t9:=rappd(b,eq(absd(td(a,b)),td(absd(a),absd(b))),[t:posd(b)]t1(p,t),[t:zero(b)]t5(t),[t:negd(b)]t3(p,t)):eq(absd(td(a,b)),td(absd(a),absd(b)))
5993 b@[n:negd(a)]
5994 t10:=rappd(b,eq(absd(td(a,b)),td(absd(a),absd(b))),[t:posd(b)]t6(n,t),[t:zero(b)]t5(t),[t:negd(b)]t8(n,t)):eq(absd(td(a,b)),td(absd(a),absd(b)))
5995 -4d193
5996 b@satzd193:=rappd(a,eq(absd(td(a,b)),td(absd(a),absd(b))),[t:posd(a)]t9".4d193"(t),[t:zero(a)]t4".4d193"(t),[t:negd(a)]t10".4d193"(t)):eq(absd(td(a,b)),td(absd(a),absd(b)))
5997 satzd103a:=symeq(absd(td(a,b)),td(absd(a),absd(b)),satzd193):eq(td(absd(a),absd(b)),absd(td(a,b)))
5998 @1df:=pdofrp(1rp):dif
5999 +4d195
6000 a@t1:=tris(cut,pl(pl(ts(1a,pl(1rp,1rp)),ts(2a,1rp)),2a),pl(ts(1a,pl(1rp,1rp)),pl(ts(2a,1rp),2a)),pl(pl(1a,1a),pl(2a,2a)),asspl1(ts(1a,pl(1rp,1rp)),ts(2a,1rp),2a),ispl12(ts(1a,pl(1rp,1rp)),pl(1a,1a),pl(ts(2a,1rp),2a),pl(2a,2a),a2isapa(1a),ispl1(ts(2a,1rp),2a,2a,satz151(2a)))):is(pl(pl(ts(1a,pl(1rp,1rp)),ts(2a,1rp)),2a),pl(pl(1a,1a),pl(2a,2a)))
6001 t2:=tris(cut,pl(1a,pl(ts(1a,1rp),ts(2a,pl(1rp,1rp)))),pl(pl(1a,ts(1a,1rp)),ts(2a,pl(1rp,1rp))),pl(pl(1a,1a),pl(2a,2a)),asspl2(1a,ts(1a,1rp),ts(2a,pl(1rp,1rp))),ispl12(pl(1a,ts(1a,1rp)),pl(1a,1a),ts(2a,pl(1rp,1rp)),pl(2a,2a),ispl2(ts(1a,1rp),1a,1a,satz151(1a)),a2isapa(2a))):is(pl(1a,pl(ts(1a,1rp),ts(2a,pl(1rp,1rp)))),pl(pl(1a,1a),pl(2a,2a)))
6002 t3:=tris2(cut,pl(pl(ts(1a,pl(1rp,1rp)),ts(2a,1rp)),2a),pl(1a,pl(ts(1a,1rp),ts(2a,pl(1rp,1rp)))),pl(pl(1a,1a),pl(2a,2a)),t1,t2):is(pl(pl(ts(1a,pl(1rp,1rp)),ts(2a,1rp)),2a),pl(1a,pl(ts(1a,1rp),ts(2a,pl(1rp,1rp)))))
6003 -4d195
6004 a@satzd195:=treq(td(a,1df),df(pl(ts(1a,pl(1rp,1rp)),ts(2a,1rp)),pl(ts(1a,1rp),ts(2a,pl(1rp,1rp)))),a,tdeq1a(a,pl(1rp,1rp),1rp),eqi2(a,pl(ts(1a,pl(1rp,1rp)),ts(2a,1rp)),pl(ts(1a,1rp),ts(2a,pl(1rp,1rp))),t3".4d195")):eq(td(a,1df),a)
6005 satzd195a:=symeq(td(a,1df),a,satzd195):eq(a,td(a,1df))
6006 satzd195b:=treq(td(1df,a),td(a,1df),a,comtd(1df,a),satzd195):eq(td(1df,a),a)
6007 satzd195c:=symeq(td(1df,a),a,satzd195b):eq(a,td(1df,a))
6008 b@[p:posd(a)][q:posd(b)]
6009 satzd196a:=symeq(td(absd(a),absd(b)),td(a,b),eqtd12(absd(a),a,absd(b),b,absnnd(a,pnotnd(a,p)),absnnd(b,pnotnd(b,q)))):eq(td(a,b),td(absd(a),absd(b)))
6010 b@[n:negd(a)][o:negd(b)]
6011 satzd196b:=treq2(td(a,b),td(absd(a),absd(b)),td(m0d(a),m0d(b)),satzd198a(a,b),eqtd12(absd(a),m0d(a),absd(b),m0d(b),absnd(a,n),absnd(b,o))):eq(td(a,b),td(absd(a),absd(b)))
6012 b@[p:posd(a)][n:negd(b)]
6013 satzd196c:=treq1(td(a,b),m0d(td(absd(a),absd(b))),td(absd(a),m0d(absd(b))),eqtd12(absd(a),a,m0d(absd(b)),b,absnnd(a,pnotnd(a,p)),satzd177b(absd(b),b,absnd(b,n))),satzd197b(absd(a),absd(b))):eq(td(a,b),m0d(td(absd(a),absd(b))))
6014 b@[n:negd(a)][p:posd(b)]
6015 satzd196d:=tr3eq(td(a,b),td(b,a),m0d(td(absd(b),absd(a))),m0d(td(absd(a),absd(b))),comtd(a,b),satzd196c(b,a,p,n),eqm0d(td(absd(b),absd(a)),td(absd(a),absd(b)),comtd(absd(b),absd(a)))):eq(td(a,b),m0d(td(absd(a),absd(b))))
6016 +4d196
6017 b@p1p2:=and(posd(a),posd(b)):'prop'
6018 p1n2:=and(posd(a),negd(b)):'prop'
6019 n1p2:=and(negd(a),posd(b)):'prop'
6020 n1n2:=and(negd(a),negd(b)):'prop'
6021 -4d196
6022 b@[n:not(zero(a))][o:not(zero(b))][e:eq(td(a,b),td(absd(a),absd(b)))]
6023 +*4d196
6024 o@t1:=ptdpp(absd(a),absd(b),satzd166e(a,n),satzd166e(b,o)):posd(td(absd(a),absd(b)))
6025 e@t2:=pnotnd(td(a,b),eqposd(td(absd(a),absd(b)),td(a,b),symeq(td(a,b),td(absd(a),absd(b)),e),t1)):not(negd(td(a,b)))
6026 [p:posd(a)]
6027 t3:=th3"l.imp"(negd(b),negd(td(a,b)),t2,[t:negd(b)]ntdpn(a,b,p,t)):not(negd(b))
6028 t4:=or3e2(zero(b),posd(b),negd(b),axrdo(b),t3,o):posd(b)
6029 t5:=andi(posd(a),posd(b),p,t4):p1p2
6030 t6:=ori1(p1p2,n1n2,t5):or(p1p2,n1n2)
6031 e@[m:negd(a)]
6032 t7:=th3"l.imp"(posd(b),negd(td(a,b)),t2,[t:posd(b)]ntdnp(a,b,m,t)):not(posd(b))
6033 t8:=or3e3(zero(b),posd(b),negd(b),axrdo(b),o,t7):negd(b)
6034 t9:=andi(negd(a),negd(b),m,t8):n1n2
6035 t10:=ori2(p1p2,n1n2,t9):or(p1p2,n1n2)
6036 -4d196
6037 e@satzd196e:=rappd(a,or(p1p2".4d196",n1n2".4d196"),[t:posd(a)]t6".4d196"(t),th2"l.imp"(zero(a),or(p1p2".4d196",n1n2".4d196"),n),[t:negd(a)]t10".4d196"(t)):or(and(posd(a),posd(b)),and(negd(a),negd(b)))
6038 o@[e:eq(td(a,b),m0d(td(absd(a),absd(b))))]
6039 +*4d196
6040 o@t11:=satzd176a(td(absd(a),absd(b)),t1):negd(m0d(td(absd(a),absd(b))))
6041 e@t12:=nnotpd(td(a,b),eqnegd(m0d(td(absd(a),absd(b))),td(a,b),symeq(td(a,b),m0d(td(absd(a),absd(b))),e),t11)):not(posd(td(a,b)))
6042 [p:posd(a)]
6043 t13:=th3"l.imp"(posd(b),posd(td(a,b)),t12,[t:posd(b)]ptdpp(a,b,p,t)):not(posd(b))
6044 t14:=or3e3(zero(b),posd(b),negd(b),axrdo(b),o,t13):negd(b)
6045 t15:=andi(posd(a),negd(b),p,t14):p1n2
6046 t16:=ori1(p1n2,n1p2,t15):or(p1n2,n1p2)
6047 e@[m:negd(a)]
6048 t17:=th3"l.imp"(negd(b),posd(td(a,b)),t12,[t:negd(b)]ptdnn(a,b,m,t)):not(negd(b))
6049 t18:=or3e2(zero(b),posd(b),negd(b),axrdo(b),t17,o):posd(b)
6050 t19:=andi(negd(a),posd(b),m,t18):n1p2
6051 t20:=ori2(p1n2,n1p2,t19):or(p1n2,n1p2)
6052 -4d196
6053 e@satzd196f:=rappd(a,or(p1n2".4d196",n1p2".4d196"),[t:posd(a)]t16".4d196"(t),th2"l.imp"(zero(a),or(p1n2".4d196",n1p2".4d196"),n),[t:negd(a)]t20".4d196"(t)):or(and(posd(a),negd(b)),and(negd(a),posd(b)))
6054 +4d199
6055 @[p:cut][q:cut][r:cut][s:cut][t:cut][u:cut]
6056 t1:=tris(cut,ts(pl(ts(p,r),ts(q,s)),t),pl(ts(ts(p,r),t),ts(ts(q,s),t)),pl(ts(p,ts(r,t)),ts(q,ts(s,t))),disttp1(ts(p,r),ts(q,s),t),ispl12(ts(ts(p,r),t),ts(p,ts(r,t)),ts(ts(q,s),t),ts(q,ts(s,t)),assts1(p,r,t),assts1(q,s,t))):is(ts(pl(ts(p,r),ts(q,s)),t),pl(ts(p,ts(r,t)),ts(q,ts(s,t))))
6057 t2:=tris(cut,pl(ts(q,ts(s,t)),ts(q,ts(r,u))),pl(ts(q,ts(r,u)),ts(q,ts(s,t))),ts(q,pl(ts(r,u),ts(s,t))),compl(ts(q,ts(s,t)),ts(q,ts(r,u))),distpt2(q,ts(r,u),ts(s,t))):is(pl(ts(q,ts(s,t)),ts(q,ts(r,u))),ts(q,pl(ts(r,u),ts(s,t))))
6058 t3:=tr3is(cut,pl(ts(pl(ts(p,r),ts(q,s)),t),ts(pl(ts(p,s),ts(q,r)),u)),pl(pl(ts(p,ts(r,t)),ts(q,ts(s,t))),pl(ts(p,ts(s,u)),ts(q,ts(r,u)))),pl(pl(ts(p,ts(r,t)),ts(p,ts(s,u))),pl(ts(q,ts(s,t)),ts(q,ts(r,u)))),pl(ts(p,pl(ts(r,t),ts(s,u))),ts(q,pl(ts(r,u),ts(s,t)))),ispl12(ts(pl(ts(p,r),ts(q,s)),t),pl(ts(p,ts(r,t)),ts(q,ts(s,t))),ts(pl(ts(p,s),ts(q,r)),u),pl(ts(p,ts(s,u)),ts(q,ts(r,u))),t1,t1(p,q,s,r,u,t)),4pl23(ts(p,ts(r,t)),ts(q,ts(s,t)),ts(p,ts(s,u)),ts(q,ts(r,u))),ispl12(pl(ts(p,ts(r,t)),ts(p,ts(s,u))),ts(p,pl(ts(r,t),ts(s,u))),pl(ts(q,ts(s,t)),ts(q,ts(r,u))),ts(q,pl(ts(r,u),ts(s,t))),distpt2(p,ts(r,t),ts(s,u)),t2)):is(pl(ts(pl(ts(p,r),ts(q,s)),t),ts(pl(ts(p,s),ts(q,r)),u)),pl(ts(p,pl(ts(r,t),ts(s,u))),ts(q,pl(ts(r,u),ts(s,t)))))
6059 -4d199
6060 c@satzd199:=tr3eq(td(td(a,b),c),df(pl(ts(pl(ts(1a,1b),ts(2a,2b)),1c),ts(pl(ts(1a,2b),ts(2a,1b)),2c)),pl(ts(pl(ts(1a,1b),ts(2a,2b)),2c),ts(pl(ts(1a,2b),ts(2a,1b)),1c))),df(pl(ts(1a,pl(ts(1b,1c),ts(2b,2c))),ts(2a,pl(ts(1b,2c),ts(2b,1c)))),pl(ts(1a,pl(ts(1b,2c),ts(2b,1c))),ts(2a,pl(ts(1b,1c),ts(2b,2c))))),td(a,td(b,c)),tdeq2a(c,pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b))),eqsmsd(pl(ts(pl(ts(1a,1b),ts(2a,2b)),1c),ts(pl(ts(1a,2b),ts(2a,1b)),2c)),pl(ts(pl(ts(1a,1b),ts(2a,2b)),2c),ts(pl(ts(1a,2b),ts(2a,1b)),1c)),pl(ts(1a,pl(ts(1b,1c),ts(2b,2c))),ts(2a,pl(ts(1b,2c),ts(2b,1c)))),pl(ts(1a,pl(ts(1b,2c),ts(2b,1c))),ts(2a,pl(ts(1b,1c),ts(2b,2c)))),t3".4d199"(1a,2a,1b,2b,1c,2c),t3".4d199"(1a,2a,1b,2b,2c,1c)),tdeq1b(a,pl(ts(1b,1c),ts(2b,2c)),pl(ts(1b,2c),ts(2b,1c)))):eq(td(td(a,b),c),td(a,td(b,c)))
6061 asstd1:=satzd199:eq(td(td(a,b),c),td(a,td(b,c)))
6062 asstd2:=symeq(td(td(a,b),c),td(a,td(b,c)),satzd199):eq(td(a,td(b,c)),td(td(a,b),c))
6063 +4d201
6064 @[p:cut][q:cut][r:cut][s:cut][t:cut][u:cut]
6065 t1:=tris(cut,pl(ts(p,pl(r,t)),ts(q,pl(s,u))),pl(pl(ts(p,r),ts(p,t)),pl(ts(q,s),ts(q,u))),pl(pl(ts(p,r),ts(q,s)),pl(ts(p,t),ts(q,u))),ispl12(ts(p,pl(r,t)),pl(ts(p,r),ts(p,t)),ts(q,pl(s,u)),pl(ts(q,s),ts(q,u)),disttp2(p,r,t),disttp2(q,s,u)),4pl23(ts(p,r),ts(p,t),ts(q,s),ts(q,u))):is(pl(ts(p,pl(r,t)),ts(q,pl(s,u))),pl(pl(ts(p,r),ts(q,s)),pl(ts(p,t),ts(q,u))))
6066 -4d201
6067 satzd201:=tr3eq(td(a,pd(b,c)),df(pl(ts(1a,pl(1b,1c)),ts(2a,pl(2b,2c))),pl(ts(1a,pl(2b,2c)),ts(2a,pl(1b,1c)))),df(pl(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,1c),ts(2a,2c))),pl(pl(ts(1a,2b),ts(2a,1b)),pl(ts(1a,2c),ts(2a,1c)))),pd(td(a,b),td(a,c)),tdeq1a(a,pl(1b,1c),pl(2b,2c)),eqsmsd(pl(ts(1a,pl(1b,1c)),ts(2a,pl(2b,2c))),pl(ts(1a,pl(2b,2c)),ts(2a,pl(1b,1c))),pl(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,1c),ts(2a,2c))),pl(pl(ts(1a,2b),ts(2a,1b)),pl(ts(1a,2c),ts(2a,1c))),t1".4d201"(1a,2a,1b,2b,1c,2c),t1".4d201"(1a,2a,2b,1b,2c,1c)),pdeq12b(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b)),pl(ts(1a,1c),ts(2a,2c)),pl(ts(1a,2c),ts(2a,1c)))):eq(td(a,pd(b,c)),pd(td(a,b),td(a,c)))
6068 disttpd1:=tr3eq(td(pd(a,b),c),td(c,pd(a,b)),pd(td(c,a),td(c,b)),pd(td(a,c),td(b,c)),comtd(pd(a,b),c),satzd201(c,a,b),eqpd12(td(c,a),td(a,c),td(c,b),td(b,c),comtd(c,a),comtd(c,b))):eq(td(pd(a,b),c),pd(td(a,c),td(b,c)))
6069 disttpd2:=satzd201:eq(td(a,pd(b,c)),pd(td(a,b),td(a,c)))
6070 distptd1:=symeq(td(pd(a,b),c),pd(td(a,c),td(b,c)),disttpd1):eq(pd(td(a,c),td(b,c)),td(pd(a,b),c))
6071 distptd2:=symeq(td(a,pd(b,c)),pd(td(a,b),td(a,c)),disttpd2):eq(pd(td(a,b),td(a,c)),td(a,pd(b,c)))
6072 satzd202:=treq(td(a,md(b,c)),pd(td(a,b),td(a,m0d(c))),md(td(a,b),td(a,c)),disttpd2(a,b,m0d(c)),eqpd2(td(a,m0d(c)),m0d(td(a,c)),td(a,b),satzd197b(a,c))):eq(td(a,md(b,c)),md(td(a,b),td(a,c)))
6073 disttmd1:=treq(td(md(a,b),c),pd(td(a,c),td(m0d(b),c)),md(td(a,c),td(b,c)),disttpd1(a,m0d(b),c),eqpd2(td(m0d(b),c),m0d(td(b,c)),td(a,c),satzd197a(b,c))):eq(td(md(a,b),c),md(td(a,c),td(b,c)))
6074 disttmd2:=satzd202:eq(td(a,md(b,c)),md(td(a,b),td(a,c)))
6075 distmtd1:=symeq(td(md(a,b),c),md(td(a,c),td(b,c)),disttmd1):eq(md(td(a,c),td(b,c)),td(md(a,b),c))
6076 distmtd2:=symeq(td(a,md(b,c)),md(td(a,b),td(a,c)),disttmd2):eq(md(td(a,b),td(a,c)),td(a,md(b,c)))
6077 satzd200:=satzd202:eq(td(a,md(b,c)),md(td(a,b),td(a,c)))
6078 [m:mored(a,b)]
6079 +4d203
6080 t1:=satzd182d(a,b,m):posd(md(a,b))
6081 -4d203
6082 [p:posd(c)]
6083 +*4d203
6084 p@t2:=eqposd(td(md(a,b),c),md(td(a,c),td(b,c)),disttmd1(a,b,c),ptdpp(md(a,b),c,t1,p)):posd(md(td(a,c),td(b,c)))
6085 -4d203
6086 p@satzd203a:=satzd182a(td(a,c),td(b,c),t2".4d203"):mored(td(a,c),td(b,c))
6087 m@[z:zero(c)]
6088 satzd203b:=zeroeq(td(a,c),td(b,c),td02(a,c,z),td02(b,c,z)):eq(td(a,c),td(b,c))
6089 m@[n:negd(c)]
6090 +*4d203
6091 n@t3:=eqnegd(td(md(a,b),c),md(td(a,c),td(b,c)),disttmd1(a,b,c),ntdpn(md(a,b),c,t1,n)):negd(md(td(a,c),td(b,c)))
6092 -4d203
6093 n@satzd203c:=satzd182c(td(a,c),td(b,c),t3".4d203"):lessd(td(a,c),td(b,c))
6094 p@satzd203d:=eqmored12(td(a,c),td(c,a),td(b,c),td(c,b),comtd(a,c),comtd(b,c),satzd203a):mored(td(c,a),td(c,b))
6095 z@satzd203e:=zeroeq(td(c,a),td(c,b),td01(c,a,z),td01(c,b,z)):eq(td(c,a),td(c,b))
6096 n@satzd203f:=eqlessd12(td(a,c),td(c,a),td(b,c),td(c,b),comtd(a,c),comtd(b,c),satzd203c):lessd(td(c,a),td(c,b))
6097 c@[l:lessd(a,b)][p:posd(c)]
6098 satzd203g:=lemmad5(td(b,c),td(a,c),satzd203a(b,a,c,lemmad6(a,b,l),p)):lessd(td(a,c),td(b,c))
6099 l@[z:zero(c)]
6100 satzd203h:=zeroeq(td(a,c),td(b,c),td02(a,c,z),td02(b,c,z)):eq(td(a,c),td(b,c))
6101 l@[n:negd(c)]
6102 satzd203j:=lemmad6(td(b,c),td(a,c),satzd203c(b,a,c,lemmad6(a,b,l),n)):mored(td(a,c),td(b,c))
6103 p@satzd203k:=lemmad5(td(c,b),td(c,a),satzd203d(b,a,c,lemmad6(a,b,l),p)):lessd(td(c,a),td(c,b))
6104 z@satzd203l:=zeroeq(td(c,a),td(c,b),td01(c,a,z),td01(c,b,z)):eq(td(c,a),td(c,b))
6105 n@satzd203m:=lemmad6(td(c,b),td(c,a),satzd203f(b,a,c,lemmad6(a,b,l),n)):mored(td(c,a),td(c,b))
6106 +*iv4d
6107 @[p:cut][q:cut]
6108 t19:=tris(cut,ts(q,pl(p,1rp)),pl(ts(q,p),ts(q,1rp)),pl(ts(q,p),q),disttp2(q,p,1rp),ispl2(ts(q,1rp),q,ts(q,p),satz151(q))):is(ts(q,pl(p,1rp)),pl(ts(q,p),q))
6109 [r:cut]
6110 t20:=tris(cut,pl(ts(q,pl(p,1rp)),ts(r,1rp)),pl(pl(ts(q,p),q),r),pl(ts(q,p),pl(q,r)),ispl12(ts(q,pl(p,1rp)),pl(ts(q,p),q),ts(r,1rp),r,t19,satz151(r)),asspl1(ts(q,p),q,r)):is(pl(ts(q,pl(p,1rp)),ts(r,1rp)),pl(ts(q,p),pl(q,r)))
6111 t21:=tr3is(cut,pl(ts(r,1rp),ts(q,pl(p,1rp))),pl(ts(q,pl(p,1rp)),ts(r,1rp)),pl(ts(q,p),pl(q,r)),pl(ts(q,p),pl(r,q)),compl(ts(r,1rp),ts(q,pl(p,1rp))),t20,ispl2(pl(q,r),pl(r,q),ts(q,p),compl(q,r))):is(pl(ts(r,1rp),ts(q,pl(p,1rp))),pl(ts(q,p),pl(r,q)))
6112 a@[p:posd(a)]
6113 arp:=rpofpd(a,p):cut
6114 arpi:=ov(1rp,arp):cut
6115 ai:=pdofrp(arpi):dif
6116 t22:=tr3eq(td(a,ai),df(pl(ts(1a,pl(arpi,1rp)),ts(2a,1rp)),pl(ts(1a,1rp),ts(2a,pl(arpi,1rp)))),df(pl(ts(1a,arpi),pl(1a,2a)),pl(ts(2a,arpi),pl(1a,2a))),df(ts(1a,arpi),ts(2a,arpi)),tdeq1a(a,pl(arpi,1rp),1rp),eqsmsd(pl(ts(1a,pl(arpi,1rp)),ts(2a,1rp)),pl(ts(1a,1rp),ts(2a,pl(arpi,1rp))),pl(ts(1a,arpi),pl(1a,2a)),pl(ts(2a,arpi),pl(1a,2a)),t20(arpi,1a,2a),t21(arpi,2a,1a)),lemmad2(ts(1a,arpi),ts(2a,arpi),pl(1a,2a))):eq(td(a,ai),df(ts(1a,arpi),ts(2a,arpi)))
6117 t23:=tr3is(cut,ts(1a,arpi),ts(pl(2a,arp),arpi),pl(ts(2a,arpi),ts(arp,arpi)),pl(ts(2a,arpi),1rp),ists1(1a,pl(2a,arp),arpi,satz140d(1a,2a,p)),disttp1(2a,arp,arpi),ispl2(ts(arp,arpi),1rp,ts(2a,arpi),satz153c(1rp,arp))):is(ts(1a,arpi),pl(ts(2a,arpi),1rp))
6118 t24:=tr3is(cut,pl(ts(1a,arpi),1rp),pl(pl(ts(2a,arpi),1rp),1rp),pl(ts(2a,arpi),pl(1rp,1rp)),pl(pl(1rp,1rp),ts(2a,arpi)),ispl1(ts(1a,arpi),pl(ts(2a,arpi),1rp),1rp,t23),asspl1(ts(2a,arpi),1rp,1rp),compl(ts(2a,arpi),pl(1rp,1rp))):is(pl(ts(1a,arpi),1rp),pl(pl(1rp,1rp),ts(2a,arpi)))
6119 t25:=eqi12(ts(1a,arpi),ts(2a,arpi),pl(1rp,1rp),1rp,t24):eq(df(ts(1a,arpi),ts(2a,arpi)),1df)
6120 t26:=treq(td(a,ai),df(ts(1a,arpi),ts(2a,arpi)),1df,t22,t25):eq(td(a,ai),1df)
6121 t27:=somei(dif,[x:dif]eq(td(a,x),1df),ai,t26):some"l"(dif,[x:dif]eq(td(a,x),1df))
6122 a@[n:negd(a)]
6123 t28:=satzd176c(a,n):posd(m0d(a))
6124 [h:dif][e:eq(td(m0d(a),h),1df)]
6125 t29:=treq(td(a,m0d(h)),td(m0d(a),h),1df,satzd197d(a,h),e):eq(td(a,m0d(h)),1df)
6126 t30:=somei(dif,[x:dif]eq(td(a,x),1df),m0d(h),t29):some"l"(dif,[x:dif]eq(td(a,x),1df))
6127 n@t31:=someapp(dif,[x:dif]eq(td(m0d(a),x),1df),t27(m0d(a),t28),some"l"(dif,[x:dif]eq(td(a,x),1df)),[x:dif][t:eq(td(m0d(a),x),1df)]t30(x,t)):some"l"(dif,[x:dif]eq(td(a,x),1df))
6128 -iv4d
6129 a@[n:not(zero(a))]
6130 lemmad7:=rappd(a,some"l"(dif,[x:dif]eq(td(a,x),1df)),[t:posd(a)]t27".iv4d"(t),th2"l.imp"(zero(a),some"l"(dif,[x:dif]eq(td(a,x),1df)),n),[t:negd(a)]t31".iv4d"(t)):some"l"(dif,[x:dif]eq(td(a,x),1df))
6131 b@[n:not(zero(b))][h:dif][k:dif][e:eq(td(b,h),a)][f:eq(td(b,k),a)]
6132 +4d204
6133 t1:=treq2(td(b,h),td(b,k),a,e,f):eq(td(b,h),td(b,k))
6134 t2:=eqzero(md(td(b,h),td(b,k)),td(b,md(h,k)),distmtd2(b,h,k),satzd182e(td(b,h),td(b,k),t1)):zero(td(b,md(h,k)))
6135 t3:=ore2(zero(b),zero(md(h,k)),satzd192c(b,md(h,k),t2),n):zero(md(h,k))
6136 -4d204
6137 satzd204b:=satzd182b(h,k,t3".4d204"):eq(h,k)
6138 +*4d204
6139 n@[h:dif][e:eq(td(b,h),1df)]
6140 t4:=tr3eq(td(b,td(h,a)),td(td(b,h),a),td(1df,a),a,asstd2(b,h,a),eqtd1(td(b,h),1df,a,e),satzd195b(a)):eq(td(b,td(h,a)),a)
6141 t5:=somei(dif,[x:dif]eq(td(b,x),a),td(h,a),t4):some"l"(dif,[x:dif]eq(td(b,x),a))
6142 -4d204
6143 n@satzd204a:=someapp(dif,[x:dif]eq(td(b,x),1df),lemmad7(b,n),some"l"(dif,[x:dif]eq(td(b,x),a)),[x:dif][t:eq(td(b,x),1df)]t5".4d204"(x,t)):some"l"(dif,[x:dif]eq(td(b,x),a))
6144 @[r:cut][s:cut][m:more(r,s)]
6145 +iv5d
6146 t1:=ismore12(pl(r,pl(1rp,1rp)),pl(pl(r,1rp),1rp),pl(s,pl(1rp,1rp)),pl(pl(s,1rp),1rp),asspl2(r,1rp,1rp),asspl2(s,1rp,1rp),satz134(r,s,pl(1rp,1rp),m)):more(pl(pl(r,1rp),1rp),pl(pl(s,1rp),1rp))
6147 -iv5d
6148 morerpepd:=moredi12(pl(r,1rp),1rp,pl(s,1rp),1rp,t1".iv5d"):mored(pdofrp(r),pdofrp(s))
6149 s@[m:mored(pdofrp(r),pdofrp(s))]
6150 +*iv5d
6151 m@t2:=morede12(pl(r,1rp),1rp,pl(s,1rp),1rp,m):more(pl(pl(r,1rp),1rp),pl(pl(s,1rp),1rp))
6152 -iv5d
6153 m@morerpipd:=satz136a(r,s,pl(1rp,1rp),ismore12(pl(pl(r,1rp),1rp),pl(r,pl(1rp,1rp)),pl(pl(s,1rp),1rp),pl(s,pl(1rp,1rp)),asspl1(r,1rp,1rp),asspl1(s,1rp,1rp),t2".iv5d")):more(r,s)
6154 s@[l:less(r,s)]
6155 lessrpepd:=lemmad5(pdofrp(s),pdofrp(r),morerpepd(s,r,satz122(r,s,l))):lessd(pdofrp(r),pdofrp(s))
6156 s@[l:lessd(pdofrp(r),pdofrp(s))]
6157 lessrpipd:=satz121(s,r,morerpipd(s,r,lemmad6(pdofrp(r),pdofrp(s),l))):less(r,s)
6158 +*iv5d
6159 @i:=1rp:cut
6160 2:=pl(i,i):cut
6161 r@rp1:=pl(r,i):cut
6162 s@sp1:=pl(s,i):cut
6163 rps:=pl(r,s):cut
6164 rs:=ts(r,s):cut
6165 t3:=tris(cut,pl(pl(rp1,sp1),i),pl(pl(rps,2),i),pl(pl(rps,i),2),ispl1(pl(rp1,sp1),pl(rps,2),i,4pl23(r,i,s,i)),3pl23(rps,2,i)):is(pl(pl(rp1,sp1),i),pl(pl(rps,i),2))
6166 t4:=treq(pd(pdofrp(r),pdofrp(s)),df(pl(rp1,sp1),2),pdofrp(rps),pdeq12a(rp1,i,sp1,i),eqi12(pl(rp1,sp1),2,pl(rps,i),i,t3)):eq(pd(pdofrp(r),pdofrp(s)),pdofrp(rps))
6167 -iv5d
6168 s@lemmad8:=t4".iv5d":eq(pd(pdofrp(r),pdofrp(s)),pdofrp(pl(r,s)))
6169 +*iv5d
6170 s@t5:=tris(cut,ts(r,sp1),pl(rs,ts(r,i)),pl(rs,r),disttp2(r,s,i),ispl2(ts(r,i),r,rs,satz151(r))):is(ts(r,sp1),pl(rs,r))
6171 t6:=tr4is(cut,ts(rp1,sp1),pl(ts(r,sp1),ts(i,sp1)),pl(pl(rs,r),sp1),pl(pl(pl(rs,r),s),i),pl(pl(rs,rps),i),disttp1(r,i,sp1),ispl12(ts(r,sp1),pl(rs,r),ts(i,sp1),sp1,t5,satz151b(sp1)),asspl2(pl(rs,r),s,i),ispl1(pl(pl(rs,r),s),pl(rs,rps),i,asspl1(rs,r,s))):is(ts(rp1,sp1),pl(pl(rs,rps),i))
6172 t7:=tr3is(cut,pl(ts(rp1,sp1),ts(i,i)),pl(pl(pl(rs,rps),i),i),pl(pl(rs,rps),2),pl(rs,pl(rps,2)),ispl12(ts(rp1,sp1),pl(pl(rs,rps),i),ts(i,i),i,t6,satz151(i)),asspl1(pl(rs,rps),i,i),asspl1(rs,rps,2)):is(pl(ts(rp1,sp1),ts(i,i)),pl(rs,pl(rps,2)))
6173 t8:=tris(cut,pl(ts(rp1,i),ts(i,sp1)),pl(rp1,sp1),pl(rps,2),ispl12(ts(rp1,i),rp1,ts(i,sp1),sp1,satz151(rp1),satz151b(sp1)),4pl23(r,i,s,i)):is(pl(ts(rp1,i),ts(i,sp1)),pl(rps,2))
6174 t9:=tris(cut,pl(pl(rs,i),pl(ts(rp1,i),ts(i,sp1))),pl(pl(rs,i),pl(rps,2)),pl(pl(rs,pl(rps,2)),i),ispl2(pl(ts(rp1,i),ts(i,sp1)),pl(rps,2),pl(rs,i),t8),3pl23(rs,i,pl(rps,2))):is(pl(pl(rs,i),pl(ts(rp1,i),ts(i,sp1))),pl(pl(rs,pl(rps,2)),i))
6175 t10:=tris2(cut,pl(pl(ts(rp1,sp1),ts(i,i)),i),pl(pl(rs,i),pl(ts(rp1,i),ts(i,sp1))),pl(pl(rs,pl(rps,2)),i),ispl1(pl(ts(rp1,sp1),ts(i,i)),pl(rs,pl(rps,2)),i,t7),t9):is(pl(pl(ts(rp1,sp1),ts(i,i)),i),pl(pl(rs,i),pl(ts(rp1,i),ts(i,sp1))))
6176 t11:=treq(td(pdofrp(r),pdofrp(s)),df(pl(ts(rp1,sp1),ts(i,i)),pl(ts(rp1,i),ts(i,sp1))),pdofrp(rs),tdeq12a(rp1,i,sp1,i),eqi12(pl(ts(rp1,sp1),ts(i,i)),pl(ts(rp1,i),ts(i,sp1)),pl(rs,i),i,t10)):eq(td(pdofrp(r),pdofrp(s)),pdofrp(rs))
6177 -iv5d
6178 s@lemmad9:=t11".iv5d":eq(td(pdofrp(r),pdofrp(s)),pdofrp(ts(r,s)))
6179 @[r:cut][s0:set(cut)]
6180 in:=esti(cut,r,s0):'prop'
6181 @[s0:set(cut)][t0:set(cut)][p0:all([x:cut]or(in(x,s0),in(x,t0)))][p1a:nonempty(cut,s0)][p1b:nonempty(cut,t0)][p2:all([x:cut][t:in(x,s0)]all([y:cut][u:in(y,t0)]less(x,y)))]
6182 +5p205
6183 t0@[r:cut]
6184 prop1:=all([x:cut][t:less(x,r)]in(x,s0)):'prop'
6185 prop2:=all([x:cut][t:more(x,r)]in(x,t0)):'prop'
6186 prop3:=and(prop1,prop2):'prop'
6187 p2@[r1:cut][r2:cut][pr1:prop3(r1)][pr2:prop3(r2)]
6188 t1:=ande2(prop1(r1),prop2(r1),pr1):prop2(r1)
6189 t2:=ande1(prop1(r2),prop2(r2),pr2):prop1(r2)
6190 [l:less(r1,r2)][x0:rat]
6191 rx:=rpofrt(x0):cut
6192 [l1:less(r1,rx)][l2:less(rx,r2)]
6193 t3:=<l2><rx>t2:in(rx,s0)
6194 t4:=<satz122(r1,rx,l1)><rx>t1:in(rx,t0)
6195 t5:=<refis(cut,rx)>ec3e31(is(rx,rx),more(rx,rx),less(rx,rx),satz123b(rx,rx),<t4><rx><t3><rx>p2):con
6196 l@t6:=satz159app(r1,r2,l,con,[x:rat][t:less(r1,rpofrt(x))][u:less(rpofrt(x),r2)]t5(x,t,u)):con
6197 pr2@t7:=[t:less(r1,r2)]t6(t):not(less(r1,r2))
6198 t8:=[t:more(r1,r2)]t6(r2,r1,pr2,pr1,satz121(r1,r2,t)):not(more(r1,r2))
6199 t9:=or3e1(is(r1,r2),more(r1,r2),less(r1,r2),satz123a(r1,r2),t8,t7):is(r1,r2)
6200 p2@t10:=[x:cut][y:cut][t:prop3(x)][u:prop3(y)]t9(x,y,t,u):amone(cut,[x:cut]prop3(x))
6201 [x0:rat]
6202 schnittprop:=some([y:cut]and(in(y,s0),lrt(y,x0))):'prop'
6203 p2@schnittset:=setof(rat,[x:rat]schnittprop(x)):set(rat)
6204 [r:cut][i:in(r,s0)][x0:rat][lx:lrt(r,x0)]
6205 t11:=andi(in(r,s0),lrt(r,x0),i,lx):and(in(r,s0),lrt(r,x0))
6206 t12:=somei(cut,[y:cut]and(in(y,s0),lrt(y,x0)),r,t11):schnittprop(x0)
6207 t13:=estii(rat,[x:rat]schnittprop(x),x0,t12):in"rt"(x0,schnittset)
6208 r@[i:in(r,t0)][x0:rat][ux:urt(r,x0)][s:cut][j:in(s,s0)]
6209 t14:=satz122(s,r,<i><r><j><s>p2):more(r,s)
6210 t15:=satz158b(r,x0,ux):moreis(rpofrt(x0),r)
6211 t16:=moreisi1(rpofrt(x0),s,satz127c(rpofrt(x0),r,s,t15,t14)):moreis(rpofrt(x0),s)
6212 t17:=satz158d(s,x0,t16):urt(s,x0)
6213 s@t18:=weli(ec(in(s,s0),lrt(s,x0)),[t:in(s,s0)]t17(t)):not(and(in(s,s0),lrt(s,x0)))
6214 ux@t19:=th5"l.some"(cut,[y:cut]and(in(y,s0),lrt(y,x0)),[y:cut]t18(y)):not(schnittprop(x0))
6215 t20:=th3"l.imp"(in"rt"(x0,schnittset),schnittprop(x0),t19,[t:in"rt"(x0,schnittset)]estie(rat,[x:rat]schnittprop(x),x0,t)):not(in"rt"(x0,schnittset))
6216 p2@[x0:rat][i:in"rt"(x0,schnittset)][y0:rat][l:less"rt"(y0,x0)]
6217 i@t21:=estie(rat,[x:rat]schnittprop(x),x0,i):schnittprop(x0)
6218 l@[r:cut][a:and(in(r,s0),lrt(r,x0))]
6219 t22:=ande1(in(r,s0),lrt(r,x0),a):in(r,s0)
6220 t23:=ande2(in(r,s0),lrt(r,x0),a):lrt(r,x0)
6221 t24:=satz120(r,x0,t23,y0,l):lrt(r,y0)
6222 t25:=andi(in(r,s0),lrt(r,y0),t22,t24):and(in(r,s0),lrt(r,y0))
6223 t26:=somei(cut,[y:cut]and(in(y,s0),lrt(y,y0)),r,t25):schnittprop(y0)
6224 l@t27:=someapp(cut,[y:cut]and(in(y,s0),lrt(y,x0)),t21,schnittprop(y0),[y:cut][r:and(in(y,s0),lrt(y,x0))]t26(y,r)):schnittprop(y0)
6225 t28:=estii(rat,[x:rat]schnittprop(x),y0,t27):in"rt"(y0,schnittset)
6226 i@[r:cut][a:and(in(r,s0),lrt(r,x0))]
6227 t29:=ande1(in(r,s0),lrt(r,x0),a):in(r,s0)
6228 t30:=ande2(in(r,s0),lrt(r,x0),a):lrt(r,x0)
6229 [y0:rat][ly:lrt(r,y0)][l:less"rt"(x0,y0)]
6230 t31:=andi(in(r,s0),lrt(r,y0),t29,ly):and(in(r,s0),lrt(r,y0))
6231 t32:=somei(cut,[y:cut]and(in(y,s0),lrt(y,y0)),r,t31):schnittprop(y0)
6232 t33:=estii(rat,[x:rat]schnittprop(x),y0,t32):in"rt"(y0,schnittset)
6233 t34:=satz83(x0,y0,l):more"rt"(y0,x0)
6234 t35:=andi(in"rt"(y0,schnittset),more"rt"(y0,x0),t33,t34):and(in"rt"(y0,schnittset),more"rt"(y0,x0))
6235 t36:=somei(rat,[y:rat]and(in"rt"(y,schnittset),more"rt"(y,x0)),y0,t35):some"rt"([y:rat]and(in"rt"(y,schnittset),more"rt"(y,x0)))
6236 a@t37:=cutapp3(r,x0,t30,some"rt"([y:rat]and(in"rt"(y,schnittset),more"rt"(y,x0))),[y:rat][t:lrt(r,y)][u:less"rt"(x0,y)]t36(y,t,u)):some"rt"([y:rat]and(in"rt"(y,schnittset),more"rt"(y,x0)))
6237 i@t38:=someapp(cut,[y:cut]and(in(y,s0),lrt(y,x0)),t21,some"rt"([y:rat]and(in"rt"(y,schnittset),more"rt"(y,x0))),[y:cut][t:and(in(y,s0),lrt(y,x0))]t37(y,t)):some"rt"([y:rat]and(in"rt"(y,schnittset),more"rt"(y,x0)))
6238 p2@[r:cut][i:in(r,s0)][x0:rat][lx:lrt(r,x0)][s:cut][j:in(s,t0)][y0:rat][uy:urt(s,y0)]
6239 t39:=cut2(schnittset,x0,t13(r,i,x0,lx),y0,t20(s,j,y0,uy),[x:rat][t:in"rt"(x,schnittset)][y:rat][u:less"rt"(y,x)]t28(x,t,y,u),[x:rat][t:in"rt"(x,schnittset)]t38(x,t)):cutprop(schnittset)
6240 j@t40:=cutapp1b(s,cutprop(schnittset),[x:rat][t:urt(s,x)]t39(x,t)):cutprop(schnittset)
6241 lx@t41:=nonemptyapp(cut,t0,p1b,cutprop(schnittset),[y:cut][t:in(y,t0)]t40(y,t)):cutprop(schnittset)
6242 i@t42:=cutapp1a(r,cutprop(schnittset),[x:rat][t:lrt(r,x)]t41(x,t)):cutprop(schnittset)
6243 p2@t43:=nonemptyapp(cut,s0,p1a,cutprop(schnittset),[y:cut][t:in(y,s0)]t42(y,t)):cutprop(schnittset)
6244 snt:=cutof(schnittset,t43):cut
6245 [r:cut][l:less(r,snt)][x0:rat][ux:urt(r,x0)][lx:lrt(snt,x0)]
6246 t44:=ini(schnittset,t43,x0,lx):in"rt"(x0,schnittset)
6247 t45:=estie(rat,[x:rat]schnittprop(x),x0,t44):schnittprop(x0)
6248 [s:cut][a:and(in(s,s0),lrt(s,x0))]
6249 t46:=ande1(in(s,s0),lrt(s,x0),a):in(s,s0)
6250 t47:=ande2(in(s,s0),lrt(s,x0),a):lrt(s,x0)
6251 t48:=andi(urt(r,x0),lrt(s,x0),ux,t47):and(urt(r,x0),lrt(s,x0))
6252 t49:=somei(rat,[x:rat]and(urt(r,x),lrt(s,x)),x0,t48):less(r,s)
6253 t50:=ec3e23(is(s,r),more(s,r),less(s,r),satz123b(s,r),satz122(r,s,t49)):not(less(s,r))
6254 t51:=th3"l.imp"(in(r,t0),less(s,r),t50,<r><t46><s>p2):not(in(r,t0))
6255 t52:=ore1(in(r,s0),in(r,t0),<r>p0,t51):in(r,s0)
6256 lx@t53:=someapp(cut,[y:cut]and(in(y,s0),lrt(y,x0)),t45,in(r,s0),[y:cut][t:and(in(y,s0),lrt(y,x0))]t52(y,t)):in(r,s0)
6257 l@t54:=lessapp(r,snt,l,in(r,s0),[x:rat][t:urt(r,x)][u:lrt(snt,x)]t53(x,t,u)):in(r,s0)
6258 r@[m:more(r,snt)][x0:rat][lx:lrt(r,x0)][ux:urt(snt,x0)][i:in(r,s0)]
6259 t55:=andi(in(r,s0),lrt(r,x0),i,lx):and(in(r,s0),lrt(r,x0))
6260 t56:=somei(cut,[y:cut]and(in(y,s0),lrt(y,x0)),r,t55):schnittprop(x0)
6261 t57:=estii(rat,[x:rat]schnittprop(x),x0,t56):in"rt"(x0,schnittset)
6262 t58:=ine(schnittset,t43,x0,t57):lrt(snt,x0)
6263 ux@t59:=th3"l.imp"(in(r,s0),lrt(snt,x0),ux,[t:in(r,s0)]t58(t)):not(in(r,s0))
6264 t60:=ore2(in(r,s0),in(r,t0),<r>p0,t59):in(r,t0)
6265 m@t61:=moreapp(r,snt,m,in(r,t0),[x:rat][t:lrt(r,x)][u:urt(snt,x)]t60(x,t,u)):in(r,t0)
6266 p2@t62:=andi(prop1(snt),prop2(snt),[x:cut][t:less(x,snt)]t54(x,t),[x:cut][t:more(x,snt)]t61(x,t)):prop3(snt)
6267 t63:=somei(cut,[x:cut]prop3(x),snt,t62):some([x:cut]prop3(x))
6268 -5p205
6269 satzp205:=onei(cut,[x:cut]prop3".5p205"(x),t10".5p205",t63".5p205"):one([x:cut]and(all([y:cut][t:less(y,x)]in(y,s0)),all([y:cut][t:more(y,x)]in(y,t0))))
6270 schnitt:=ind(cut,[x:cut]prop3".5p205"(x),satzp205):cut
6271 satzp205a:=ande1(prop1".5p205"(schnitt),prop2".5p205"(schnitt),oneax(cut,[x:cut]prop3".5p205"(x),satzp205)):all([x:cut][t:less(x,schnitt)]in(x,s0))
6272 satzp205b:=ande2(prop1".5p205"(schnitt),prop2".5p205"(schnitt),oneax(cut,[x:cut]prop3".5p205"(x),satzp205)):all([x:cut][t:more(x,schnitt)]in(x,t0))
6273 @[r:cut][s:cut]
6274 +ivad
6275 @i:=1rp:cut
6276 r@r1:=pl(r,i):cut
6277 s@s1:=pl(s,i):cut
6278 rps:=pl(r,s):cut
6279 @2:=pl(i,i):cut
6280 s@t1:=pdeq12a(r1,i,s1,i):eq(pd(pdofrp(r),pdofrp(s)),df(pl(r1,s1),2))
6281 t2:=tris(cut,pl(r1,s1),pl(rps,2),pl(pl(rps,i),i),4pl23(r,i,s,i),asspl2(rps,i,i)):is(pl(r1,s1),pl(pl(rps,i),i))
6282 t3:=ispl1(pl(r1,s1),pl(pl(rps,i),i),i,t2):is(pl(pl(r1,s1),i),pl(pl(pl(rps,i),i),i))
6283 t4:=tris(cut,pl(pl(r1,s1),i),pl(pl(pl(rps,i),i),i),pl(pl(rps,i),2),t3,asspl1(pl(rps,i),i,i)):is(pl(pl(r1,s1),i),pl(pl(rps,i),2))
6284 t5:=eqi12(pl(r1,s1),2,pl(rps,i),i,t4):eq(df(pl(r1,s1),2),pdofrp(rps))
6285 -ivad
6286 lemmaivad1:=treq(pd(pdofrp(r),pdofrp(s)),df(pl(pl(r,1rp),pl(s,1rp)),pl(1rp,1rp)),pdofrp(pl(r,s)),t1".ivad",t5".ivad"):eq(pd(pdofrp(r),pdofrp(s)),pdofrp(pl(r,s)))
6287 +*ivad
6288 s@rs:=ts(r,s):cut
6289 t6:=tdeq12a(r1,i,s1,i):eq(td(pdofrp(r),pdofrp(s)),df(pl(ts(r1,s1),ts(i,i)),pl(ts(r1,i),ts(i,s1))))
6290 t7:=tris(cut,ts(r1,s),pl(rs,ts(i,s)),pl(rs,s),disttp1(r,i,s),ispl2(ts(i,s),s,rs,satz151b(s))):is(ts(r1,s),pl(rs,s))
6291 t8:=tr3is(cut,ts(r1,s1),pl(ts(r1,s),ts(r1,i)),pl(pl(rs,s),r1),pl(pl(rs,i),rps),disttp2(r1,s,i),ispl12(ts(r1,s),pl(rs,s),ts(r1,i),r1,t7,satz151(r1)),4pl24(rs,s,r,i)):is(ts(r1,s1),pl(pl(rs,i),rps))
6292 t9:=tris(cut,pl(ts(r1,s1),ts(i,i)),pl(pl(pl(rs,i),rps),i),pl(pl(rs,i),pl(rps,i)),ispl12(ts(r1,s1),pl(pl(rs,i),rps),ts(i,i),i,t8,satz151(i)),asspl1(pl(rs,i),rps,i)):is(pl(ts(r1,s1),ts(i,i)),pl(pl(rs,i),pl(rps,i)))
6293 t10:=tris(cut,pl(pl(ts(r1,s1),ts(i,i)),i),pl(pl(pl(rs,i),pl(rps,i)),i),pl(pl(rs,i),pl(pl(rps,i),i)),ispl1(pl(ts(r1,s1),ts(i,i)),pl(pl(rs,i),pl(rps,i)),i,t9),asspl1(pl(rs,i),pl(rps,i),i)):is(pl(pl(ts(r1,s1),ts(i,i)),i),pl(pl(rs,i),pl(pl(rps,i),i)))
6294 t11:=tr3is(cut,pl(pl(rps,i),i),pl(rps,2),pl(r1,s1),pl(ts(r1,i),ts(i,s1)),asspl1(rps,i,i),4pl23(r,s,i,i),ispl12(r1,ts(r1,i),s1,ts(i,s1),satz151a(r1),satz151c(s1))):is(pl(pl(rps,i),i),pl(ts(r1,i),ts(i,s1)))
6295 t12:=tris(cut,pl(pl(ts(r1,s1),ts(i,i)),i),pl(pl(rs,i),pl(pl(rps,i),i)),pl(pl(rs,i),pl(ts(r1,i),ts(i,s1))),t10,ispl2(pl(pl(rps,i),i),pl(ts(r1,i),ts(i,s1)),pl(rs,i),t11)):is(pl(pl(ts(r1,s1),ts(i,i)),i),pl(pl(rs,i),pl(ts(r1,i),ts(i,s1))))
6296 t13:=eqi12(pl(ts(r1,s1),ts(i,i)),pl(ts(r1,i),ts(i,s1)),pl(rs,i),i,t12):eq(df(pl(ts(r1,s1),ts(i,i)),pl(ts(r1,i),ts(i,s1))),pdofrp(rs))
6297 -ivad
6298 s@lemmaivad2:=treq(td(pdofrp(r),pdofrp(s)),df(pl(ts(pl(r,1rp),pl(s,1rp)),ts(1rp,1rp)),pl(ts(pl(r,1rp),1rp),ts(1rp,pl(s,1rp)))),pdofrp(ts(r,s)),t6".ivad",t13".ivad"):eq(td(pdofrp(r),pdofrp(s)),pdofrp(ts(r,s)))
6299 [m:mored(pdofrp(r),pdofrp(s))]
6300 +*ivad
6301 m@t14:=morede12(r1,i,s1,i,m):more(pl(r1,i),pl(s1,i))
6302 t15:=satz136a(r1,s1,i,t14):more(r1,s1)
6303 -ivad
6304 m@lemmaivad3:=satz136a(r,s,1rp,t15".ivad"):more(r,s)
6305 @[c:dif][a:dif][b:dif][n:not(negd(a))][o:not(negd(b))][e:eq(td(a,a),c)][f:eq(td(b,b),c)]
6306 +d161
6307 t1:=treq2(td(a,a),td(b,b),c,e,f):eq(td(a,a),td(b,b))
6308 t2:=treq(pd(md(td(a,a),td(a,b)),td(b,a)),pd(md(td(a,a),td(a,b)),td(a,b)),td(a,a),eqpd2(td(b,a),td(a,b),md(td(a,a),td(a,b)),comtd(b,a)),pdmd(td(a,a),td(a,b))):eq(pd(md(td(a,a),td(a,b)),td(b,a)),td(a,a))
6309 t3:=tr4eq(td(pd(a,b),md(a,b)),pd(td(a,md(a,b)),td(b,md(a,b))),pd(md(td(a,a),td(a,b)),md(td(b,a),td(b,b))),md(pd(md(td(a,a),td(a,b)),td(b,a)),td(b,b)),md(td(a,a),td(b,b)),disttpd1(a,b,md(a,b)),eqpd12(td(a,md(a,b)),md(td(a,a),td(a,b)),td(b,md(a,b)),md(td(b,a),td(b,b)),disttmd2(a,a,b),disttmd2(b,a,b)),asspd2(md(td(a,a),td(a,b)),td(b,a),m0d(td(b,b))),eqmd1(pd(md(td(a,a),td(a,b)),td(b,a)),td(a,a),td(b,b),t2)):eq(td(pd(a,b),md(a,b)),md(td(a,a),td(b,b)))
6310 t4:=eqzero(md(td(a,a),td(b,b)),td(pd(a,b),md(a,b)),symeq(td(pd(a,b),md(a,b)),md(td(a,a),td(b,b)),t3),satzd182e(td(a,a),td(b,b),t1)):zero(td(pd(a,b),md(a,b)))
6311 t5:=satzd192c(pd(a,b),md(a,b),t4):or(zero(pd(a,b)),zero(md(a,b)))
6312 [z:zero(a)]
6313 t6:=eqzero(td(a,a),td(b,b),t1,td01(a,a,z)):zero(td(b,b))
6314 t7:=th1"l.imp"(zero(b),zero(b),refimp(zero(b)),satzd192c(b,b,t6)):zero(b)
6315 t8:=zeroeq(a,b,z,t7):eq(a,b)
6316 f@[p:not(zero(a))]
6317 t9:=or3e2(zero(a),posd(a),negd(a),axrdo(a),n,p):posd(a)
6318 t10:=th3"l.imp"(zero(b),zero(a),p,[t:zero(b)]t7(b,a,o,n,f,e,t)):not(zero(b))
6319 t11:=t9(b,a,o,n,f,e,t10):posd(b)
6320 t12:=pnot0d(pd(a,b),ppd(a,b,t9,t11)):not(zero(pd(a,b)))
6321 t13:=ore2(zero(pd(a,b)),zero(md(a,b)),t5,t12):zero(md(a,b))
6322 t14:=satzd182b(a,b,t13):eq(a,b)
6323 -d161
6324 satzd161b:=th1"l.imp"(zero(a),eq(a,b),[t:zero(a)]t8".d161"(t),[t:not(zero(a))]t14".d161"(t)):eq(a,b)
6325 c@[n:not(negd(c))]
6326 +*d161
6327 n@[z:zero(c)]
6328 t15:=zeroeq(td(c,c),c,td01(c,c,z),z):eq(td(c,c),c)
6329 t16:=andi(not(negd(c)),eq(td(c,c),c),n,t15):and(not(negd(c)),eq(td(c,c),c))
6330 t17:=somei(dif,[x:dif]and(not(negd(x)),eq(td(x,x),c)),c,t16):some"l"(dif,[x:dif]and(not(negd(x)),eq(td(x,x),c)))
6331 n@[o:not(zero(c))]
6332 t18:=or3e2(zero(c),posd(c),negd(c),axrdo(c),n,o):posd(c)
6333 crp:=rpofpd(c,t18):cut
6334 srp:=sqrt(crp):cut
6335 s:=pdofrp(srp):dif
6336 t19:=tr3eq(td(s,s),pdofrp(ts(srp,srp)),pdofrp(crp),c,lemmaivad2(srp,srp),isrpepd(ts(srp,srp),crp,thsqrt1(crp)),eqpdrp2(c,t18)):eq(td(s,s),c)
6337 t20:=andi(not(negd(s)),eq(td(s,s),c),pnotnd(s,posdirp(srp)),t19):and(not(negd(s)),eq(td(s,s),c))
6338 t21:=somei(dif,[x:dif]and(not(negd(x)),eq(td(x,x),c)),s,t20):some"l"(dif,[x:dif]and(not(negd(x)),eq(td(x,x),c)))
6339 -d161
6340 n@satzd161a:=th1"l.imp"(zero(c),some"l"(dif,[x:dif]and(not(negd(x)),eq(td(x,x),c))),[t:zero(c)]t17".d161"(t),[t:not(zero(c))]t21".d161"(t)):some"l"(dif,[x:dif]and(not(negd(x)),eq(td(x,x),c)))
6341 @[a:dif][i:intd(a)]
6342 +intd
6343 [z:zero(a)]
6344 t1:=ori1(zero(absd(a)),natd(absd(absd(a))),satzd166f(a,z)):intd(absd(a))
6345 i@[n:natd(absd(a))]
6346 t2:=natintd(absd(a),n):intd(absd(a))
6347 -intd
6348 intabsd:=orapp(zero(a),natd(absd(a)),intd(absd(a)),i,[t:zero(a)]t1".intd"(t),[t:natd(absd(a))]t2".intd"(t)):intd(absd(a))
6349 +*intd
6350 n@t4:=eqnatd(absd(a),absd(m0d(a)),satzd178a(a),n):natd(absd(m0d(a)))
6351 -intd
6352 i@intm0d:=th9"l.or"(zero(a),natd(absd(a)),zero(m0d(a)),natd(absd(m0d(a))),i,[t:zero(a)]satzd176b(a,t),[t:natd(absd(a))]t4".intd"(t)):intd(m0d(a))
6353 a@[b:dif][i:intd(a)][j:intd(b)]
6354 +*intd
6355 j@[z:zero(a)]
6356 t5:=symeq(pd(a,b),b,pd01(a,b,z)):eq(b,pd(a,b))
6357 t6:=eqintd(b,pd(a,b),t5,j):intd(pd(a,b))
6358 j@[z:zero(b)]
6359 t7:=symeq(pd(a,b),a,pd02(a,b,z)):eq(a,pd(a,b))
6360 t8:=eqintd(a,pd(a,b),t7,i):intd(pd(a,b))
6361 a@[i:intd(a)][p:posd(a)]
6362 t9:=<p>ande2(posd(a),[t:posd(a)]natrp(rpofpd(a,t)),posintnatd(a,p,i)):natrp(rpofpd(a,p))
6363 j@[pp:posd(pd(a,b))]
6364 apb1:=rpofpd(pd(a,b),pp):cut
6365 [p:posd(a)]
6366 a1:=rpofpd(a,p):cut
6367 [q:posd(b)]
6368 b1:=rpofpd(b,q):cut
6369 t10:=natpl(a1,t9(a,i,p),b1,t9(b,j,q)):natrp(pl(a1,b1))
6370 t11:=eqpd12(a,pdofrp(a1),b,pdofrp(b1),eqpdrp1(a,p),eqpdrp1(b,q)):eq(pd(a,b),pd(pdofrp(a1),pdofrp(b1)))
6371 t12:=treq(pd(a,b),pd(pdofrp(a1),pdofrp(b1)),pdofrp(pl(a1,b1)),t11,lemmaivad1(a1,b1)):eq(pd(a,b),pdofrp(pl(a1,b1)))
6372 t13:=tris(cut,apb1,rpofpd(pdofrp(pl(a1,b1)),posdirp(pl(a1,b1))),pl(a1,b1),eqpderp(pd(a,b),pp,pdofrp(pl(a1,b1)),posdirp(pl(a1,b1)),t12),isrppd2(pl(a1,b1))):is(apb1,pl(a1,b1))
6373 t14:=isp1(cut,[t:cut]natrp(t),pl(a1,b1),apb1,t10,t13):natrp(apb1)
6374 t15:=andi(posd(pd(a,b)),[t:posd(pd(a,b))]natrp(apb1(t)),pp,[t:posd(pd(a,b))]t14(t,p,q)):natd(pd(a,b))
6375 t16:=natintd(pd(a,b),t15):intd(pd(a,b))
6376 p@[n:negd(b)]
6377 t17:=satzd176c(b,n):posd(m0d(b))
6378 b2:=rpofpd(m0d(b),t17):cut
6379 t18:=eqpd2(b,m0d(m0d(b)),a,satzd177a(b)):eq(pd(a,b),md(a,m0d(b)))
6380 t19:=eqposd(pd(a,b),md(a,m0d(b)),t18,pp):posd(md(a,m0d(b)))
6381 t20:=satzd182a(a,m0d(b),t19):mored(a,m0d(b))
6382 t21:=eqmored12(a,pdofrp(a1),m0d(b),pdofrp(b2),eqpdrp1(a,p),eqpdrp1(m0d(b),t17),t20):mored(pdofrp(a1),pdofrp(b2))
6383 t22:=lemmaivad3(a1,b2,t21):more(a1,b2)
6384 t23:=natmn(a1,t9(a,i,p),b2,t9(m0d(b),intm0d(b,j),t17),t22):natrp(mn(a1,b2,t22))
6385 t24:=eqpd12(m0d(b),pdofrp(b2),pd(a,b),pdofrp(apb1),eqpdrp1(m0d(b),t17),eqpdrp1(pd(a,b),pp)):eq(pd(m0d(b),pd(a,b)),pd(pdofrp(b2),pdofrp(apb1)))
6386 t25:=tr4eq(a,md(pd(a,b),b),pd(m0d(b),pd(a,b)),pd(pdofrp(b2),pdofrp(apb1)),pdofrp(pl(b2,apb1)),symeq(md(pd(a,b),b),a,mdpd(a,b)),compd(pd(a,b),m0d(b)),t24,lemmaivad1(b2,apb1)):eq(a,pdofrp(pl(b2,apb1)))
6387 t26:=tris2(cut,pl(b2,apb1),a1,rpofpd(pdofrp(pl(b2,apb1)),posdirp(pl(b2,apb1))),isrppd1(pl(b2,apb1)),eqpderp(a,p,pdofrp(pl(b2,apb1)),posdirp(pl(b2,apb1)),t25)):is(pl(b2,apb1),a1)
6388 t27:=satz140g(a1,b2,apb1,t22,t26):is(apb1,mn(a1,b2,t22))
6389 t28:=isp1(cut,[t:cut]natrp(t),mn(a1,b2,t22),apb1,t23,t27):natrp(apb1)
6390 t29:=andi(posd(pd(a,b)),[t:posd(pd(a,b))]natrp(apb1(t)),pp,[t:posd(pd(a,b))]t28(t,p,n)):natd(pd(a,b))
6391 t30:=natintd(pd(a,b),t29):intd(pd(a,b))
6392 p@t31:=rappd(b,intd(pd(a,b)),[t:posd(b)]t16(t),[t:zero(b)]t8(t),[t:negd(b)]t30(t)):intd(pd(a,b))
6393 pp@[n:negd(a)]
6394 t31a:=th3"l.imp"(negd(b),negd(pd(a,b)),pnotnd(pd(a,b),pp),[t:negd(b)]npd(a,b,n,t)):not(negd(b))
6395 t32:=th3"l.imp"(zero(b),negd(pd(a,b)),pnotnd(pd(a,b),pp),[t:zero(b)]eqnegd(a,pd(a,b),symeq(pd(a,b),a,pd02(a,b,t)),n)):not(zero(b))
6396 t33:=or3e2(zero(b),posd(b),negd(b),axrdo(b),t31a,t32):posd(b)
6397 t34:=eqposd(pd(a,b),pd(b,a),compd(a,b),pp):posd(pd(b,a))
6398 t35:=t30(b,a,j,i,t34,t33,n):intd(pd(b,a))
6399 t36:=eqintd(pd(b,a),pd(a,b),compd(b,a),t35):intd(pd(a,b))
6400 pp@t37:=rappd(a,intd(pd(a,b)),[t:posd(a)]t31(t),[t:zero(a)]t6(t),[t:negd(a)]t36(t)):intd(pd(a,b))
6401 j@[0p:zero(pd(a,b))]
6402 t38:=intdi0(pd(a,b),0p):intd(pd(a,b))
6403 j@[np:negd(pd(a,b))]
6404 t39:=satzd176c(pd(a,b),np):posd(m0d(pd(a,b)))
6405 t40:=eqposd(m0d(pd(a,b)),pd(m0d(a),m0d(b)),satzd180(a,b),t39):posd(pd(m0d(a),m0d(b)))
6406 t41:=t37(m0d(a),m0d(b),intm0d(a,i),intm0d(b,j),t40):intd(pd(m0d(a),m0d(b)))
6407 t42:=eqintd(pd(m0d(a),m0d(b)),m0d(pd(a,b)),satzd180a(a,b),t41):intd(m0d(pd(a,b)))
6408 t43:=intm0d(m0d(pd(a,b)),t42):intd(m0d(m0d(pd(a,b))))
6409 t44:=eqintd(m0d(m0d(pd(a,b))),pd(a,b),satzd177(pd(a,b)),t43):intd(pd(a,b))
6410 -intd
6411 j@intpd:=rappd(pd(a,b),intd(pd(a,b)),[t:posd(pd(a,b))]t37".intd"(t),[t:zero(pd(a,b))]t38".intd"(t),[t:negd(pd(a,b))]t44".intd"(t)):intd(pd(a,b))
6412 intmd:=intpd(a,m0d(b),i,intm0d(b,j)):intd(md(a,b))
6413 +*intd
6414 j@[n:not(zero(td(a,b)))]
6415 t45:=th3"l.imp"(zero(a),zero(td(a,b)),n,[t:zero(a)]td01(a,b,t)):not(zero(a))
6416 t46:=th3"l.imp"(zero(b),zero(td(a,b)),n,[t:zero(b)]td02(a,b,t)):not(zero(b))
6417 t47:=satzd166e(a,t45):posd(absd(a))
6418 a3:=rpofpd(absd(a),t47):cut
6419 t48:=satzd166e(b,t46):posd(absd(b))
6420 b3:=rpofpd(absd(b),t48):cut
6421 t49:=natts(a3,t9(absd(a),intabsd(a,i),t47),b3,t9(absd(b),intabsd(b,j),t48)):natrp(ts(a3,b3))
6422 t50:=satzd166e(td(a,b),n):posd(absd(td(a,b)))
6423 [p:posd(absd(td(a,b)))]
6424 atb3:=rpofpd(absd(td(a,b)),p):cut
6425 t51:=eqtd12(absd(a),pdofrp(a3),absd(b),pdofrp(b3),eqpdrp1(absd(a),t47),eqpdrp1(absd(b),t48)):eq(td(absd(a),absd(b)),td(pdofrp(a3),pdofrp(b3)))
6426 t52:=tr3eq(absd(td(a,b)),td(absd(a),absd(b)),td(pdofrp(a3),pdofrp(b3)),pdofrp(ts(a3,b3)),satzd193(a,b),t51,lemmaivad2(a3,b3)):eq(absd(td(a,b)),pdofrp(ts(a3,b3)))
6427 t53:=tris2(cut,atb3,ts(a3,b3),rpofpd(pdofrp(ts(a3,b3)),posdirp(ts(a3,b3))),eqpderp(absd(td(a,b)),p,pdofrp(ts(a3,b3)),posdirp(ts(a3,b3)),t52),isrppd1(ts(a3,b3))):is(atb3,ts(a3,b3))
6428 t54:=isp1(cut,[t:cut]natrp(t),ts(a3,b3),atb3,t49,t53):natrp(atb3)
6429 n@t55:=andi(posd(absd(td(a,b))),[t:posd(absd(td(a,b)))]natrp(atb3(t)),t50,[t:posd(absd(td(a,b)))]t54(t)):natd(absd(td(a,b)))
6430 -intd
6431 j@inttd:=[t:not(zero(td(a,b)))]t55".intd"(t):intd(td(a,b))
6432 +r
6433 @eq:=[x:dif][y:dif]eq"rp"(x,y):[x:dif][y:dif]'prop'
6434 refeq:=[x:dif]refeq"rp"(x):[x:dif]<x><x>eq
6435 symeq:=[x:dif][y:dif][t:<y><x>eq]symeq"rp"(x,y,t):[x:dif][y:dif][t:<y><x>eq]<x><y>eq
6436 treq:=[x:dif][y:dif][z:dif][t:<y><x>eq][u:<z><y>eq]treq"rp"(x,y,z,t,u):[x:dif][y:dif][z:dif][t:<y><x>eq][u:<z><y>eq]<z><x>eq
6437 [a:dif][s:set(dif)]
6438 inn:=esti(dif,a,s):'prop'
6439 @real:=ect"eq"(dif,eq,refeq,symeq,treq):'type'
6440 [r:real][s:real]
6441 is:=is"e"(real,r,s):'prop'
6442 nis:=not(is(r,s)):'prop'
6443 @[p:[x:real]'prop']
6444 some:=some"l"(real,p):'prop'
6445 all:=all"l"(real,p):'prop'
6446 one:=one"e"(real,p):'prop'
6447 r@[s0:set(real)]
6448 in:=esti(real,r,s0):'prop'
6449 a@realof:=ectelt"eq"(dif,eq,refeq,symeq,treq,a):real
6450 r@class:=ecect"eq"(dif,eq,refeq,symeq,treq,r):set(dif)
6451 a@innclass:=th5"eq.4"(dif,eq,refeq,symeq,treq,a):inn(a,class(realof(a)))
6452 r@[a:dif][b:dif][e:eq"rp"(a,b)][air:inn(a,class(r))]
6453 eqinn:=th8"eq.4"(dif,eq,refeq,symeq,treq,r,a,air,b,e):inn(b,class(r))
6454 r@[p:'prop'][p1:[x:dif][xi:inn(x,class(r))]p]
6455 realapp1:=th3"eq.4"(dif,eq,refeq,symeq,treq,r,p,p1):p
6456 r@[s:real][p:'prop'][p1:[x:dif][y:dif][xi:inn(x,class(r))][yi:inn(y,class(s))]p]
6457 +ivr1
6458 [a:dif][air:inn(a,class(r))]
6459 t1:=realapp1(s,p,[y:dif]<air><y><a>p1):p
6460 -ivr1
6461 realapp2:=realapp1(r,p,[x:dif][xi:inn(x,class(r))]t1".ivr1"(x,xi)):p
6462 s@[t:real][p:'prop'][p1:[x:dif][y:dif][z:dif][xi:inn(x,class(r))][yi:inn(y,class(s))][zi:inn(z,class(t))]p]
6463 +*ivr1
6464 p1@[a:dif][air:inn(a,class(r))]
6465 t2:=realapp2(s,t,p,[y:dif][z:dif]<air><z><y><a>p1):p
6466 -ivr1
6467 p1@realapp3:=realapp1(r,p,[x:dif][xi:inn(x,class(r))]t2".ivr1"(x,xi)):p
6468 t@[u:real][p:'prop'][p1:[x:dif][y:dif][z:dif][v:dif][xi:inn(x,class(r))][yi:inn(y,class(s))][zi:inn(z,class(t))][vi:inn(v,class(u))]p]
6469 +*ivr1
6470 p1@[a:dif][air:inn(a,class(r))]
6471 t3:=realapp3(s,t,u,p,[y:dif][z:dif][v:dif]<air><v><z><y><a>p1):p
6472 -ivr1
6473 p1@realapp4:=realapp1(r,p,[x:dif][xi:inn(x,class(r))]t3".ivr1"(x,xi)):p
6474 s@[a1:dif][b1:dif][a1ir:inn(a1,class(r))][b1is:inn(b1,class(s))][e:eq"rp"(a1,b1)]
6475 isin:=th3"eq.5"(dif,eq,refeq,symeq,treq,r,s,a1,a1ir,b1,b1is,e):is(r,s)
6476 b1is@[i:is(r,s)]
6477 isex:=th5"eq.5"(dif,eq,refeq,symeq,treq,r,s,a1,a1ir,b1,b1is,i):eq"rp"(a1,b1)
6478 b1is@[n:not(eq"rp"(a1,b1))]
6479 nisin:=th3"l.imp"(is(r,s),eq"rp"(a1,b1),n,[t:is(r,s)]isex(t)):nis(r,s)
6480 b1is@[n:nis(r,s)]
6481 nisex:=th3"l.imp"(eq"rp"(a1,b1),is(r,s),n,[t:eq"rp"(a1,b1)]isin(t)):not(eq"rp"(a1,b1))
6482 @[alpha:'type'][f:[x:dif]alpha]
6483 fixf:=fixfu"eq"(dif,eq,refeq,symeq,treq,alpha,f):'prop'
6484 [ff:fixf(alpha,f)][r0:real]
6485 indreal:=indeq"eq"(dif,eq,refeq,symeq,treq,alpha,f,ff,r0):alpha
6486 [a:dif][air:inn(a,class(r0))]
6487 isindreal:=th2"eq.10"(dif,eq,refeq,symeq,treq,alpha,f,ff,r0,a,air):is"e"(alpha,<a>f,indreal)
6488 alpha@[g:[x:dif][y:dif]alpha]
6489 fixf2:=fixfu2"eq"(dif,eq,refeq,symeq,treq,alpha,g):'prop'
6490 [ff2:fixf2(alpha,g)][r0:real][s0:real]
6491 indreal2:=indeq2"eq"(dif,eq,refeq,symeq,treq,alpha,g,ff2,r0,s0):alpha
6492 [a:dif][b:dif][air:inn(a,class(r0))][bis:inn(b,class(s0))]
6493 isindreal2:=th1"eq.11"(dif,eq,refeq,symeq,treq,alpha,g,ff2,r0,s0,a,air,b,bis):is"e"(alpha,<b><a>g,indreal2)
6494 @0:=realof(df(1rp,1rp)):real
6495 r@[a0:dif][a0ir:inn(a0,class(r))][z:zero(a0)]
6496 0in:=isin(r,0,a0,df(1rp,1rp),a0ir,innclass(df(1rp,1rp)),zeroeq(a0,df(1rp,1rp),z,zeroi(1rp,1rp,refis(cut,1rp)))):is(r,0)
6497 a0ir@[i:is(r,0)]
6498 0ex:=eqzero(df(1rp,1rp),a0,isex(0,r,df(1rp,1rp),a0,innclass(df(1rp,1rp)),a0ir,symis(real,r,0,i)),tris(cut,stm(df(1rp,1rp)),1rp,std(df(1rp,1rp)),stmis(1rp,1rp),isstd(1rp,1rp))):zero(a0)
6499 +*ivr1
6500 a0@propp:=and(inn(a0,class(r)),posd(a0)):'prop'
6501 -ivr1
6502 r@pos:=some"l"(dif,[x:dif]propp".ivr1"(x)):'prop'
6503 a0ir@[p:posd(a0)]
6504 +*ivr1
6505 p@t4:=andi(inn(a0,class(r)),posd(a0),a0ir,p):propp(a0)
6506 -ivr1
6507 p@posin:=somei(dif,[x:dif]propp".ivr1"(x),a0,t4".ivr1"):pos(r)
6508 a0ir@[p:pos(r)]
6509 +*ivr1
6510 p@[a:dif][q1:propp(a)]
6511 t5:=ande1(inn(a,class(r)),posd(a),q1):inn(a,class(r))
6512 t6:=ande2(inn(a,class(r)),posd(a),q1):posd(a)
6513 t7:=eqposd(a,a0,isex(r,r,a,a0,t5,a0ir,refis(real,r)),t6):posd(a0)
6514 -ivr1
6515 p@posex:=someapp(dif,[x:dif]propp".ivr1"(x),p,posd(a0),[x:dif][t:propp".ivr1"(x)]t7".ivr1"(x,t)):posd(a0)
6516 +*ivr1
6517 a0@propn:=and(inn(a0,class(r)),negd(a0)):'prop'
6518 -ivr1
6519 r@neg:=some"l"(dif,[x:dif]propn".ivr1"(x)):'prop'
6520 a0ir@[n:negd(a0)]
6521 +*ivr1
6522 n@t8:=andi(inn(a0,class(r)),negd(a0),a0ir,n):propn(a0)
6523 -ivr1
6524 n@negin:=somei(dif,[x:dif]propn".ivr1"(x),a0,t8".ivr1"):neg(r)
6525 a0ir@[n:neg(r)]
6526 +*ivr1
6527 n@[a:dif][pl:propn(a)]
6528 t9:=ande1(inn(a,class(r)),negd(a),pl):inn(a,class(r))
6529 t10:=ande2(inn(a,class(r)),negd(a),pl):negd(a)
6530 t11:=eqnegd(a,a0,isex(r,r,a,a0,t9,a0ir,refis(real,r)),t10):negd(a0)
6531 -ivr1
6532 n@negex:=someapp(dif,[x:dif]propn".ivr1"(x),n,negd(a0),[x:dif][t:propn".ivr1"(x)]t11".ivr1"(x,t)):negd(a0)
6533 +*ivr1
6534 a0ir@[p:posd(a0)]
6535 t12:=or3i2(is(r,0),pos(r),neg(r),posin(p)):or3(is(r,0),pos(r),neg(r))
6536 a0ir@[z:zero(a0)]
6537 t13:=or3i1(is(r,0),pos(r),neg(r),0in(z)):or3(is(r,0),pos(r),neg(r))
6538 a0ir@[n:negd(a0)]
6539 t14:=or3i3(is(r,0),pos(r),neg(r),negin(n)):or3(is(r,0),pos(r),neg(r))
6540 a0ir@t15:=rappd(a0,or3(is(r,0),pos(r),neg(r)),[t:posd(a0)]t12(t),[t:zero(a0)]t13(t),[t:negd(a0)]t14(t)):or3(is(r,0),pos(r),neg(r))
6541 -ivr1
6542 r@axrlo:=realapp1(r,or3(is(r,0),pos(r),neg(r)),[x:dif][xi:inn(x,class(r))]t15".ivr1"(x,xi)):or3(is(r,0),pos(r),neg(r))
6543 +*ivr1
6544 a0ir@[i:is(r,0)]
6545 t16:=th3"l.imp"(pos(r),posd(a0),0notpd(a0,0ex(i)),[t:pos(r)]posex(t)):not(pos(r))
6546 a0ir@[p:pos(r)]
6547 t17:=th3"l.imp"(neg(r),negd(a0),pnotnd(a0,posex(p)),[t:neg(r)]negex(t)):not(neg(r))
6548 a0ir@[n:neg(r)]
6549 t18:=th3"l.imp"(is(r,0),zero(a0),nnot0d(a0,negex(n)),[t:is(r,0)]0ex(t)):not(is(r,0))
6550 a0ir@t19:=th6"l.ec3"(is(r,0),pos(r),neg(r),[t:is(r,0)]t16(t),[t:pos(r)]t17(t),[t:neg(r)]t18(t)):ec3(is(r,0),pos(r),neg(r))
6551 -ivr1
6552 r@axrle:=realapp1(r,ec3(is(r,0),pos(r),neg(r)),[x:dif][xi:inn(x,class(r))]t19".ivr1"(x,xi)):ec3(is(r,0),pos(r),neg(r))
6553 axrl:=orec3i(is(r,0),pos(r),neg(r),axrlo,axrle):orec3(is(r,0),pos(r),neg(r))
6554 [p:'prop'][p1:[t:pos(r)]p][p2:[t:is(r,0)]p][p3:[t:neg(r)]p]
6555 rapp:=or3app(is(r,0),pos(r),neg(r),p,axrlo,p2,p1,p3):p
6556 r@[p:pos(r)]
6557 pnotn:=ec3e23(is(r,0),pos(r),neg(r),axrle,p):not(neg(r))
6558 pnot0:=ec3e21(is(r,0),pos(r),neg(r),axrle,p):nis(r,0)
6559 r@[i:is(r,0)]
6560 0notp:=ec3e12(is(r,0),pos(r),neg(r),axrle,i):not(pos(r))
6561 0notn:=ec3e13(is(r,0),pos(r),neg(r),axrle,i):not(neg(r))
6562 r@[n:neg(r)]
6563 nnotp:=ec3e32(is(r,0),pos(r),neg(r),axrle,n):not(pos(r))
6564 nnot0:=ec3e31(is(r,0),pos(r),neg(r),axrle,n):nis(r,0)
6565 s@[i:is(r,s)][p:pos(r)]
6566 ispos:=isp(real,[x:real]pos(x),r,s,p,i):pos(s)
6567 i@[n:neg(r)]
6568 isneg:=isp(real,[x:real]neg(x),r,s,n,i):neg(s)
6569 @[r0:cut]
6570 pofrp:=realof(pdofrp(r0)):real
6571 nofrp:=realof(ndofrp(r0)):real
6572 [s0:cut][i:is"rp"(r0,s0)]
6573 isrpep:=isf(cut,real,[x:cut]pofrp(x),r0,s0,i):is(pofrp(r0),pofrp(s0))
6574 isrpen:=isf(cut,real,[x:cut]nofrp(x),r0,s0,i):is(nofrp(r0),nofrp(s0))
6575 s0@[i:is(pofrp(r0),pofrp(s0))]
6576 +*ivr1
6577 i"r"@t20:=isex(pofrp(r0),pofrp(s0),pdofrp(r0),pdofrp(s0),innclass(pdofrp(r0)),innclass(pdofrp(s0)),i):eq"rp"(pdofrp(r0),pdofrp(s0))
6578 -ivr1
6579 i@isrpip:=isrpipd(r0,s0,t20".ivr1"):is"rp"(r0,s0)
6580 s0@[i:is(nofrp(r0),nofrp(s0))]
6581 +*ivr1
6582 i"r"@t21:=isex(nofrp(r0),nofrp(s0),ndofrp(r0),ndofrp(s0),innclass(ndofrp(r0)),innclass(ndofrp(s0)),i):eq"rp"(ndofrp(r0),ndofrp(s0))
6583 -ivr1
6584 i@isrpin:=isrpind(r0,s0,t21".ivr1"):is"rp"(r0,s0)
6585 r0@posi:=posin(pofrp(r0),pdofrp(r0),innclass(pdofrp(r0)),posdirp(r0)):pos(pofrp(r0))
6586 negi:=negin(nofrp(r0),ndofrp(r0),innclass(ndofrp(r0)),negdirp(r0)):neg(nofrp(r0))
6587 s@[r0:cut][s0:cut][i:is(r,pofrp(r0))][j:is(s,pofrp(s0))][k:is(r,s)]
6588 +*ivr1
6589 k@t22:=isrpip(r0,s0,tr3is(real,pofrp(r0),r,s,pofrp(s0),symis(real,r,pofrp(r0),i),k,j)):is"rp"(r0,s0)
6590 r@t23:=[x:cut][y:cut][t:is(r,pofrp(x))][u:is(r,pofrp(y))]t22(r,r,x,y,t,u,refis(real,r)):amone(cut,[x:cut]is(r,pofrp(x)))
6591 a0ir@[p1:pos(r)]
6592 t24:=posex(p1):posd(a0)
6593 pr:=rpofpd(a0,t24):cut
6594 t25:=isin(r,pofrp(pr),a0,pdofrp(pr),a0ir,innclass(pdofrp(pr)),eqpdrp1(a0,t24)):is(r,pofrp(pr))
6595 t26:=somei(cut,[x:cut]is(r,pofrp(x)),pr,t25):some"rp"([x:cut]is(r,pofrp(x)))
6596 -ivr1
6597 r@[p:pos(r)]
6598 +*ivr1
6599 p"r"@t27:=realapp1(some"rp"([x:cut]is(r,pofrp(x))),[x:dif][t:inn(x,class(r))]t26(x,t,p)):some"rp"([x:cut]is(r,pofrp(x)))
6600 t28:=onei(cut,[x:cut]is(r,pofrp(x)),t23,t27):one"rp"([x:cut]is(r,pofrp(x)))
6601 -ivr1
6602 p@rpofp:=ind(cut,[x:cut]is(r,pofrp(x)),t28".ivr1"):cut
6603 isprp1:=oneax(cut,[x:cut]is(r,pofrp(x)),t28".ivr1"):is(r,pofrp(rpofp(r,p)))
6604 isprp2:=symis(real,r,pofrp(rpofp(r,p)),isprp1):is(pofrp(rpofp(r,p)),r)
6605 @[r1:real][p:pos(r1)][s1:real][q:pos(s1)][i:is(r1,s1)]
6606 isperp:=t22".ivr1"(r1,s1,rpofp(r1,p),rpofp(s1,q),isprp1(r1,p),isprp1(s1,q),i):is"rp"(rpofp(r1,p),rpofp(s1,q))
6607 q@[i:is"rp"(rpofp(r1,p),rpofp(s1,q))]
6608 ispirp:=tr3is(real,r1,pofrp(rpofp(r1,p)),pofrp(rpofp(s1,q)),s1,isprp1(r1,p),isrpep(rpofp(r1,p),rpofp(s1,q),i),isprp2(s1,q)):is(r1,s1)
6609 @[r0:cut]
6610 isrpp1:=t22".ivr1"(pofrp(r0),pofrp(r0),r0,rpofp(pofrp(r0),posi(r0)),refis(real,pofrp(r0)),isprp1(pofrp(r0),posi(r0)),refis(real,pofrp(r0))):is"rp"(r0,rpofp(pofrp(r0),posi(r0)))
6611 isrpp2:=symis(cut,r0,rpofp(pofrp(r0),posi(r0)),isrpp1):is"rp"(rpofp(pofrp(r0),posi(r0)),r0)
6612 s@[r0:cut][s0:cut][i:is(r,nofrp(r0))][j:is(s,nofrp(s0))][k:is(r,s)]
6613 +*ivr1
6614 k@t29:=isrpin(r0,s0,tr3is(real,nofrp(r0),r,s,nofrp(s0),symis(real,r,nofrp(r0),i),k,j)):is"rp"(r0,s0)
6615 r@t30:=[x:cut][y:cut][t:is(r,nofrp(x))][u:is(r,nofrp(y))]t29(r,r,x,y,t,u,refis(real,r)):amone(cut,[x:cut]is(r,nofrp(x)))
6616 a0ir@[n1:neg(r)]
6617 t31:=negex(n1):negd(a0)
6618 nr:=rpofnd(a0,t31):cut
6619 t32:=isin(r,nofrp(nr),a0,ndofrp(nr),a0ir,innclass(ndofrp(nr)),eqndrp1(a0,t31)):is(r,nofrp(nr))
6620 t33:=somei(cut,[x:cut]is(r,nofrp(x)),nr,t32):some"rp"([x:cut]is(r,nofrp(x)))
6621 -ivr1
6622 r@[n:neg(r)]
6623 +*ivr1
6624 n"r"@t34:=realapp1(some"rp"([x:cut]is(r,nofrp(x))),[x:dif][t:inn(x,class(r))]t33(x,t,n)):some"rp"([x:cut]is(r,nofrp(x)))
6625 t35:=onei(cut,[x:cut]is(r,nofrp(x)),t30,t34):one"rp"([x:cut]is(r,nofrp(x)))
6626 -ivr1
6627 n@rpofn:=ind(cut,[x:cut]is(r,nofrp(x)),t35".ivr1"):cut
6628 isnrp1:=oneax(cut,[x:cut]is(r,nofrp(x)),t35".ivr1"):is(r,nofrp(rpofn(r,n)))
6629 isnrp2:=symis(real,r,nofrp(rpofn(r,n)),isnrp1):is(nofrp(rpofn(r,n)),r)
6630 @[r1:real][n:neg(r1)][s1:real][m:neg(s1)][i:is(r1,s1)]
6631 isnerp:=t29".ivr1"(r1,s1,rpofn(r1,n),rpofn(s1,m),isnrp1(r1,n),isnrp1(s1,m),i):is"rp"(rpofn(r1,n),rpofn(s1,m))
6632 m@[i:is"rp"(rpofn(r1,n),rpofn(s1,m))]
6633 isnirp:=tr3is(real,r1,nofrp(rpofn(r1,n)),nofrp(rpofn(s1,m)),s1,isnrp1(r1,n),isrpen(rpofn(r1,n),rpofn(s1,m),i),isnrp2(s1,m)):is(r1,s1)
6634 @[r0:cut]
6635 isrpn1:=t29".ivr1"(nofrp(r0),nofrp(r0),r0,rpofn(nofrp(r0),negi(r0)),refis(real,nofrp(r0)),isnrp1(nofrp(r0),negi(r0)),refis(real,nofrp(r0))):is"rp"(r0,rpofn(nofrp(r0),negi(r0)))
6636 isrpn2:=symis(cut,r0,rpofn(nofrp(r0),negi(r0)),isrpn1):is"rp"(rpofn(nofrp(r0),negi(r0)),r0)
6637 r@satz163:=refis(real,r):is(r,r)
6638 s@[i:is(r,s)]
6639 satz164:=symis(real,r,s,i):is(s,r)
6640 t@[i:is(r,s)][j:is(s,t)]
6641 satz165:=tris(real,r,s,t,i,j):is(r,t)
6642 @absdr:=[x:dif]realof(absd(x)):[x:dif]real
6643 +ivr2
6644 [a:dif][b:dif][e:eq"rp"(a,b)]
6645 t1:=isin(realof(absd(a)),realof(absd(b)),absd(a),absd(b),innclass(absd(a)),innclass(absd(b)),eqabsd(a,b,e)):is(<a>absdr,<b>absdr)
6646 -ivr2
6647 fabsdr:=[x:dif][y:dif][t:<y><x>eq]t1".ivr2"(x,y,t):fixf(real,absdr)
6648 r@abs:=indreal(real,absdr,fabsdr,r):real
6649 +*ivr2
6650 a0ir@t2:=isindreal(real,absdr,fabsdr,r,a0,a0ir):is(realof(absd(a0)),abs(r))
6651 -ivr2
6652 a0ir@aica:=isp(real,[x:real]inn(absd(a0),class(x)),realof(absd(a0)),abs(r),innclass(absd(a0)),t2".ivr2"):inn(absd(a0),class(abs(r)))
6653 s@[i:is(r,s)]
6654 isabs:=isf(real,real,[x:real]abs(x),r,s,i):is(abs(r),abs(s))
6655 +2r166
6656 a0ir@[p:pos(r)]
6657 t1:=satzd166a(a0,posex(p)):posd(absd(a0))
6658 t2:=posin(abs(r),absd(a0),aica,t1):pos(abs(r))
6659 -2r166
6660 r@[p:pos(r)]
6661 satz166a:=realapp1(pos(abs(r)),[x:dif][t:inn(x,class(r))]t2".2r166"(x,t,p)):pos(abs(r))
6662 +*2r166
6663 a0ir@[n:neg(r)]
6664 t3:=satzd166b(a0,negex(n)):posd(absd(a0))
6665 t4:=posin(abs(r),absd(a0),aica,t3):pos(abs(r))
6666 -2r166
6667 r@[n:neg(r)]
6668 satz166b:=realapp1(pos(abs(r)),[x:dif][t:inn(x,class(r))]t4".2r166"(x,t,n)):pos(abs(r))
6669 +*2r166
6670 b1is@[p:pos(r)][q:pos(s)][i:is(abs(r),abs(s))]
6671 t5:=satzd166c(a1,b1,posex(r,a1,a1ir,p),posex(s,b1,b1is,q),isex(abs(r),abs(s),absd(a1),absd(b1),aica(r,a1,a1ir),aica(s,b1,b1is),i)):eq"rp"(a1,b1)
6672 t6:=isin(t5):is(r,s)
6673 -2r166
6674 s@[p:pos(r)][q:pos(s)][i:is(abs(r),abs(s))]
6675 satz166c:=realapp2(is(r,s),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t6".2r166"(x,y,t,u,p,q,i)):is(r,s)
6676 +*2r166
6677 b1is@[n:neg(r)][o:neg(s)][i:is(abs(r),abs(s))]
6678 t7:=satzd166d(a1,b1,negex(r,a1,a1ir,n),negex(s,b1,b1is,o),isex(abs(r),abs(s),absd(a1),absd(b1),aica(r,a1,a1ir),aica(s,b1,b1is),i)):eq"rp"(a1,b1)
6679 t8:=isin(t7):is(r,s)
6680 -2r166
6681 s@[n:neg(r)][o:neg(s)][i:is(abs(r),abs(s))]
6682 satz166d:=realapp2(is(r,s),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t8".2r166"(x,y,t,u,n,o,i)):is(r,s)
6683 r@[n:nis(r,0)]
6684 satz166e:=rapp(r,pos(abs(r)),[t:pos(r)]satz166a(t),th2"l.imp"(is(r,0),pos(abs(r)),n),[t:neg(r)]satz166b(t)):pos(abs(r))
6685 +*2r166
6686 a0ir@[i:is(r,0)]
6687 t9:=satzd166f(a0,0ex(i)):zero(absd(a0))
6688 t10:=0in(abs(r),absd(a0),aica,t9):is(abs(r),0)
6689 -2r166
6690 r@[i:is(r,0)]
6691 satz166f:=realapp1(is(abs(r),0),[x:dif][t:inn(x,class(r))]t10".2r166"(x,t,i)):is(abs(r),0)
6692 s@more:=some"l"(dif,[x:dif]some"l"(dif,[y:dif]and3(inn(x,class(r)),inn(y,class(s)),mored(x,y)))):'prop'
6693 +*ivr2
6694 b1@propm:=and3(inn(a1,class(r)),inn(b1,class(s)),mored(a1,b1)):'prop'
6695 -ivr2
6696 b1is@[m:mored(a1,b1)]
6697 +*ivr2
6698 m@t3:=and3i(inn(a1,class(r)),inn(b1,class(s)),mored(a1,b1),a1ir,b1is,m):propm(a1,b1)
6699 t4:=somei(dif,[x:dif]propm(a1,x),b1,t3):some"l"(dif,[x:dif]propm(a1,x))
6700 -ivr2
6701 m@morein:=somei(dif,[x:dif]some"l"(dif,[y:dif]propm".ivr2"(x,y)),a1,t4".ivr2"):more(r,s)
6702 b1is@[m:more(r,s)]
6703 +*ivr2
6704 m@[a:dif][sa:some"l"(dif,[x:dif]propm(a,x))][b:dif][p2:propm(a,b)]
6705 t5:=and3e1(inn(a,class(r)),inn(b,class(s)),mored(a,b),p2):inn(a,class(r))
6706 t6:=and3e2(inn(a,class(r)),inn(b,class(s)),mored(a,b),p2):inn(b,class(s))
6707 t7:=and3e3(inn(a,class(r)),inn(b,class(s)),mored(a,b),p2):mored(a,b)
6708 t8:=eqmored12(a,a1,b,b1,isex(r,r,a,a1,t5,a1ir,refis(real,r)),isex(s,s,b,b1,t6,b1is,refis(real,s)),t7):mored(a1,b1)
6709 sa@t9:=someapp(dif,[x:dif]propm(a,x),sa,mored(a1,b1),[x:dif][t:propm(a,x)]t8(x,t)):mored(a1,b1)
6710 -ivr2
6711 m@moreex:=someapp(dif,[x:dif]some"l"(dif,[y:dif]propm".ivr2"(x,y)),m,mored(a1,b1),[x:dif][t:some"l"(dif,[y:dif]propm".ivr2"(x,y))]t9".ivr2"(x,t)):mored(a1,b1)
6712 s@less:=some"l"(dif,[x:dif]some"l"(dif,[y:dif]and3(inn(x,class(r)),inn(y,class(s)),lessd(x,y)))):'prop'
6713 +*ivr2
6714 b1@propl:=and3(inn(a1,class(r)),inn(b1,class(s)),lessd(a1,b1)):'prop'
6715 -ivr2
6716 b1is@[l:lessd(a1,b1)]
6717 +*ivr2
6718 l@t10:=and3i(inn(a1,class(r)),inn(b1,class(s)),lessd(a1,b1),a1ir,b1is,l):propl(a1,b1)
6719 t11:=somei(dif,[x:dif]propl(a1,x),b1,t10):some"l"(dif,[x:dif]propl(a1,x))
6720 -ivr2
6721 l@lessin:=somei(dif,[x:dif]some"l"(dif,[y:dif]propl".ivr2"(x,y)),a1,t11".ivr2"):less(r,s)
6722 b1is@[l:less(r,s)]
6723 +*ivr2
6724 l@[a:dif][sa:some"l"(dif,[x:dif]propl(a,x))][b:dif][p2:propl(a,b)]
6725 t12:=and3e1(inn(a,class(r)),inn(b,class(s)),lessd(a,b),p2):inn(a,class(r))
6726 t13:=and3e2(inn(a,class(r)),inn(b,class(s)),lessd(a,b),p2):inn(b,class(s))
6727 t14:=and3e3(inn(a,class(r)),inn(b,class(s)),lessd(a,b),p2):lessd(a,b)
6728 t15:=eqlessd12(a,a1,b,b1,isex(r,r,a,a1,t12,a1ir,refis(real,r)),isex(s,s,b,b1,t13,b1is,refis(real,s)),t14):lessd(a1,b1)
6729 sa@t16:=someapp(dif,[x:dif]propl(a,x),sa,lessd(a1,b1),[x:dif][t:propl(a,x)]t15(x,t)):lessd(a1,b1)
6730 -ivr2
6731 l@lessex:=someapp(dif,[x:dif]some"l"(dif,[y:dif]propl".ivr2"(x,y)),l,lessd(a1,b1),[x:dif][t:some"l"(dif,[y:dif]propl".ivr2"(x,y))]t16".ivr2"(x,t)):lessd(a1,b1)
6732 t@[i:is(r,s)][m:more(r,t)]
6733 ismore1:=isp(real,[x:real]more(x,t),r,s,m,i):more(s,t)
6734 i@[m:more(t,r)]
6735 ismore2:=isp(real,[x:real]more(t,x),r,s,m,i):more(t,s)
6736 i@[l:less(r,t)]
6737 isless1:=isp(real,[x:real]less(x,t),r,s,l,i):less(s,t)
6738 i@[l:less(t,r)]
6739 isless2:=isp(real,[x:real]less(t,x),r,s,l,i):less(t,s)
6740 u@[i:is(r,s)][j:is(t,u)][m:more(r,t)]
6741 ismore12:=ismore2(t,u,s,j,ismore1(r,s,t,i,m)):more(s,u)
6742 j@[l:less(r,t)]
6743 isless12:=isless2(t,u,s,j,isless1(r,s,t,i,l)):less(s,u)
6744 +*ivr2
6745 b1is@[m:more(r,s)]
6746 t17:=lemmad5(a1,b1,moreex(m)):lessd(b1,a1)
6747 t18:=lessin(s,r,b1,a1,b1is,a1ir,t17):less(s,r)
6748 -ivr2
6749 s@[m:more(r,s)]
6750 lemma1:=realapp2(less(s,r),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t18".ivr2"(x,y,t,u,m)):less(s,r)
6751 +*ivr2
6752 b1is@[l:less(r,s)]
6753 t19:=lemmad6(a1,b1,lessex(l)):mored(b1,a1)
6754 t20:=morein(s,r,b1,a1,b1is,a1ir,t19):more(s,r)
6755 -ivr2
6756 s@[l:less(r,s)]
6757 lemma2:=realapp2(more(s,r),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t20".ivr2"(x,y,t,u,l)):more(s,r)
6758 +2r167
6759 b1is@t1:=satzd167a(a1,b1):or3(eq"rp"(a1,b1),mored(a1,b1),lessd(a1,b1))
6760 [e:eq"rp"(a1,b1)]
6761 t2:=or3i1(is(r,s),more(r,s),less(r,s),isin(e)):or3(is(r,s),more(r,s),less(r,s))
6762 b1is@[m:mored(a1,b1)]
6763 t3:=or3i2(is(r,s),more(r,s),less(r,s),morein(m)):or3(is(r,s),more(r,s),less(r,s))
6764 b1is@[l:lessd(a1,b1)]
6765 t4:=or3i3(is(r,s),more(r,s),less(r,s),lessin(l)):or3(is(r,s),more(r,s),less(r,s))
6766 b1is@t5:=or3app(eq"rp"(a1,b1),mored(a1,b1),lessd(a1,b1),or3(is(r,s),more(r,s),less(r,s)),t1,[t:eq"rp"(a1,b1)]t2(t),[t:mored(a1,b1)]t3(t),[t:lessd(a1,b1)]t4(t)):or3(is(r,s),more(r,s),less(r,s))
6767 t6:=satzd167b(a1,b1):ec3(eq"rp"(a1,b1),mored(a1,b1),lessd(a1,b1))
6768 [i:is(r,s)]
6769 t7:=th3"l.imp"(more(r,s),mored(a1,b1),ec3e12(eq"rp"(a1,b1),mored(a1,b1),lessd(a1,b1),t6,isex(i)),[t:more(r,s)]moreex(t)):not(more(r,s))
6770 b1is@[m:more(r,s)]
6771 t8:=th3"l.imp"(less(r,s),lessd(a1,b1),ec3e23(eq"rp"(a1,b1),mored(a1,b1),lessd(a1,b1),t6,moreex(m)),[t:less(r,s)]lessex(t)):not(less(r,s))
6772 b1is@[l:less(r,s)]
6773 t9:=th3"l.imp"(is(r,s),eq"rp"(a1,b1),ec3e31(eq"rp"(a1,b1),mored(a1,b1),lessd(a1,b1),t6,lessex(l)),[t:is(r,s)]isex(t)):not(is(r,s))
6774 b1is@t10:=th6"l.ec3"(is(r,s),more(r,s),less(r,s),th1"l.ec"(is(r,s),more(r,s),[t:is(r,s)]t7(t)),th1"l.ec"(more(r,s),less(r,s),[t:more(r,s)]t8(t)),th1"l.ec"(less(r,s),is(r,s),[t:less(r,s)]t9(t))):ec3(is(r,s),more(r,s),less(r,s))
6775 t11:=orec3i(is(r,s),more(r,s),less(r,s),t5,t10):orec3(is(r,s),more(r,s),less(r,s))
6776 -2r167
6777 s@satz167:=realapp2(orec3(is(r,s),more(r,s),less(r,s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t11".2r167"(x,y,t,u)):orec3(is(r,s),more(r,s),less(r,s))
6778 satz167a:=orec3e1(is(r,s),more(r,s),less(r,s),satz167):or3(is(r,s),more(r,s),less(r,s))
6779 satz167b:=orec3e2(is(r,s),more(r,s),less(r,s),satz167):ec3(is(r,s),more(r,s),less(r,s))
6780 moreis:=or(more(r,s),is(r,s)):'prop'
6781 lessis:=or(less(r,s),is(r,s)):'prop'
6782 [m:moreis(r,s)]
6783 satz168a:=th9"l.or"(more(r,s),is(r,s),less(s,r),is(s,r),m,[t:more(r,s)]lemma1(t),[t:is(r,s)]symis(real,r,s,t)):lessis(s,r)
6784 s@[l:lessis(r,s)]
6785 satz168b:=th9"l.or"(less(r,s),is(r,s),more(s,r),is(s,r),l,[t:less(r,s)]lemma2(t),[t:is(r,s)]symis(real,r,s,t)):moreis(s,r)
6786 t@[i:is(r,s)][m:moreis(r,t)]
6787 ismoreis1:=isp(real,[x:real]moreis(x,t),r,s,m,i):moreis(s,t)
6788 i@[m:moreis(t,r)]
6789 ismoreis2:=isp(real,[x:real]moreis(t,x),r,s,m,i):moreis(t,s)
6790 i@[l:lessis(r,t)]
6791 islessis1:=isp(real,[x:real]lessis(x,t),r,s,l,i):lessis(s,t)
6792 i@[l:lessis(t,r)]
6793 islessis2:=isp(real,[x:real]lessis(t,x),r,s,l,i):lessis(t,s)
6794 u@[i:is(r,s)][j:is(t,u)][m:moreis(r,t)]
6795 ismoreis12:=ismoreis2(t,u,s,j,ismoreis1(r,s,t,i,m)):moreis(s,u)
6796 j@[l:lessis(r,t)]
6797 islessis12:=islessis2(t,u,s,j,islessis1(r,s,t,i,l)):lessis(s,u)
6798 s@[m:more(r,s)]
6799 moreisi1:=ori1(more(r,s),is(r,s),m):moreis(r,s)
6800 s@[l:less(r,s)]
6801 lessisi1:=ori1(less(r,s),is(r,s),l):lessis(r,s)
6802 s@[i:is(r,s)]
6803 moreisi2:=ori2(more(r,s),is(r,s),i):moreis(r,s)
6804 lessisi2:=ori2(less(r,s),is(r,s),i):lessis(r,s)
6805 b1is@[m:moreq(a1,b1)]
6806 moreisin:=orapp(mored(a1,b1),eq"rp"(a1,b1),moreis(r,s),m,[t:mored(a1,b1)]moreisi1(morein(t)),[t:eq"rp"(a1,b1)]moreisi2(isin(t))):moreis(r,s)
6807 b1is@[m:moreis(r,s)]
6808 moreisex:=orapp(more(r,s),is(r,s),moreq(a1,b1),m,[t:more(r,s)]moreqi1(a1,b1,moreex(t)),[t:is(r,s)]moreqi2(a1,b1,isex(t))):moreq(a1,b1)
6809 b1is@[l:lesseq(a1,b1)]
6810 lessisin:=orapp(lessd(a1,b1),eq"rp"(a1,b1),lessis(r,s),l,[t:lessd(a1,b1)]lessisi1(lessin(t)),[t:eq"rp"(a1,b1)]lessisi2(isin(t))):lessis(r,s)
6811 b1is@[l:lessis(r,s)]
6812 lessisex:=orapp(less(r,s),is(r,s),lesseq(a1,b1),l,[t:less(r,s)]lesseqi1(a1,b1,lessex(t)),[t:is(r,s)]lesseqi2(a1,b1,isex(t))):lesseq(a1,b1)
6813 s@[m:moreis(r,s)]
6814 satz167c:=th7"l.ec3"(is(r,s),more(r,s),less(r,s),satz167b,comor(more(r,s),is(r,s),m)):not(less(r,s))
6815 s@[l:lessis(r,s)]
6816 satz167d:=th9"l.ec3"(is(r,s),more(r,s),less(r,s),satz167b,l):not(more(r,s))
6817 s@[n:not(more(r,s))]
6818 satz167e:=th2"l.or3"(is(r,s),more(r,s),less(r,s),satz167a,n):lessis(r,s)
6819 s@[n:not(less(r,s))]
6820 s@[n:not(less(r,s))]
6821 satz167f:=comor(is(r,s),more(r,s),th3"l.or3"(is(r,s),more(r,s),less(r,s),satz167a,n)):moreis(r,s)
6822 s@[m:more(r,s)]
6823 satz167g:=th3"l.imp"(lessis(r,s),not(more(r,s)),weli(more(r,s),m),[t:lessis(r,s)]satz167d(t)):not(lessis(r,s))
6824 s@[l:less(r,s)]
6825 satz167h:=th3"l.imp"(moreis(r,s),not(less(r,s)),weli(less(r,s),l),[t:moreis(r,s)]satz167c(t)):not(moreis(r,s))
6826 s@[n:not(moreis(r,s))]
6827 satz167j:=or3e3(is(r,s),more(r,s),less(r,s),satz167a,th5"l.or"(more(r,s),is(r,s),n),th4"l.or"(more(r,s),is(r,s),n)):less(r,s)
6828 s@[n:not(lessis(r,s))]
6829 satz167k:=or3e2(is(r,s),more(r,s),less(r,s),satz167a,th4"l.or"(less(r,s),is(r,s),n),th5"l.or"(less(r,s),is(r,s),n)):more(r,s)
6830 r@[p:pos(r)]
6831 +2r169
6832 [a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))]
6833 t1:=satzd169a(a,b,0ex(0,b,bi0,refis(real,0)),posex(a,air,p)):mored(a,b)
6834 t2:=morein(r,0,a,b,air,bi0,t1):more(r,0)
6835 -2r169
6836 satz169a:=realapp2(r,0,more(r,0),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(0))]t2".2r169"(x,y,t,u)):more(r,0)
6837 r@[m:more(r,0)]
6838 +*2r169
6839 m@[a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))]
6840 t3:=satzd169b(a,b,0ex(0,b,bi0,refis(real,0)),moreex(r,0,a,b,air,bi0,m)):posd(a)
6841 t4:=posin(r,a,air,t3):pos(r)
6842 -2r169
6843 m@satz169b:=realapp2(r,0,pos(r),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(0))]t4".2r169"(x,y,t,u)):pos(r)
6844 r@[n:neg(r)]
6845 +*2r169
6846 n@[a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))]
6847 t5:=satzd169c(a,b,0ex(0,b,bi0,refis(real,0)),negex(a,air,n)):lessd(a,b)
6848 t6:=lessin(r,0,a,b,air,bi0,t5):less(r,0)
6849 -2r169
6850 n@satz169c:=realapp2(r,0,less(r,0),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(0))]t6".2r169"(x,y,t,u)):less(r,0)
6851 r@[l:less(r,0)]
6852 +*2r169
6853 l@[a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))]
6854 t7:=satzd169d(a,b,0ex(0,b,bi0,refis(real,0)),lessex(r,0,a,b,air,bi0,l)):negd(a)
6855 t8:=negin(r,a,air,t7):neg(r)
6856 -2r169
6857 l@satz169d:=realapp2(r,0,neg(r),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(0))]t8".2r169"(x,y,t,u)):neg(r)
6858 +2r170
6859 r@[a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))]
6860 t1:=satzd170(a,b,0ex(0,b,bi0,refis(real,0))):moreq(absd(a),b)
6861 t2:=moreisin(abs(r),0,absd(a),b,aica(r,a,air),bi0,t1):moreis(abs(r),0)
6862 -2r170
6863 r@satz170:=realapp2(r,0,moreis(abs(r),0),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(0))]t2".2r170"(x,y,t,u)):moreis(abs(r),0)
6864 satz170a:=th3"l.imp"(neg(abs(r)),less(abs(r),0),satz167c(abs(r),0,satz170),[t:neg(abs(r))]satz169c(abs(r),t)):not(neg(abs(r)))
6865 t@[l:less(r,s)][k:less(s,t)]
6866 +2r171
6867 [a:dif][b:dif][c:dif][air:inn(a,class(r))][bis:inn(b,class(s))][cit:inn(c,class(t))]
6868 t1:=satzd171(a,b,c,lessex(r,s,a,b,air,bis,l),lessex(s,t,b,c,bis,cit,k)):lessd(a,c)
6869 t2:=lessin(r,t,a,c,air,cit,t1):less(r,t)
6870 -2r171
6871 satz171:=realapp3(r,s,t,less(r,t),[x:dif][y:dif][z:dif][w:inn(x,class(r))][u:inn(y,class(s))][v:inn(z,class(t))]t2".2r171"(x,y,z,w,u,v)):less(r,t)
6872 trless:=satz171:less(r,t)
6873 t@[m:more(r,s)][n:more(s,t)]
6874 trmore:=lemma2(t,r,trless(t,s,r,lemma1(s,t,n),lemma1(r,s,m))):more(r,t)
6875 t@[a2:dif][b2:dif][c2:dif][a2ir:inn(a2,class(r))][b2is:inn(b2,class(s))][c2it:inn(c2,class(t))]
6876 +2r172
6877 [l:lessis(r,s)][k:less(s,t)]
6878 t1:=satzd172a(a2,b2,c2,lessisex(a2,b2,a2ir,b2is,l),lessex(s,t,b2,c2,b2is,c2it,k)):lessd(a2,c2)
6879 t2:=lessin(r,t,a2,c2,a2ir,c2it,t1):less(r,t)
6880 -2r172
6881 t@[l:lessis(r,s)][k:less(s,t)]
6882 satz172a:=realapp3(less(r,t),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t2".2r172"(x,y,z,u,v,w,l,k)):less(r,t)
6883 +*2r172
6884 c2it@[l:less(r,s)][k:lessis(s,t)]
6885 t3:=satzd172b(a2,b2,c2,lessex(a2,b2,a2ir,b2is,l),lessisex(s,t,b2,c2,b2is,c2it,k)):lessd(a2,c2)
6886 t4:=lessin(r,t,a2,c2,a2ir,c2it,t3):less(r,t)
6887 -2r172
6888 t@[l:less(r,s)][k:lessis(s,t)]
6889 satz172b:=realapp3(less(r,t),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t4".2r172"(x,y,z,u,v,w,l,k)):less(r,t)
6890 t@[m:moreis(r,s)][n:more(s,t)]
6891 satz172c:=lemma2(t,r,satz172b(t,s,r,lemma1(s,t,n),satz168a(m))):more(r,t)
6892 t@[m:more(r,s)][n:moreis(s,t)]
6893 satz172d:=lemma2(t,r,satz172a(t,s,r,satz168a(s,t,n),lemma1(m))):more(r,t)
6894 +2r173
6895 c2it@[l:lessis(r,s)][k:lessis(s,t)]
6896 t1:=satzd173(a2,b2,c2,lessisex(a2,b2,a2ir,b2is,l),lessisex(s,t,b2,c2,b2is,c2it,k)):lesseq(a2,c2)
6897 t2:=lessisin(r,t,a2,c2,a2ir,c2it,t1):lessis(r,t)
6898 -2r173
6899 t@[l:lessis(r,s)][k:lessis(s,t)]
6900 satz173:=realapp3(lessis(r,t),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t2".2r173"(x,y,z,u,v,w,l,k)):lessis(r,t)
6901 trlessis:=satz173:lessis(r,t)
6902 t@[m:moreis(r,s)][n:moreis(s,t)]
6903 trmoreis:=satz168b(t,r,trlessis(t,s,r,satz168a(s,t,n),satz168a(m))):moreis(r,t)
6904 r@ratrl:=some"l"(dif,[x:dif]and(inn(x,class(r)),ratd(x))):'prop'
6905 a0ir@[r1:ratd(a0)]
6906 +*ivr2
6907 r1@t21:=andi(inn(a0,class(r)),ratd(a0),a0ir,r1):and(inn(a0,class(r)),ratd(a0))
6908 -ivr2
6909 r1@ratrlin:=somei(dif,[x:dif]and(inn(x,class(r)),ratd(x)),a0,t21".ivr2"):ratrl(r)
6910 a0ir@[rr:ratrl(r)]
6911 +*ivr2
6912 rr@[a:dif][b:and(inn(a,class(r)),ratd(a))]
6913 t22:=ande1(inn(a,class(r)),ratd(a),b):inn(a,class(r))
6914 t23:=ande2(inn(a,class(r)),ratd(a),b):ratd(a)
6915 t24:=eqratd(a,a0,isex(r,r,a,a0,t22,a0ir,refis(real,r)),t23):ratd(a0)
6916 -ivr2
6917 rr@ratrlex:=someapp(dif,[x:dif]and(inn(x,class(r)),ratd(x)),rr,ratd(a0),[x:dif][t:and(inn(x,class(r)),ratd(x))]t24".ivr2"(x,t)):ratd(a0)
6918 r@irratrl:=not(ratrl(r)):'prop'
6919 @[r0:cut][rr:ratrp(r0)]
6920 remark2:=ratrlin(pofrp(r0),pdofrp(r0),innclass(pdofrp(r0)),remark2a(r0,rr)):ratrl(pofrp(r0))
6921 remark3:=ratrlin(nofrp(r0),ndofrp(r0),innclass(ndofrp(r0)),remark3a(r0,rr)):ratrl(nofrp(r0))
6922 r0@[ir:irratrp(r0)]
6923 remark4:=th3"l.imp"(ratrl(pofrp(r0)),ratd(pdofrp(r0)),remark4a(r0,ir),[t:ratrl(pofrp(r0))]ratrlex(pofrp(r0),pdofrp(r0),innclass(pdofrp(r0)),t)):irratrl(pofrp(r0))
6924 remark5:=th3"l.imp"(ratrl(nofrp(r0)),ratd(ndofrp(r0)),remark5a(r0,ir),[t:ratrl(nofrp(r0))]ratrlex(nofrp(r0),ndofrp(r0),innclass(ndofrp(r0)),t)):irratrl(nofrp(r0))
6925 r@natrl:=some"l"(dif,[x:dif]and(inn(x,class(r)),natd(x))):'prop'
6926 a0ir@[n:natd(a0)]
6927 +*ivr2
6928 n@t25:=andi(inn(a0,class(r)),natd(a0),a0ir,n):and(inn(a0,class(r)),natd(a0))
6929 -ivr2
6930 n@natrlin:=somei(dif,[x:dif]and(inn(x,class(r)),natd(x)),a0,t25".ivr2"):natrl(r)
6931 a0ir@[n:natrl(r)]
6932 +*ivr2
6933 n@[a:dif][b:and(inn(a,class(r)),natd(a))]
6934 t26:=ande1(inn(a,class(r)),natd(a),b):inn(a,class(r))
6935 t27:=ande2(inn(a,class(r)),natd(a),b):natd(a)
6936 t28:=eqnatd(a,a0,isex(r,r,a,a0,t26,a0ir,refis(real,r)),t27):natd(a0)
6937 -ivr2
6938 n@natrlex:=someapp(dif,[x:dif]and(inn(x,class(r)),natd(x)),n,natd(a0),[x:dif][t:and(inn(x,class(r)),natd(x))]t28".ivr2"(x,t)):natd(a0)
6939 +*ivr2
6940 a0ir@[n:natrl(r)]
6941 t29:=natposd(a0,natrlex(n)):posd(a0)
6942 t30:=posin(t29):pos(r)
6943 -ivr2
6944 r@[n:natrl(r)]
6945 natpos:=realapp1(pos(r),[x:dif][t:inn(x,class(r))]t30".ivr2"(x,t,n)):pos(r)
6946 @[x:nat]
6947 rlofnt:=realof(pdofnt(x)):real
6948 natrli:=natrlin(rlofnt(x),pdofnt(x),innclass(pdofnt(x)),natdi(x)):natrl(rlofnt(x))
6949 [y:nat][i:is"n"(x,y)]
6950 isnterl:=isf(nat,real,[u:nat]rlofnt(u),x,y,i):is(rlofnt(x),rlofnt(y))
6951 y@[i:is(rlofnt(x),rlofnt(y))]
6952 isntirl:=isntirp(x,y,isrpip(rpofnt(x),rpofnt(y),i)):is"n"(x,y)
6953 +*ivr2
6954 @t31:=[x:nat][y:nat][t:is(rlofnt(x),rlofnt(y))]isntirl(x,y,t):injective(nat,real,[x:nat]rlofnt(x))
6955 a0ir@[n:natrl(r)]
6956 t32:=natposd(a0,natrlex(n)):posd(a0)
6957 ap:=rpofpd(a0,t32):cut
6958 t33:=natderp(a0,natrlex(n)):natrp(ap)
6959 x0:=ntofrp(ap,t33):nat
6960 t34:=isrpepd(ap,rpofnt(x0),isrpnt1(ap,t33)):eq"rp"(pdofrp(ap),pdofnt(x0))
6961 t35:=treq"rp"(a0,pdofrp(ap),pdofnt(x0),eqpdrp1(a0,t32),t34):eq"rp"(a0,pdofnt(x0))
6962 t36:=isin(r,rlofnt(x0),a0,pdofnt(x0),a0ir,innclass(pdofnt(x0)),t35):is(r,rlofnt(x0))
6963 t37:=somei(nat,[x:nat]is(r,rlofnt(x)),x0,t36):image(nat,real,[x:nat]rlofnt(x),r)
6964 -ivr2
6965 r@[n:natrl(r)]
6966 natimage:=realapp1(image(nat,real,[x:nat]rlofnt(x),r),[x:dif][t:inn(x,class(r))]t37".ivr2"(x,t,n)):image(nat,real,[x:nat]rlofnt(x),r)
6967 r@[i:image(nat,real,[x:nat]rlofnt(x),r)]
6968 +*ivr2
6969 i"r"@[x:nat][j:is(r,rlofnt(x))]
6970 t38:=isp1(real,[u:real]natrl(u),rlofnt(x),r,natrli(x),j):natrl(r)
6971 -ivr2
6972 i@imagenat:=someapp(nat,[u:nat]is(r,rlofnt(u)),i,natrl(r),[u:nat][v:is(r,rlofnt(u))]t38".ivr2"(u,v)):natrl(r)
6973 r@[n:natrl(r)]
6974 ntofrl:=soft(nat,real,[x:nat]rlofnt(x),t31".ivr2",r,natimage(r,n)):nat
6975 @[r1:real][n:natrl(r1)][s1:real][m:natrl(s1)][i:is(r1,s1)]
6976 isrlent:=isinv(nat,real,[x:nat]rlofnt(x),t31".ivr2",r1,natimage(r1,n),s1,natimage(s1,m),i):is"n"(ntofrl(r1,n),ntofrl(s1,m))
6977 m@[i:is"n"(ntofrl(r1,n),ntofrl(s1,m))]
6978 isrlint:=isinve(nat,real,[x:nat]rlofnt(x),t31".ivr2",r1,natimage(r1,n),s1,natimage(s1,m),i):is(r1,s1)
6979 r@[n:natrl(r)]
6980 isrlnt1:=ists1"e"(nat,real,[x:nat]rlofnt(x),t31".ivr2",r,natimage(r,n)):is(r,rlofnt(ntofrl(r,n)))
6981 isrlnt2:=symis(real,r,rlofnt(ntofrl(r,n)),isrlnt1):is(rlofnt(ntofrl(r,n)),r)
6982 @[x:nat]
6983 +*ivr2
6984 x"r"@xn:=soft(nat,real,[u:nat]rlofnt(u),t31,rlofnt(x),imagei(nat,real,[u:nat]rlofnt(u),x)):nat
6985 t39:=isinv(nat,real,[u:nat]rlofnt(u),t31,rlofnt(x),imagei(nat,real,[u:nat]rlofnt(u),x),rlofnt(x),natimage(rlofnt(x),natrli(x)),refis(real,rlofnt(x))):is"n"(xn,ntofrl(rlofnt(x),natrli(x)))
6986 -ivr2
6987 x@isntrl1:=tris(nat,x,xn".ivr2",ntofrl(rlofnt(x),natrli(x)),isst1(nat,real,[u:nat]rlofnt(u),t31".ivr2",x),t39".ivr2"):is"n"(x,ntofrl(rlofnt(x),natrli(x)))
6988 isntrl2:=symis(nat,x,ntofrl(rlofnt(x),natrli(x)),isntrl1):is"n"(ntofrl(rlofnt(x),natrli(x)),x)
6989 r@intrl:=some"l"(dif,[x:dif]and(inn(x,class(r)),intd(x))):'prop'
6990 a0ir@[i:intd(a0)]
6991 +*ivr2
6992 i@t40:=andi(inn(a0,class(r)),intd(a0),a0ir,i):and(inn(a0,class(r)),intd(a0))
6993 -ivr2
6994 i@intrlin:=somei(dif,[x:dif]and(inn(x,class(r)),intd(x)),a0,t40".ivr2"):intrl(r)
6995 a0ir@[i:intrl(r)]
6996 +*ivr2
6997 i@[a:dif][b:and(inn(a,class(r)),intd(a))]
6998 t41:=ande1(inn(a,class(r)),intd(a),b):inn(a,class(r))
6999 t42:=ande2(inn(a,class(r)),intd(a),b):intd(a)
7000 t43:=eqintd(a,a0,isex(r,r,a,a0,t41,a0ir,refis(real,r)),t42):intd(a0)
7001 -ivr2
7002 i@intrlex:=someapp(dif,[x:dif]and(inn(x,class(r)),intd(x)),i,intd(a0),[x:dif][t:and(inn(x,class(r)),intd(x))]t43".ivr2"(x,t)):intd(a0)
7003 +*ivr2
7004 a0ir@[n:natrl(r)]
7005 t44:=natintd(a0,natrlex(n)):intd(a0)
7006 t45:=intrlin(t44):intrl(r)
7007 -ivr2
7008 r@[n:natrl(r)]
7009 natintrl:=realapp1(intrl(r),[x:dif][t:inn(x,class(r))]t45".ivr2"(x,t,n)):intrl(r)
7010 +*ivr2
7011 a0ir@[p:pos(r)][i:intrl(r)]
7012 t46:=posintnatd(a0,posex(p),intrlex(i)):natd(a0)
7013 t47:=natrlin(t46):natrl(r)
7014 -ivr2
7015 r@[p:pos(r)][i:intrl(r)]
7016 posintnatrl:=realapp1(natrl(r),[x:dif][t:inn(x,class(r))]t47".ivr2"(x,t,p,i)):natrl(r)
7017 +*ivr2
7018 a0ir@[i2:is(r,0)]
7019 t48:=intdi0(a0,0ex(i2)):intd(a0)
7020 t49:=intrlin(t48):intrl(r)
7021 -ivr2
7022 r@[i:is(r,0)]
7023 intrli0:=realapp1(intrl(r),[x:dif][t:inn(x,class(r))]t49".ivr2"(x,t,i)):intrl(r)
7024 r0@[n:natrp(r0)]
7025 remark6:=intrlin(pofrp(r0),pdofrp(r0),innclass(pdofrp(r0)),remark6"rp"(r0,n)):intrl(pofrp(r0))
7026 remark7:=intrlin(nofrp(r0),ndofrp(r0),innclass(ndofrp(r0)),remark7"rp"(r0,n)):intrl(nofrp(r0))
7027 +2r174
7028 a0ir@[i:intrl(r)]
7029 t1:=satzd174(a0,intrlex(i)):ratd(a0)
7030 t2:=ratrlin(t1):ratrl(r)
7031 -2r174
7032 r@[i:intrl(r)]
7033 satz174:=realapp1(ratrl(r),[x:dif][t:inn(x,class(r))]t2".2r174"(x,t,i)):ratrl(r)
7034 @plusdr:=[x:dif][y:dif]realof(pd(x,y)):[x:dif][y:dif]real
7035 +ivr3
7036 [a:dif][b:dif][c:dif][d:dif][e:eq"rp"(a,b)][f:eq"rp"(c,d)]
7037 t1:=isin(realof(pd(a,c)),realof(pd(b,d)),pd(a,c),pd(b,d),innclass(pd(a,c)),innclass(pd(b,d)),eqpd12(a,b,c,d,e,f)):is(<c><a>plusdr,<d><b>plusdr)
7038 -ivr3
7039 fplusdr:=[x:dif][y:dif][z:dif][v:dif][t:<y><x>eq][u:<v><z>eq]t1".ivr3"(x,y,z,v,t,u):fixf2(real,plusdr)
7040 s@pl:=indreal2(real,plusdr,fplusdr,r,s):real
7041 +*ivr3
7042 b1is@t2:=isindreal2(real,plusdr,fplusdr,r,s,a1,b1,a1ir,b1is):is(realof(pd(a1,b1)),pl(r,s))
7043 -ivr3
7044 b1is@picp:=isp(real,[x:real]inn(pd(a1,b1),class(x)),realof(pd(a1,b1)),pl(r,s),innclass(pd(a1,b1)),t2".ivr3"):inn(pd(a1,b1),class(pl(r,s)))
7045 t@[i:is(r,s)]
7046 ispl1:=isf(real,real,[x:real]pl(x,t),r,s,i):is(pl(r,t),pl(s,t))
7047 ispl2:=isf(real,real,[x:real]pl(t,x),r,s,i):is(pl(t,r),pl(t,s))
7048 u@[i:is(r,s)][j:is(t,u)]
7049 ispl12:=tris(real,pl(r,t),pl(s,t),pl(s,u),ispl1(i),ispl2(t,u,s,j)):is(pl(r,t),pl(s,u))
7050 +3r175
7051 b1is@t1:=satzd175(a1,b1):eq"rp"(pd(a1,b1),pd(b1,a1))
7052 t2:=isin(pl(r,s),pl(s,r),pd(a1,b1),pd(b1,a1),picp,picp(s,r,b1,a1,b1is,a1ir),t1):is(pl(r,s),pl(s,r))
7053 -3r175
7054 s@satz175:=realapp2(is(pl(r,s),pl(s,r)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t2".3r175"(x,y,t,u)):is(pl(r,s),pl(s,r))
7055 compl:=satz175:is(pl(r,s),pl(s,r))
7056 +*ivr3
7057 b1is@[i:is(r,0)]
7058 t3:=pd01(a1,b1,0ex(r,a1,a1ir,i)):eq"rp"(pd(a1,b1),b1)
7059 t4:=isin(pl(r,s),s,pd(a1,b1),b1,picp,b1is,t3):is(pl(r,s),s)
7060 -ivr3
7061 s@[i:is(r,0)]
7062 pl01:=realapp2(is(pl(r,s),s),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t4".ivr3"(x,y,t,u,i)):is(pl(r,s),s)
7063 s@[i:is(s,0)]
7064 pl02:=tris(real,pl(r,s),pl(s,r),r,compl,pl01(s,r,i)):is(pl(r,s),r)
7065 +*ivr3
7066 b1is@[p:pos(r)][q:pos(s)]
7067 t5:=ppd(a1,b1,posex(r,a1,a1ir,p),posex(s,b1,b1is,q)):posd(pd(a1,b1))
7068 t6:=posin(pl(r,s),pd(a1,b1),picp,t5):pos(pl(r,s))
7069 -ivr3
7070 s@[p:pos(r)][q:pos(s)]
7071 pospl:=realapp2(pos(pl(r,s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t6".ivr3"(x,y,t,u,p,q)):pos(pl(r,s))
7072 +*ivr3
7073 b1is@[n:neg(r)][o:neg(s)]
7074 t7:=npd(a1,b1,negex(r,a1,a1ir,n),negex(s,b1,b1is,o)):negd(pd(a1,b1))
7075 t8:=negin(pl(r,s),pd(a1,b1),picp,t7):neg(pl(r,s))
7076 -ivr3
7077 s@[n:neg(r)][o:neg(s)]
7078 negpl:=realapp2(neg(pl(r,s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t8".ivr3"(x,y,t,u,n,o)):neg(pl(r,s))
7079 @m0dr:=[x:dif]realof(m0d(x)):[x:dif]real
7080 +*ivr3
7081 @[a:dif][b:dif][e:eq"rp"(a,b)]
7082 t5a:=isin(realof(m0d(a)),realof(m0d(b)),m0d(a),m0d(b),innclass(m0d(a)),innclass(m0d(b)),eqm0d(a,b,e)):is(<a>m0dr,<b>m0dr)
7083 -ivr3
7084 @fm0dr:=[x:dif][y:dif][t:<y><x>eq]t5a".ivr3"(x,y,t):fixf(real,m0dr)
7085 r@m0:=indreal(real,m0dr,fm0dr,r):real
7086 +*ivr3
7087 a0ir@t6a:=isindreal(real,m0dr,fm0dr,r,a0,a0ir):is(realof(m0d(a0)),m0(r))
7088 -ivr3
7089 a0ir@micm0:=isp(real,[x:real]inn(m0d(a0),class(x)),realof(m0d(a0)),m0(r),innclass(m0d(a0)),t6a".ivr3"):inn(m0d(a0),class(m0(r)))
7090 s@[i:is(r,s)]
7091 ism0:=isf(real,real,[x:real]m0(x),r,s,i):is(m0(r),m0(s))
7092 +*ivr3
7093 a0ir@[n:neg(r)]
7094 t7a:=absnd(a0,negex(n)):eq"rp"(absd(a0),m0d(a0))
7095 t8a:=isin(abs(r),m0(r),absd(a0),m0d(a0),aica,micm0,t7a):is(abs(r),m0(r))
7096 -ivr3
7097 r@[n:neg(r)]
7098 absn:=realapp1(is(abs(r),m0(r)),[x:dif][t:inn(x,class(r))]t8a".ivr3"(x,t,n)):is(abs(r),m0(r))
7099 +*ivr3
7100 a0ir@[nn:not(neg(r))]
7101 t9:=absnnd(a0,th3"l.imp"(negd(a0),neg(r),nn,[t:negd(a0)]negin(t))):eq"rp"(absd(a0),a0)
7102 t10:=isin(abs(r),r,absd(a0),a0,aica,a0ir,t9):is(abs(r),r)
7103 -ivr3
7104 r@[nn:not(neg(r))]
7105 absnn:=realapp1(is(abs(r),r),[x:dif][t:inn(x,class(r))]t10".ivr3"(x,t,nn)):is(abs(r),r)
7106 r@[p:pos(r)]
7107 absp:=absnn(r,pnotn(r,p)):is(abs(r),r)
7108 r@[i:is(r,0)]
7109 abs0:=tris(real,abs(r),r,0,absnn(r,0notn(r,i)),i):is(abs(r),0)
7110 +3r176
7111 a0ir@[p:pos(r)]
7112 t1:=satzd176a(a0,posex(p)):negd(m0d(a0))
7113 t2:=negin(m0(r),m0d(a0),micm0,t1):neg(m0(r))
7114 -3r176
7115 r@[p:pos(r)]
7116 satz176a:=realapp1(neg(m0(r)),[x:dif][t:inn(x,class(r))]t2".3r176"(x,t,p)):neg(m0(r))
7117 +*3r176
7118 a0ir@[i:is(r,0)]
7119 t3:=satzd176b(a0,0ex(i)):zero(m0d(a0))
7120 t4:=0in(m0(r),m0d(a0),micm0,t3):is(m0(r),0)
7121 -3r176
7122 r@[i:is(r,0)]
7123 satz176b:=realapp1(is(m0(r),0),[x:dif][t:inn(x,class(r))]t4".3r176"(x,t,i)):is(m0(r),0)
7124 +*3r176
7125 a0ir@[n:neg(r)]
7126 t5:=satzd176c(a0,negex(n)):posd(m0d(a0))
7127 t6:=posin(m0(r),m0d(a0),micm0,t5):pos(m0(r))
7128 -3r176
7129 r@[n:neg(r)]
7130 satz176c:=realapp1(pos(m0(r)),[x:dif][t:inn(x,class(r))]t6".3r176"(x,t,n)):pos(m0(r))
7131 +*3r176
7132 a0ir@[n:neg(m0(r))]
7133 t7:=satzd176d(a0,negex(m0(r),m0d(a0),micm0,n)):posd(a0)
7134 t8:=posin(t7):pos(r)
7135 -3r176
7136 r@[n:neg(m0(r))]
7137 satz176d:=realapp1(pos(r),[x:dif][t:inn(x,class(r))]t8".3r176"(x,t,n)):pos(r)
7138 +*3r176
7139 a0ir@[i:is(m0(r),0)]
7140 t9:=satzd176e(a0,0ex(m0(r),m0d(a0),micm0,i)):zero(a0)
7141 t10:=0in(t9):is(r,0)
7142 -3r176
7143 r@[i:is(m0(r),0)]
7144 satz176e:=realapp1(is(r,0),[x:dif][t:inn(x,class(r))]t10".3r176"(x,t,i)):is(r,0)
7145 +*3r176
7146 a0ir@[p:pos(m0(r))]
7147 t11:=satzd176f(a0,posex(m0(r),m0d(a0),micm0,p)):negd(a0)
7148 t12:=negin(t11):neg(r)
7149 -3r176
7150 r@[p:pos(m0(r))]
7151 satz176f:=realapp1(neg(r),[x:dif][t:inn(x,class(r))]t12".3r176"(x,t,p)):neg(r)
7152 +3r177
7153 a0ir@t1:=isin(m0(m0(r)),r,m0d(m0d(a0)),a0,micm0(m0(r),m0d(a0),micm0),a0ir,satzd177(a0)):is(m0(m0(r)),r)
7154 -3r177
7155 r@satz177:=realapp1(is(m0(m0(r)),r),[x:dif][t:inn(x,class(r))]t1".3r177"(x,t)):is(m0(m0(r)),r)
7156 satz177a:=symis(real,m0(m0(r)),r,satz177):is(r,m0(m0(r)))
7157 s@[i:is(r,m0(s))]
7158 satz177b:=tris(real,m0(r),m0(m0(s)),s,ism0(r,m0(s),i),satz177(s)):is(m0(r),s)
7159 satz177c:=symis(real,m0(r),s,satz177b):is(s,m0(r))
7160 s@[i:is(m0(r),s)]
7161 satz177d:=satz177c(s,r,symis(real,m0(r),s,i)):is(r,m0(s))
7162 satz177e:=symis(real,r,m0(s),satz177d):is(m0(s),r)
7163 +3r178
7164 a0ir@t1:=isin(abs(m0(r)),abs(r),absd(m0d(a0)),absd(a0),aica(m0(r),m0d(a0),micm0),aica,satzd178(a0)):is(abs(m0(r)),abs(r))
7165 -3r178
7166 r@satz178:=realapp1(is(abs(m0(r)),abs(r)),[x:dif][t:inn(x,class(r))]t1".3r178"(x,t)):is(abs(m0(r)),abs(r))
7167 satz178a:=symis(real,abs(m0(r)),abs(r),satz178):is(abs(r),abs(m0(r)))
7168 +3r179
7169 a0ir@t1:=0in(pl(r,m0(r)),pd(a0,m0d(a0)),picp(r,m0(r),a0,m0d(a0),a0ir,micm0),satzd179(a0)):is(pl(r,m0(r)),0)
7170 -3r179
7171 satz179:=realapp1(is(pl(r,m0(r)),0),[x:dif][t:inn(x,class(r))]t1".3r179"(x,t)):is(pl(r,m0(r)),0)
7172 satz179a:=tris(real,pl(m0(r),r),pl(r,m0(r)),0,compl(m0(r),r),satz179):is(pl(m0(r),r),0)
7173 +3r180
7174 b1is@t1:=isin(m0(pl(r,s)),pl(m0(r),m0(s)),m0d(pd(a1,b1)),pd(m0d(a1),m0d(b1)),micm0(pl(r,s),pd(a1,b1),picp),picp(m0(r),m0(s),m0d(a1),m0d(b1),micm0(r,a1,a1ir),micm0(s,b1,b1is)),satzd180(a1,b1)):is(m0(pl(r,s)),pl(m0(r),m0(s)))
7175 -3r180
7176 s@satz180:=realapp2(is(m0(pl(r,s)),pl(m0(r),m0(s))),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t1".3r180"(x,y,t,u)):is(m0(pl(r,s)),pl(m0(r),m0(s)))
7177 satz180a:=symis(real,m0(pl(r,s)),pl(m0(r),m0(s)),satz180):is(pl(m0(r),m0(s)),m0(pl(r,s)))
7178 mn:=pl(r,m0(s)):real
7179 b1is@micmn:=picp(r,m0(s),a1,m0d(b1),a1ir,micm0(s,b1,b1is)):inn(md(a1,b1),class(mn(r,s)))
7180 t@[i:is(r,s)]
7181 ismn1:=ispl1(r,s,m0(t),i):is(mn(r,t),mn(s,t))
7182 ismn2:=ispl2(m0(r),m0(s),t,ism0(r,s,i)):is(mn(t,r),mn(t,s))
7183 u@[i:is(r,s)][j:is(t,u)]
7184 ismn12:=ispl12(r,s,m0(t),m0(u),i,ism0(t,u,j)):is(mn(r,t),mn(s,u))
7185 s@satz181:=tr3is(real,m0(mn(r,s)),pl(m0(r),m0(m0(s))),pl(m0(r),s),mn(s,r),satz180(r,m0(s)),ispl2(m0(m0(s)),s,m0(r),satz177(s)),compl(m0(r),s)):is(m0(mn(r,s)),mn(s,r))
7186 satz181a:=symis(real,m0(mn(s,r)),mn(r,s),satz181(s,r)):is(mn(r,s),m0(mn(s,r)))
7187 +3r182
7188 b1is@[p:pos(mn(r,s))]
7189 t1:=satzd182a(a1,b1,posex(mn(r,s),md(a1,b1),micmn,p)):mored(a1,b1)
7190 t2:=morein(t1):more(r,s)
7191 -3r182
7192 [p:pos(mn(r,s))]
7193 satz182a:=realapp2(more(r,s),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t2".3r182"(x,y,t,u,p)):more(r,s)
7194 +*3r182
7195 b1is@[i:is(mn(r,s),0)]
7196 t3:=satzd182b(a1,b1,0ex(mn(r,s),md(a1,b1),micmn,i)):eq"rp"(a1,b1)
7197 t4:=isin(t3):is(r,s)
7198 -3r182
7199 s@[i:is(mn(r,s),0)]
7200 satz182b:=realapp2(is(r,s),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t4".3r182"(x,y,t,u,i)):is(r,s)
7201 +*3r182
7202 b1is@[n:neg(mn(r,s))]
7203 t5:=satzd182c(a1,b1,negex(mn(r,s),md(a1,b1),micmn,n)):lessd(a1,b1)
7204 t6:=lessin(t5):less(r,s)
7205 -3r182
7206 s@[n:neg(mn(r,s))]
7207 satz182c:=realapp2(less(r,s),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t6".3r182"(x,y,t,u,n)):less(r,s)
7208 +*3r182
7209 b1is@[m:more(r,s)]
7210 t7:=satzd182d(a1,b1,moreex(m)):posd(md(a1,b1))
7211 t8:=posin(mn(r,s),md(a1,b1),micmn,t7):pos(mn(r,s))
7212 -3r182
7213 s@[m:more(r,s)]
7214 satz182d:=realapp2(pos(mn(r,s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t8".3r182"(x,y,t,u,m)):pos(mn(r,s))
7215 +*3r182
7216 b1is@[i:is(r,s)]
7217 t9:=satzd182e(a1,b1,isex(i)):zero(md(a1,b1))
7218 t10:=0in(mn(r,s),md(a1,b1),micmn,t9):is(mn(r,s),0)
7219 -3r182
7220 s@[i:is(r,s)]
7221 satz182e:=realapp2(is(mn(r,s),0),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t10".3r182"(x,y,t,u,i)):is(mn(r,s),0)
7222 +*3r182
7223 b1is@[l:less(r,s)]
7224 t11:=satzd182f(a1,b1,lessex(l)):negd(md(a1,b1))
7225 t12:=negin(mn(r,s),md(a1,b1),micmn,t11):neg(mn(r,s))
7226 -3r182
7227 s@[l:less(r,s)]
7228 satz182f:=realapp2(neg(mn(r,s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t12".3r182"(x,y,t,u,l)):neg(mn(r,s))
7229 +3r183
7230 b1is@[m:more(r,s)]
7231 t1:=satzd183a(a1,b1,moreex(m)):lessd(m0d(a1),m0d(b1))
7232 t2:=lessin(m0(r),m0(s),m0d(a1),m0d(b1),micm0(r,a1,a1ir),micm0(s,b1,b1is),t1):less(m0(r),m0(s))
7233 -3r183
7234 s@[m:more(r,s)]
7235 satz183a:=realapp2(less(m0(r),m0(s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t2".3r183"(x,y,t,u,m)):less(m0(r),m0(s))
7236 s@[i:is(r,s)]
7237 satz183b:=ism0(r,s,i):is(m0(r),m0(s))
7238 +*3r183
7239 b1is@[l:less(r,s)]
7240 t3:=satzd183c(a1,b1,lessex(l)):mored(m0d(a1),m0d(b1))
7241 t4:=morein(m0(r),m0(s),m0d(a1),m0d(b1),micm0(r,a1,a1ir),micm0(s,b1,b1is),t3):more(m0(r),m0(s))
7242 -3r183
7243 s@[l:less(r,s)]
7244 satz183c:=realapp2(more(m0(r),m0(s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t4".3r183"(x,y,t,u,l)):more(m0(r),m0(s))
7245 s@[l:less(m0(r),m0(s))]
7246 satz183d:=ismore12(m0(m0(r)),r,m0(m0(s)),s,satz177(r),satz177(s),satz183c(m0(r),m0(s),l)):more(r,s)
7247 s@[i:is(m0(r),m0(s))]
7248 satz183e:=tr3is(real,r,m0(m0(r)),m0(m0(s)),s,satz177a(r),ism0(m0(r),m0(s),i),satz177(s)):is(r,s)
7249 s@[m:more(m0(r),m0(s))]
7250 satz183f:=isless12(m0(m0(r)),r,m0(m0(s)),s,satz177(r),satz177(s),satz183a(m0(r),m0(s),m)):less(r,s)
7251 +3r184
7252 t@prop1:=and3(pos(s),pos(t),is(r,mn(s,t))):'prop'
7253 s@prop2:=some([x:real]prop1(x)):'prop'
7254 r@prop3:=some([x:real]prop2(x)):'prop'
7255 a0ir@[a:dif][b:dif]
7256 prop1d:=and3(posd(a),posd(b),eq"rp"(a0,md(a,b))):'prop'
7257 a@prop2d:=some"l"(dif,[x:dif]prop1d(x)):'prop'
7258 [p2:prop2d(a)][b:dif][p1:prop1d(a,b)]
7259 t1:=and3e1(posd(a),posd(b),eq"rp"(a0,md(a,b)),p1):posd(a)
7260 t2:=and3e2(posd(a),posd(b),eq"rp"(a0,md(a,b)),p1):posd(b)
7261 t3:=and3e3(posd(a),posd(b),eq"rp"(a0,md(a,b)),p1):eq"rp"(a0,md(a,b))
7262 p2@ra:=realof(a):real
7263 p1@rb:=realof(b):real
7264 t4:=innclass(a):inn(a,class(ra))
7265 t5:=innclass(b):inn(b,class(rb))
7266 t6:=isin(r,mn(ra,rb),a0,md(a,b),a0ir,micmn(ra,rb,a,b,t4,t5),t3):is(r,mn(ra,rb))
7267 t7:=and3i(pos(ra),pos(rb),is(r,mn(ra,rb)),posin(ra,a,t4,t1),posin(rb,b,t5,t2),t6):prop1(ra,rb)
7268 t8:=somei(real,[x:real]prop1(ra,x),rb,t7):prop2(ra)
7269 p2@t9:=someapp(dif,[x:dif]prop1d(a,x),p2,prop2(ra),[x:dif][t:prop1d(a,x)]t8(x,t)):prop2(ra)
7270 t10:=somei(real,[x:real]prop2(x),ra,t9):prop3
7271 a0ir@t11:=someapp(dif,[x:dif]prop2d(x),satzd184(a0),prop3,[x:dif][t:prop2d(x)]t10(x,t)):prop3
7272 -3r184
7273 r@satz184:=realapp1(prop3".3r184",[x:dif][t:inn(x,class(r))]t11".3r184"(x,t)):some([x:real]some([y:real]and3(pos(x),pos(y),is(r,mn(x,y)))))
7274 u@[a3:dif][b3:dif][c3:dif][d3:dif][a3ir:inn(a3,class(r))][b3is:inn(b3,class(s))][c3it:inn(c3,class(t))][d3iu:inn(d3,class(u))]
7275 +3r185
7276 t1:=satzd185(a3,b3,c3,d3):eq"rp"(pd(md(a3,b3),md(c3,d3)),md(pd(a3,c3),pd(b3,d3)))
7277 t2:=isin(pl(mn(r,s),mn(t,u)),mn(pl(r,t),pl(s,u)),pd(md(a3,b3),md(c3,d3)),md(pd(a3,c3),pd(b3,d3)),picp(mn(r,s),mn(t,u),md(a3,b3),md(c3,d3),micmn(r,s,a3,b3,a3ir,b3is),micmn(t,u,c3,d3,c3it,d3iu)),micmn(pl(r,t),pl(s,u),pd(a3,c3),pd(b3,d3),picp(r,t,a3,c3,a3ir,c3it),picp(s,u,b3,d3,b3is,d3iu)),t1):is(pl(mn(r,s),mn(t,u)),mn(pl(r,t),pl(s,u)))
7278 -3r185
7279 u@satz185:=realapp4(is(pl(mn(r,s),mn(t,u)),mn(pl(r,t),pl(s,u))),[x:dif][y:dif][z:dif][v:dif][xi:inn(x,class(r))][yi:inn(y,class(s))][zi:inn(z,class(t))][vi:inn(v,class(u))]t2".3r185"(x,y,z,v,xi,yi,zi,vi)):is(pl(mn(r,s),mn(t,u)),mn(pl(r,t),pl(s,u)))
7280 +3r186
7281 c2it@t1:=satzd186(a2,b2,c2):eq"rp"(pd(pd(a2,b2),c2),pd(a2,pd(b2,c2)))
7282 t2:=isin(pl(pl(r,s),t),pl(r,pl(s,t)),pd(pd(a2,b2),c2),pd(a2,pd(b2,c2)),picp(pl(r,s),t,pd(a2,b2),c2,picp(r,s,a2,b2,a2ir,b2is),c2it),picp(r,pl(s,t),a2,pd(b2,c2),a2ir,picp(s,t,b2,c2,b2is,c2it)),t1):is(pl(pl(r,s),t),pl(r,pl(s,t)))
7283 -3r186
7284 t@satz186:=realapp3(is(pl(pl(r,s),t),pl(r,pl(s,t))),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t2".3r186"(x,y,z,u,v,w)):is(pl(pl(r,s),t),pl(r,pl(s,t)))
7285 asspl1:=satz186:is(pl(pl(r,s),t),pl(r,pl(s,t)))
7286 asspl2:=symis(real,pl(pl(r,s),t),pl(r,pl(s,t)),satz186):is(pl(r,pl(s,t)),pl(pl(r,s),t))
7287 s@plmn:=tris(real,pl(mn(r,s),s),pl(r,pl(m0(s),s)),r,asspl1(r,m0(s),s),pl02(r,pl(m0(s),s),satz179a(s))):is(pl(mn(r,s),s),r)
7288 mnpl:=tris(real,mn(pl(r,s),s),pl(r,pl(s,m0(s))),r,asspl1(r,s,m0(s)),pl02(r,pl(s,m0(s)),satz179(s))):is(mn(pl(r,s),s),r)
7289 satz187a:=tris(real,pl(s,mn(r,s)),pl(mn(r,s),s),r,compl(s,mn(r,s)),plmn):is(pl(s,mn(r,s)),r)
7290 satz187b:=somei(real,[x:real]is(pl(s,x),r),mn(r,s),satz187a):some([x:real]is(pl(s,x),r))
7291 [x:real][i:is(pl(s,x),r)]
7292 satz187c:=tris(real,mn(r,s),mn(pl(x,s),s),x,ismn1(r,pl(x,s),s,tris1(real,r,pl(x,s),pl(s,x),i,compl(s,x))),mnpl(x,s)):is(mn(r,s),x)
7293 satz187d:=symis(real,mn(r,s),x,satz187c):is(x,mn(r,s))
7294 x@[i:is(pl(x,s),r)]
7295 satz187e:=satz187c(tris(real,pl(s,x),pl(x,s),r,compl(s,x),i)):is(mn(r,s),x)
7296 satz187f:=symis(real,mn(r,s),x,satz187e):is(x,mn(r,s))
7297 +3r187
7298 s@[x:real][y:real][i:is(pl(s,x),r)][j:is(pl(s,y),r)]
7299 t1:=tris1(real,x,y,mn(r,s),satz187c(x,i),satz187c(y,j)):is(x,y)
7300 s@t2:=[x:real][y:real][t:is(pl(s,x),r)][u:is(pl(s,y),r)]t1(x,y,t,u):amone(real,[x:real]is(pl(s,x),r))
7301 -3r187
7302 s@satz187:=onei(real,[x:real]is(pl(s,x),r),t2".3r187",satz187b):one([x:real]is(pl(s,x),r))
7303 +3r188
7304 c2it@[m:more(pl(r,t),pl(s,t))]
7305 t1:=satzd188a(a2,b2,c2,moreex(pl(r,t),pl(s,t),pd(a2,c2),pd(b2,c2),picp(r,t,a2,c2,a2ir,c2it),picp(s,t,b2,c2,b2is,c2it),m)):mored(a2,b2)
7306 t2:=morein(r,s,a2,b2,a2ir,b2is,t1):more(r,s)
7307 -3r188
7308 t@[m:more(pl(r,t),pl(s,t))]
7309 satz188a:=realapp3(more(r,s),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t2".3r188"(x,y,z,u,v,w,m)):more(r,s)
7310 +*3r188
7311 c2it@[i:is(pl(r,t),pl(s,t))]
7312 t3:=satzd188b(a2,b2,c2,isex(pl(r,t),pl(s,t),pd(a2,c2),pd(b2,c2),picp(r,t,a2,c2,a2ir,c2it),picp(s,t,b2,c2,b2is,c2it),i)):eq"rp"(a2,b2)
7313 t4:=isin(r,s,a2,b2,a2ir,b2is,t3):is(r,s)
7314 -3r188
7315 t@[i:is(pl(r,t),pl(s,t))]
7316 satz188b:=realapp3(is(r,s),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t4".3r188"(x,y,z,u,v,w,i)):is(r,s)
7317 +*3r188
7318 c2it@[l:less(pl(r,t),pl(s,t))]
7319 t5:=satzd188c(a2,b2,c2,lessex(pl(r,t),pl(s,t),pd(a2,c2),pd(b2,c2),picp(r,t,a2,c2,a2ir,c2it),picp(s,t,b2,c2,b2is,c2it),l)):lessd(a2,b2)
7320 t6:=lessin(r,s,a2,b2,a2ir,b2is,t5):less(r,s)
7321 -3r188
7322 t@[l:less(pl(r,t),pl(s,t))]
7323 satz188c:=realapp3(less(r,s),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t6".3r188"(x,y,z,u,v,w,l)):less(r,s)
7324 +*3r188
7325 c2it@[m:more(r,s)]
7326 t7:=satzd188d(a2,b2,c2,moreex(r,s,a2,b2,a2ir,b2is,m)):mored(pd(a2,c2),pd(b2,c2))
7327 t8:=morein(pl(r,t),pl(s,t),pd(a2,c2),pd(b2,c2),picp(r,t,a2,c2,a2ir,c2it),picp(s,t,b2,c2,b2is,c2it),t7):more(pl(r,t),pl(s,t))
7328 -3r188
7329 t@[m:more(r,s)]
7330 satz188d:=realapp3(more(pl(r,t),pl(s,t)),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t8".3r188"(x,y,z,u,v,w,m)):more(pl(r,t),pl(s,t))
7331 t@[i:is(r,s)]
7332 satz188e:=ispl1(r,s,t,i):is(pl(r,t),pl(s,t))
7333 +*3r188
7334 c2it@[l:less(r,s)]
7335 t9:=satzd188f(a2,b2,c2,lessex(r,s,a2,b2,a2ir,b2is,l)):lessd(pd(a2,c2),pd(b2,c2))
7336 t10:=lessin(pl(r,t),pl(s,t),pd(a2,c2),pd(b2,c2),picp(r,t,a2,c2,a2ir,c2it),picp(s,t,b2,c2,b2is,c2it),t9):less(pl(r,t),pl(s,t))
7337 -3r188
7338 t@[l:less(r,s)]
7339 satz188f:=realapp3(less(pl(r,t),pl(s,t)),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t10".3r188"(x,y,z,u,v,w,l)):less(pl(r,t),pl(s,t))
7340 t@[m:more(pl(t,r),pl(t,s))]
7341 satz188g:=satz188a(ismore12(pl(t,r),pl(r,t),pl(t,s),pl(s,t),compl(t,r),compl(t,s),m)):more(r,s)
7342 t@[i:is(pl(t,r),pl(t,s))]
7343 satz188h:=satz188b(tr3is(real,pl(r,t),pl(t,r),pl(t,s),pl(s,t),compl(r,t),i,compl(t,s))):is(r,s)
7344 t@[l:less(pl(t,r),pl(t,s))]
7345 satz188j:=satz188c(isless12(pl(t,r),pl(r,t),pl(t,s),pl(s,t),compl(t,r),compl(t,s),l)):less(r,s)
7346 t@[m:more(r,s)]
7347 satz188k:=ismore12(pl(r,t),pl(t,r),pl(s,t),pl(t,s),compl(r,t),compl(s,t),satz188d(m)):more(pl(t,r),pl(t,s))
7348 t@[i:is(r,s)]
7349 satz188l:=ispl2(r,s,t,i):is(pl(t,r),pl(t,s))
7350 t@[l:less(r,s)]
7351 satz188m:=isless12(pl(r,t),pl(t,r),pl(s,t),pl(t,s),compl(r,t),compl(s,t),satz188f(l)):less(pl(t,r),pl(t,s))
7352 u@[i:is(r,s)][m:more(t,u)]
7353 satz188n:=ismore2(pl(r,u),pl(s,u),pl(r,t),ispl1(r,s,u,i),satz188k(t,u,r,m)):more(pl(r,t),pl(s,u))
7354 satz188o:=ismore12(pl(r,t),pl(t,r),pl(s,u),pl(u,s),compl(r,t),compl(s,u),satz188n):more(pl(t,r),pl(u,s))
7355 i@[l:less(t,u)]
7356 satz188p:=isless2(pl(r,u),pl(s,u),pl(r,t),ispl1(r,s,u,i),satz188m(t,u,r,l)):less(pl(r,t),pl(s,u))
7357 satz188q:=isless12(pl(r,t),pl(t,r),pl(s,u),pl(u,s),compl(r,t),compl(s,u),satz188p):less(pl(t,r),pl(u,s))
7358 u@[m:more(r,s)][n:more(t,u)]
7359 satz189:=trmore(pl(r,t),pl(s,t),pl(s,u),satz188d(m),satz188k(t,u,s,n)):more(pl(r,t),pl(s,u))
7360 u@[l:less(r,s)][k:less(t,u)]
7361 satz189a:=lemma1(pl(s,u),pl(r,t),satz189(s,r,u,t,lemma2(r,s,l),lemma2(t,u,k))):less(pl(r,t),pl(s,u))
7362 u@[m:moreis(r,s)][n:more(t,u)]
7363 satz190a:=orapp(more(r,s),is(r,s),more(pl(r,t),pl(s,u)),m,[v:more(r,s)]satz189(v,n),[v:is(r,s)]satz188n(v,n)):more(pl(r,t),pl(s,u))
7364 u@[m:more(r,s)][n:moreis(t,u)]
7365 satz190b:=ismore12(pl(t,r),pl(r,t),pl(u,s),pl(s,u),compl(t,r),compl(u,s),satz190a(t,u,r,s,n,m)):more(pl(r,t),pl(s,u))
7366 u@[l:lessis(r,s)][k:less(t,u)]
7367 satz190c:=lemma1(pl(s,u),pl(r,t),satz190a(s,r,u,t,satz168b(l),lemma2(t,u,k))):less(pl(r,t),pl(s,u))
7368 u@[l:less(r,s)][k:lessis(t,u)]
7369 satz190d:=lemma1(pl(s,u),pl(r,t),satz190b(s,r,u,t,lemma2(l),satz168b(t,u,k))):less(pl(r,t),pl(s,u))
7370 +3r191
7371 d3iu@[m:moreis(r,s)][n:moreis(t,u)]
7372 t1:=satzd191(a3,b3,c3,d3,moreisex(r,s,a3,b3,a3ir,b3is,m),moreisex(t,u,c3,d3,c3it,d3iu,n)):moreq(pd(a3,c3),pd(b3,d3))
7373 t2:=moreisin(pl(r,t),pl(s,u),pd(a3,c3),pd(b3,d3),picp(r,t,a3,c3,a3ir,c3it),picp(s,u,b3,d3,b3is,d3iu),t1):moreis(pl(r,t),pl(s,u))
7374 -3r191
7375 u@[m:moreis(r,s)][n:moreis(t,u)]
7376 satz191:=realapp4(moreis(pl(r,t),pl(s,u)),[x:dif][y:dif][z:dif][v:dif][xi:inn(x,class(r))][yi:inn(y,class(s))][zi:inn(z,class(t))][vi:inn(v,class(u))]t2".3r191"(x,y,z,v,xi,yi,zi,vi,m,n)):moreis(pl(r,t),pl(s,u))
7377 u@[l:lessis(r,s)][k:lessis(t,u)]
7378 satz191a:=satz168a(pl(s,u),pl(r,t),satz191(s,r,u,t,satz168b(l),satz168b(t,u,k))):lessis(pl(r,t),pl(s,u))
7379 @timesdr:=[x:dif][y:dif]realof(td(x,y)):[x:dif][y:dif]real
7380 +ivr4
7381 [a:dif][b:dif][c:dif][d:dif][e:eq"rp"(a,b)][f:eq"rp"(c,d)]
7382 t1:=isin(realof(td(a,c)),realof(td(b,d)),td(a,c),td(b,d),innclass(td(a,c)),innclass(td(b,d)),eqtd12(a,b,c,d,e,f)):is(<c><a>timesdr,<d><b>timesdr)
7383 -ivr4
7384 ftimesdr:=[x:dif][y:dif][z:dif][v:dif][t:<y><x>eq][u:<v><z>eq]t1".ivr4"(x,y,z,v,t,u):fixf2(real,timesdr)
7385 s@ts:=indreal2(real,timesdr,ftimesdr,r,s):real
7386 +*ivr4
7387 b1is@t2:=isindreal2(real,timesdr,ftimesdr,r,s,a1,b1,a1ir,b1is):is(realof(td(a1,b1)),ts(r,s))
7388 -ivr4
7389 b1is@tict:=isp(real,[x:real]inn(td(a1,b1),class(x)),realof(td(a1,b1)),ts(r,s),innclass(td(a1,b1)),t2".ivr4"):inn(td(a1,b1),class(ts(r,s)))
7390 t@[i:is(r,s)]
7391 ists1:=isf(real,real,[x:real]ts(x,t),r,s,i):is(ts(r,t),ts(s,t))
7392 ists2:=isf(real,real,[x:real]ts(t,x),r,s,i):is(ts(t,r),ts(t,s))
7393 u@[i:is(r,s)][j:is(t,u)]
7394 ists12:=tris(real,ts(r,t),ts(s,t),ts(s,u),ists1(i),ists2(t,u,s,j)):is(ts(r,t),ts(s,u))
7395 +4r192
7396 b1is@[i:is(r,0)]
7397 t1:=satzd192a(a1,b1,0ex(r,a1,a1ir,i)):zero(td(a1,b1))
7398 t2:=0in(ts(r,s),td(a1,b1),tict,t1):is(ts(r,s),0)
7399 -4r192
7400 s@[i:is(r,0)]
7401 satz192a:=realapp2(is(ts(r,s),0),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t2".4r192"(x,y,t,u,i)):is(ts(r,s),0)
7402 +*4r192
7403 b1is@[i:is(s,0)]
7404 t3:=satzd192b(a1,b1,0ex(s,b1,b1is,i)):zero(td(a1,b1))
7405 t4:=0in(ts(r,s),td(a1,b1),tict,t3):is(ts(r,s),0)
7406 -4r192
7407 s@[i:is(s,0)]
7408 satz192b:=realapp2(is(ts(r,s),0),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t4".4r192"(x,y,t,u,i)):is(ts(r,s),0)
7409 +*4r192
7410 b1is@[i:is(ts(r,s),0)]
7411 t5:=satzd192c(a1,b1,0ex(ts(r,s),td(a1,b1),tict,i)):or(zero(a1),zero(b1))
7412 t6:=th9"l.or"(zero(a1),zero(b1),is(r,0),is(s,0),t5,[t:zero(a1)]0in(r,a1,a1ir,t),[t:zero(b1)]0in(s,b1,b1is,t)):or(is(r,0),is(s,0))
7413 -4r192
7414 s@[i:is(ts(r,s),0)]
7415 satz192c:=realapp2(or(is(r,0),is(s,0)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t6".4r192"(x,y,t,u,i)):or(is(r,0),is(s,0))
7416 s@[n:nis(r,0)][o:nis(s,0)]
7417 satz192d:=th3"l.imp"(is(ts(r,s),0),or(is(r,0),is(s,0)),th3"l.or"(is(r,0),is(s,0),n,o),[t:is(ts(r,s),0)]satz192c(t)):nis(ts(r,s),0)
7418 s@[i:is(r,0)]
7419 ts01:=satz192a(i):is(ts(r,s),0)
7420 s@[i:is(s,0)]
7421 ts02:=satz192b(i):is(ts(r,s),0)
7422 +4r193
7423 b1is@t1:=satzd193(a1,b1):eq"rp"(absd(td(a1,b1)),td(absd(a1),absd(b1)))
7424 t2:=isin(abs(ts(r,s)),ts(abs(r),abs(s)),absd(td(a1,b1)),td(absd(a1),absd(b1)),aica(ts(r,s),td(a1,b1),tict),tict(abs(r),abs(s),absd(a1),absd(b1),aica(r,a1,a1ir),aica(s,b1,b1is)),t1):is(abs(ts(r,s)),ts(abs(r),abs(s)))
7425 -4r193
7426 s@satz193:=realapp2(is(abs(ts(r,s)),ts(abs(r),abs(s))),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t2".4r193"(x,y,t,u)):is(abs(ts(r,s)),ts(abs(r),abs(s)))
7427 satz193a:=symis(real,abs(ts(r,s)),ts(abs(r),abs(s)),satz193):is(ts(abs(r),abs(s)),abs(ts(r,s)))
7428 +4r194
7429 b1is@t1:=satzd194(a1,b1):eq"rp"(td(a1,b1),td(b1,a1))
7430 t2:=isin(ts(r,s),ts(s,r),td(a1,b1),td(b1,a1),tict,tict(s,r,b1,a1,b1is,a1ir),t1):is(ts(r,s),ts(s,r))
7431 -4r194
7432 satz194:=realapp2(is(ts(r,s),ts(s,r)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t2".4r194"(x,y,t,u)):is(ts(r,s),ts(s,r))
7433 comts:=satz194:is(ts(r,s),ts(s,r))
7434 @1rl:=realof(1df):real
7435 pos1:=posin(1rl,1df,innclass(1df),posdirp(1rp)):pos(1rl)
7436 natrl1:=natrli(1):natrl(1rl)
7437 intrl1:=natintrl(1rl,natrl1):intrl(1rl)
7438 +4r195
7439 a0ir@t1:=satzd195(a0):eq"rp"(td(a0,1df),a0)
7440 t2:=isin(ts(r,1rl),r,td(a0,1df),a0,tict(r,1rl,a0,1df,a0ir,innclass(1df)),a0ir,t1):is(ts(r,1rl),r)
7441 -4r195
7442 r@satz195:=realapp1(is(ts(r,1rl),r),[x:dif][t:inn(x,class(r))]t2".4r195"(x,t)):is(ts(r,1rl),r)
7443 satz195a:=symis(real,ts(r,1rl),r,satz195):is(r,ts(r,1rl))
7444 satz195b:=tris(real,ts(1rl,r),ts(r,1rl),r,comts(1rl,r),satz195):is(ts(1rl,r),r)
7445 satz195c:=symis(real,ts(1rl,r),r,satz195b):is(r,ts(1rl,r))
7446 s@[p:pos(r)][q:pos(s)]
7447 satz196a:=symis(real,ts(abs(r),abs(s)),ts(r,s),ists12(abs(r),r,abs(s),s,absp(r,p),absp(s,q))):is(ts(r,s),ts(abs(r),abs(s)))
7448 +4r196
7449 b1is@[n:neg(r)][o:neg(s)]
7450 t1:=satzd196b(a1,b1,negex(r,a1,a1ir,n),negex(s,b1,b1is,o)):eq"rp"(td(a1,b1),td(absd(a1),absd(b1)))
7451 t2:=isin(ts(r,s),ts(abs(r),abs(s)),td(a1,b1),td(absd(a1),absd(b1)),tict,tict(abs(r),abs(s),absd(a1),absd(b1),aica(r,a1,a1ir),aica(s,b1,b1is)),t1):is(ts(r,s),ts(abs(r),abs(s)))
7452 -4r196
7453 s@[n:neg(r)][o:neg(s)]
7454 satz196b:=realapp2(is(ts(r,s),ts(abs(r),abs(s))),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t2".4r196"(x,y,t,u,n,o)):is(ts(r,s),ts(abs(r),abs(s)))
7455 +*4r196
7456 b1is@[p:pos(r)][n:neg(s)]
7457 t1a:=satzd196c(a1,b1,posex(r,a1,a1ir,p),negex(s,b1,b1is,n)):eq"rp"(td(a1,b1),m0d(td(absd(a1),absd(b1))))
7458 t2a:=isin(ts(r,s),m0(ts(abs(r),abs(s))),td(a1,b1),m0d(td(absd(a1),absd(b1))),tict,micm0(ts(abs(r),abs(s)),td(absd(a1),absd(b1)),tict(abs(r),abs(s),absd(a1),absd(b1),aica(r,a1,a1ir),aica(s,b1,b1is))),t1a):is(ts(r,s),m0(ts(abs(r),abs(s))))
7459 -4r196
7460 s@[p:pos(r)][n:neg(s)]
7461 satz196c:=realapp2(is(ts(r,s),m0(ts(abs(r),abs(s)))),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t2a".4r196"(x,y,t,u,p,n)):is(ts(r,s),m0(ts(abs(r),abs(s))))
7462 s@[n:neg(r)][p:pos(s)]
7463 satz196d:=tr3is(real,ts(r,s),ts(s,r),m0(ts(abs(s),abs(r))),m0(ts(abs(r),abs(s))),comts(r,s),satz196c(s,r,p,n),ism0(ts(abs(s),abs(r)),ts(abs(r),abs(s)),comts(abs(s),abs(r)))):is(ts(r,s),m0(ts(abs(r),abs(s))))
7464 +*4r196
7465 a0ir@[n:not(is(r,0))]
7466 t3:=th3"l.imp"(zero(a0),is(r,0),n,[t:zero(a0)]0in(t)):not(zero(a0))
7467 b1is@[n:not(is(r,0))][o:not(is(s,0))][i:is(ts(r,s),ts(abs(r),abs(s)))]
7468 t4:=satzd196e(a1,b1,t3(r,a1,a1ir,n),t3(s,b1,b1is,o),isex(ts(r,s),ts(abs(r),abs(s)),td(a1,b1),td(absd(a1),absd(b1)),tict,tict(abs(r),abs(s),absd(a1),absd(b1),aica(r,a1,a1ir),aica(s,b1,b1is)),i)):or(and(posd(a1),posd(b1)),and(negd(a1),negd(b1)))
7469 [a:and(posd(a1),posd(b1))]
7470 t5:=andi(pos(r),pos(s),posin(r,a1,a1ir,ande1(posd(a1),posd(b1),a)),posin(s,b1,b1is,ande2(posd(a1),posd(b1),a))):and(pos(r),pos(s))
7471 i@[a:and(negd(a1),negd(b1))]
7472 t6:=andi(neg(r),neg(s),negin(r,a1,a1ir,ande1(negd(a1),negd(b1),a)),negin(s,b1,b1is,ande2(negd(a1),negd(b1),a))):and(neg(r),neg(s))
7473 i@t7:=th9"l.or"(and(posd(a1),posd(b1)),and(negd(a1),negd(b1)),and(pos(r),pos(s)),and(neg(r),neg(s)),t4,[t:and(posd(a1),posd(b1))]t5(t),[t:and(negd(a1),negd(b1))]t6(t)):or(and(pos(r),pos(s)),and(neg(r),neg(s)))
7474 -4r196
7475 s@[n:not(is(r,0))][o:not(is(s,0))][i:is(ts(r,s),ts(abs(r),abs(s)))]
7476 satz196e:=realapp2(or(and(pos(r),pos(s)),and(neg(r),neg(s))),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t7".4r196"(x,y,t,u,n,o,i)):or(and(pos(r),pos(s)),and(neg(r),neg(s)))
7477 +*4r196
7478 o@[i:is(ts(r,s),m0(ts(abs(r),abs(s))))]
7479 t8:=satzd196f(a1,b1,t3(r,a1,a1ir,n),t3(s,b1,b1is,o),isex(ts(r,s),m0(ts(abs(r),abs(s))),td(a1,b1),m0d(td(absd(a1),absd(b1))),tict,micm0(ts(abs(r),abs(s)),td(absd(a1),absd(b1)),tict(abs(r),abs(s),absd(a1),absd(b1),aica(r,a1,a1ir),aica(s,b1,b1is))),i)):or(and(posd(a1),negd(b1)),and(negd(a1),posd(b1)))
7480 [a:and(posd(a1),negd(b1))]
7481 t9:=andi(pos(r),neg(s),posin(r,a1,a1ir,ande1(posd(a1),negd(b1),a)),negin(s,b1,b1is,ande2(posd(a1),negd(b1),a))):and(pos(r),neg(s))
7482 i@[a:and(negd(a1),posd(b1))]
7483 t10:=andi(neg(r),pos(s),negin(r,a1,a1ir,ande1(negd(a1),posd(b1),a)),posin(s,b1,b1is,ande2(negd(a1),posd(b1),a))):and(neg(r),pos(s))
7484 i@t11:=th9"l.or"(and(posd(a1),negd(b1)),and(negd(a1),posd(b1)),and(pos(r),neg(s)),and(neg(r),pos(s)),t8,[t:and(posd(a1),negd(b1))]t9(t),[t:and(negd(a1),posd(b1))]t10(t)):or(and(pos(r),neg(s)),and(neg(r),pos(s)))
7485 -4r196
7486 o@[i:is(ts(r,s),m0(ts(abs(r),abs(s))))]
7487 satz196f:=realapp2(or(and(pos(r),neg(s)),and(neg(r),pos(s))),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t11".4r196"(x,y,t,u,n,o,i)):or(and(pos(r),neg(s)),and(neg(r),pos(s)))
7488 s@[p:pos(ts(r,s))]
7489 +*4r196
7490 p"r"@t12:=th3"l.imp"(is(r,0),is(ts(r,s),0),pnot0(ts(r,s),p),[t:is(r,0)]ts01(r,s,t)):nis(r,0)
7491 t13:=th3"l.imp"(is(s,0),is(ts(r,s),0),pnot0(ts(r,s),p),[t:is(s,0)]ts02(r,s,t)):nis(s,0)
7492 t14:=tris1(real,ts(r,s),ts(abs(r),abs(s)),abs(ts(r,s)),absp(ts(r,s),p),satz193(r,s)):is(ts(r,s),ts(abs(r),abs(s)))
7493 -4r196
7494 p@satz196g:=satz196e(t12".4r196",t13".4r196",t14".4r196"):or(and(pos(r),pos(s)),and(neg(r),neg(s)))
7495 s@[n:neg(ts(r,s))]
7496 +*4r196
7497 n"r"@t15:=th3"l.imp"(is(r,0),is(ts(r,s),0),nnot0(ts(r,s),n),[t:is(r,0)]ts01(r,s,t)):nis(r,0)
7498 t16:=th3"l.imp"(is(s,0),is(ts(r,s),0),nnot0(ts(r,s),n),[t:is(s,0)]ts02(r,s,t)):nis(s,0)
7499 t17:=satz177c(ts(abs(r),abs(s)),ts(r,s),tris(real,ts(abs(r),abs(s)),abs(ts(r,s)),m0(ts(r,s)),satz193a(r,s),absn(ts(r,s),n))):is(ts(r,s),m0(ts(abs(r),abs(s))))
7500 -4r196
7501 n@satz196h:=satz196f(t15".4r196",t16".4r196",t17".4r196"):or(and(pos(r),neg(s)),and(neg(r),pos(s)))
7502 +4r197
7503 b1is@t1:=satzd197a(a1,b1):eq"rp"(td(m0d(a1),b1),m0d(td(a1,b1)))
7504 t2:=isin(ts(m0(r),s),m0(ts(r,s)),td(m0d(a1),b1),m0d(td(a1,b1)),tict(m0(r),s,m0d(a1),b1,micm0(r,a1,a1ir),b1is),micm0(ts(r,s),td(a1,b1),tict),t1):is(ts(m0(r),s),m0(ts(r,s)))
7505 -4r197
7506 s@satz197a:=realapp2(is(ts(m0(r),s),m0(ts(r,s))),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t2".4r197"(x,y,t,u)):is(ts(m0(r),s),m0(ts(r,s)))
7507 satz197b:=tr3is(real,ts(r,m0(s)),ts(m0(s),r),m0(ts(s,r)),m0(ts(r,s)),comts(r,m0(s)),satz197a(s,r),ism0(ts(s,r),ts(r,s),comts(s,r))):is(ts(r,m0(s)),m0(ts(r,s)))
7508 satz197c:=tris2(real,ts(m0(r),s),ts(r,m0(s)),m0(ts(r,s)),satz197a,satz197b):is(ts(m0(r),s),ts(r,m0(s)))
7509 satz197d:=symis(real,ts(m0(r),s),ts(r,m0(s)),satz197c):is(ts(r,m0(s)),ts(m0(r),s))
7510 satz197e:=symis(real,ts(m0(r),s),m0(ts(r,s)),satz197a):is(m0(ts(r,s)),ts(m0(r),s))
7511 satz197f:=symis(real,ts(r,m0(s)),m0(ts(r,s)),satz197b):is(m0(ts(r,s)),ts(r,m0(s)))
7512 satz198:=tris(real,ts(m0(r),m0(s)),ts(r,m0(m0(s))),ts(r,s),satz197c(r,m0(s)),ists2(m0(m0(s)),s,r,satz177(s))):is(ts(m0(r),m0(s)),ts(r,s))
7513 satz198a:=symis(real,ts(m0(r),m0(s)),ts(r,s),satz198):is(ts(r,s),ts(m0(r),m0(s)))
7514 +*ivr4
7515 b1is@[p:pos(r)][q:pos(s)]
7516 t3:=ptdpp(a1,b1,posex(r,a1,a1ir,p),posex(s,b1,b1is,q)):posd(td(a1,b1))
7517 t4:=posin(ts(r,s),td(a1,b1),tict,t3):pos(ts(r,s))
7518 -ivr4
7519 s@[p:pos(r)][q:pos(s)]
7520 postspp:=realapp2(pos(ts(r,s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t4".ivr4"(x,y,t,u,p,q)):pos(ts(r,s))
7521 +*ivr4
7522 p@[n:neg(s)]
7523 t5:=ntdpn(a1,b1,posex(r,a1,a1ir,p),negex(s,b1,b1is,n)):negd(td(a1,b1))
7524 t6:=negin(ts(r,s),td(a1,b1),tict,t5):neg(ts(r,s))
7525 -ivr4
7526 s@[p:pos(r)][n:neg(s)]
7527 negtspn:=realapp2(neg(ts(r,s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t6".ivr4"(x,y,t,u,p,n)):neg(ts(r,s))
7528 s@[n:neg(r)][p:pos(s)]
7529 negtsnp:=isneg(ts(s,r),ts(r,s),comts(s,r),negtspn(s,r,p,n)):neg(ts(r,s))
7530 s@[n:neg(r)][o:neg(s)]
7531 postsnn:=ispos(ts(m0(r),m0(s)),ts(r,s),satz198,postspp(m0(r),m0(s),satz176c(r,n),satz176c(s,o))):pos(ts(r,s))
7532 r@[n:nis(r,0)]
7533 possq:=rapp(r,pos(ts(r,r)),[t:pos(r)]postspp(r,r,t,t),th2"l.imp"(is(r,0),pos(ts(r,r)),n),[t:neg(r)]postsnn(r,r,t,t)):pos(ts(r,r))
7534 r@nnegsq:=th1"l.imp"(is(r,0),not(neg(ts(r,r))),[t:is(r,0)]0notn(ts(r,r),satz192a(r,r,t)),[t:nis(r,0)]pnotn(ts(r,r),possq(r,t))):not(neg(ts(r,r)))
7535 +4r199
7536 c2it@t1:=satzd199(a2,b2,c2):eq"rp"(td(td(a2,b2),c2),td(a2,td(b2,c2)))
7537 t2:=isin(ts(ts(r,s),t),ts(r,ts(s,t)),td(td(a2,b2),c2),td(a2,td(b2,c2)),tict(ts(r,s),t,td(a2,b2),c2,tict(r,s,a2,b2,a2ir,b2is),c2it),tict(r,ts(s,t),a2,td(b2,c2),a2ir,tict(s,t,b2,c2,b2is,c2it)),t1):is(ts(ts(r,s),t),ts(r,ts(s,t)))
7538 -4r199
7539 t@satz199:=realapp3(is(ts(ts(r,s),t),ts(r,ts(s,t))),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t2".4r199"(x,y,z,u,v,w)):is(ts(ts(r,s),t),ts(r,ts(s,t)))
7540 assts1:=satz199:is(ts(ts(r,s),t),ts(r,ts(s,t)))
7541 assts2:=symis(real,ts(ts(r,s),t),ts(r,ts(s,t)),satz199):is(ts(r,ts(s,t)),ts(ts(r,s),t))
7542 +4r201
7543 c2it@t1:=satzd201(a2,b2,c2):eq"rp"(td(a2,pd(b2,c2)),pd(td(a2,b2),td(a2,c2)))
7544 t2:=isin(ts(r,pl(s,t)),pl(ts(r,s),ts(r,t)),td(a2,pd(b2,c2)),pd(td(a2,b2),td(a2,c2)),tict(r,pl(s,t),a2,pd(b2,c2),a2ir,picp(s,t,b2,c2,b2is,c2it)),picp(ts(r,s),ts(r,t),td(a2,b2),td(a2,c2),tict(r,s,a2,b2,a2ir,b2is),tict(r,t,a2,c2,a2ir,c2it)),t1):is(ts(r,pl(s,t)),pl(ts(r,s),ts(r,t)))
7545 -4r201
7546 satz201:=realapp3(is(ts(r,pl(s,t)),pl(ts(r,s),ts(r,t))),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t2".4r201"(x,y,z,u,v,w)):is(ts(r,pl(s,t)),pl(ts(r,s),ts(r,t)))
7547 disttp1:=tr3is(real,ts(pl(r,s),t),ts(t,pl(r,s)),pl(ts(t,r),ts(t,s)),pl(ts(r,t),ts(s,t)),comts(pl(r,s),t),satz201(t,r,s),ispl12(ts(t,r),ts(r,t),ts(t,s),ts(s,t),comts(t,r),comts(t,s))):is(ts(pl(r,s),t),pl(ts(r,t),ts(s,t)))
7548 disttp2:=satz201:is(ts(r,pl(s,t)),pl(ts(r,s),ts(r,t)))
7549 distpt1:=symis(real,ts(pl(r,s),t),pl(ts(r,t),ts(s,t)),disttp1):is(pl(ts(r,t),ts(s,t)),ts(pl(r,s),t))
7550 distpt2:=symis(real,ts(r,pl(s,t)),pl(ts(r,s),ts(r,t)),disttp2):is(pl(ts(r,s),ts(r,t)),ts(r,pl(s,t)))
7551 satz202:=tris(real,ts(r,mn(s,t)),pl(ts(r,s),ts(r,m0(t))),mn(ts(r,s),ts(r,t)),disttp2(r,s,m0(t)),ispl2(ts(r,m0(t)),m0(ts(r,t)),ts(r,s),satz197b(r,t))):is(ts(r,mn(s,t)),mn(ts(r,s),ts(r,t)))
7552 disttm1:=tris(real,ts(mn(r,s),t),pl(ts(r,t),ts(m0(s),t)),mn(ts(r,t),ts(s,t)),disttp1(r,m0(s),t),ispl2(ts(m0(s),t),m0(ts(s,t)),ts(r,t),satz197a(s,t))):is(ts(mn(r,s),t),mn(ts(r,t),ts(s,t)))
7553 disttm2:=satz202:is(ts(r,mn(s,t)),mn(ts(r,s),ts(r,t)))
7554 distmt1:=symis(real,ts(mn(r,s),t),mn(ts(r,t),ts(s,t)),disttm1):is(mn(ts(r,t),ts(s,t)),ts(mn(r,s),t))
7555 distmt2:=symis(real,ts(r,mn(s,t)),mn(ts(r,s),ts(r,t)),disttm2):is(mn(ts(r,s),ts(r,t)),ts(r,mn(s,t)))
7556 satz200:=satz202:is(ts(r,mn(s,t)),mn(ts(r,s),ts(r,t)))
7557 +4r203
7558 c2it@[m:more(r,s)][p:pos(t)]
7559 t1:=satzd203a(a2,b2,c2,moreex(r,s,a2,b2,a2ir,b2is,m),posex(t,c2,c2it,p)):mored(td(a2,c2),td(b2,c2))
7560 t2:=morein(ts(r,t),ts(s,t),td(a2,c2),td(b2,c2),tict(r,t,a2,c2,a2ir,c2it),tict(s,t,b2,c2,b2is,c2it),t1):more(ts(r,t),ts(s,t))
7561 -4r203
7562 [m:more(r,s)][p:pos(t)]
7563 satz203a:=realapp3(more(ts(r,t),ts(s,t)),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t2".4r203"(x,y,z,u,v,w,m,p)):more(ts(r,t),ts(s,t))
7564 m@[i:is(t,0)]
7565 satz203b:=tris2(real,ts(r,t),ts(s,t),0,ts02(r,t,i),ts02(s,t,i)):is(ts(r,t),ts(s,t))
7566 +*4r203
7567 m@[n:neg(t)]
7568 t3:=satzd203c(a2,b2,c2,moreex(r,s,a2,b2,a2ir,b2is,m),negex(t,c2,c2it,n)):lessd(td(a2,c2),td(b2,c2))
7569 t4:=lessin(ts(r,t),ts(s,t),td(a2,c2),td(b2,c2),tict(r,t,a2,c2,a2ir,c2it),tict(s,t,b2,c2,b2is,c2it),t3):less(ts(r,t),ts(s,t))
7570 -4r203
7571 m@[n:neg(t)]
7572 satz203c:=realapp3(less(ts(r,t),ts(s,t)),[x:dif][y:dif][z:dif][u:inn(x,class(r))][v:inn(y,class(s))][w:inn(z,class(t))]t4".4r203"(x,y,z,u,v,w,m,n)):less(ts(r,t),ts(s,t))
7573 p@satz203d:=ismore12(ts(r,t),ts(t,r),ts(s,t),ts(t,s),comts(r,t),comts(s,t),satz203a):more(ts(t,r),ts(t,s))
7574 i@satz203e:=tris2(real,ts(t,r),ts(t,s),0,ts01(t,r,i),ts01(t,s,i)):is(ts(t,r),ts(t,s))
7575 n@satz203f:=isless12(ts(r,t),ts(t,r),ts(s,t),ts(t,s),comts(r,t),comts(s,t),satz203c):less(ts(t,r),ts(t,s))
7576 t@[l:less(r,s)][p:pos(t)]
7577 satz203g:=lemma1(ts(s,t),ts(r,t),satz203a(s,r,t,lemma2(r,s,l),p)):less(ts(r,t),ts(s,t))
7578 l@[i:is(t,0)]
7579 satz203h:=tris2(real,ts(r,t),ts(s,t),0,ts02(r,t,i),ts02(s,t,i)):is(ts(r,t),ts(s,t))
7580 l@[n:neg(t)]
7581 satz203j:=lemma2(ts(s,t),ts(r,t),satz203c(s,r,t,lemma2(r,s,l),n)):more(ts(r,t),ts(s,t))
7582 p@satz203k:=lemma1(ts(t,s),ts(t,r),satz203d(s,r,t,lemma2(r,s,l),p)):less(ts(t,r),ts(t,s))
7583 i@satz203l:=tris2(real,ts(t,r),ts(t,s),0,ts01(t,r,i),ts01(t,s,i)):is(ts(t,r),ts(t,s))
7584 n@satz203m:=lemma2(ts(t,s),ts(t,r),satz203f(s,r,t,lemma2(r,s,l),n)):more(ts(t,r),ts(t,s))
7585 +4r204
7586 a0ir@[n1:nis(r,0)]
7587 t1:=th3"l.imp"(zero(a0),is(r,0),n1,[t:zero(a0)]0in(t)):not(zero(a0))
7588 d3iu@[n1:nis(s,0)][i:is(ts(s,t),r)][j:is(ts(s,u),r)]
7589 t2:=satzd204b(a3,b3,t1(s,b3,b3is,n1),c3,d3,isex(ts(s,t),r,td(b3,c3),a3,tict(s,t,b3,c3,b3is,c3it),a3ir,i),isex(ts(s,u),r,td(b3,d3),a3,tict(s,u,b3,d3,b3is,d3iu),a3ir,j)):eq"rp"(c3,d3)
7590 t3:=isin(t,u,c3,d3,c3it,d3iu,t2):is(t,u)
7591 -4r204
7592 s@[n:nis(s,0)][x:real][y:real][i:is(ts(s,x),r)][j:is(ts(s,y),r)]
7593 satz204b:=realapp4(x,y,is(x,y),[z:dif][u:dif][v:dif][w:dif][zi:inn(z,class(r))][ui:inn(u,class(s))][vi:inn(v,class(x))][wi:inn(w,class(y))]t3".4r204"(x,y,z,u,v,w,zi,ui,vi,wi,n,i,j)):is(x,y)
7594 +*4r204
7595 b1is@[n1:nis(s,0)]
7596 t4:=satzd204a(a1,b1,t1(s,b1,b1is,n1)):some"l"(dif,[x:dif]eq"rp"(td(b1,x),a1))
7597 [a:dif][e:eq"rp"(td(b1,a),a1)]
7598 ar:=realof(a):real
7599 t5:=isin(ts(s,ar),r,td(b1,a),a1,tict(s,ar,b1,a,b1is,innclass(a)),a1ir,e):is(ts(s,ar),r)
7600 t6:=somei(real,[x:real]is(ts(s,x),r),ar,t5):some([x:real]is(ts(s,x),r))
7601 n1@t7:=someapp(dif,[x:dif]eq"rp"(td(b1,x),a1),t4,some([x:real]is(ts(s,x),r)),[x:dif][t:eq"rp"(td(b1,x),a1)]t6(x,t)):some([x:real]is(ts(s,x),r))
7602 -4r204
7603 n@satz204a:=realapp2(some([x:real]is(ts(s,x),r)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t7".4r204"(x,y,t,u,n)):some([x:real]is(ts(s,x),r))
7604 satz204:=onei(real,[x:real]is(ts(s,x),r),[x:real][y:real][t:is(ts(s,x),r)][u:is(ts(s,y),r)]satz204b(x,y,t,u),satz204a):one([x:real]is(ts(s,x),r))
7605 ov:=ind(real,[x:real]is(ts(s,x),r),satz204):real
7606 satz204c:=oneax(real,[x:real]is(ts(s,x),r),satz204):is(ts(s,ov(r,s,n)),r)
7607 satz204d:=symis(real,ts(s,ov(r,s,n)),r,satz204c):is(r,ts(s,ov(r,s,n)))
7608 satz204e:=tris(real,ts(ov(r,s,n),s),ts(s,ov(r,s,n)),r,comts(ov(r,s,n),s),satz204c):is(ts(ov(r,s,n),s),r)
7609 satz204f:=symis(real,ts(ov(r,s,n),s),r,satz204e):is(r,ts(ov(r,s,n),s))
7610 s@[x:real][n:nis(s,0)][i:is(ts(s,x),r)]
7611 satz204g:=satz204b(n,x,ov(r,s,n),i,satz204c(n)):is(x,ov(r,s,n))
7612 s@[n:nis(s,0)][p:pos(r)][q:pos(s)]
7613 +*4r204
7614 n@ros:=ov(r,s,n):real
7615 p@t8:=ispos(r,ts(s,ros),satz204d(n),p):pos(ts(s,ros))
7616 q@t9:=th1"l.and"(neg(s),neg(ros),pnotn(s,q)):not(and(neg(s),neg(ros)))
7617 t10:=ore1(and(pos(s),pos(ros)),and(neg(s),neg(ros)),satz196g(s,ros,t8),t9):and(pos(s),pos(ros))
7618 -4r204
7619 q@posovpp:=ande2(pos(s),pos(ov(r,s,n)),t10".4r204"):pos(ov(r,s,n))
7620 p@[m:neg(s)]
7621 +*4r204
7622 m@t11:=th1"l.and"(pos(s),pos(ros),nnotp(s,m)):not(and(pos(s),pos(ros)))
7623 t12:=ore2(and(pos(s),pos(ros)),and(neg(s),neg(ros)),satz196g(s,ros,t8),t11):and(neg(s),neg(ros))
7624 -4r204
7625 m@negovpn:=ande2(neg(s),neg(ov(r,s,n)),t12".4r204"):neg(ov(r,s,n))
7626 n@[m:neg(r)][p:pos(s)]
7627 +*4r204
7628 m@t13:=isneg(r,ts(s,ros),satz204d(n),m):neg(ts(s,ros))
7629 p@t14:=th1"l.and"(neg(s),pos(ros),pnotn(s,p)):not(and(neg(s),pos(ros)))
7630 t15:=ore1(and(pos(s),neg(ros)),and(neg(s),pos(ros)),satz196h(s,ros,t13),t14):and(pos(s),neg(ros))
7631 -4r204
7632 p@negovnp:=ande2(pos(s),neg(ov(r,s,n)),t15".4r204"):neg(ov(r,s,n))
7633 m@[l:neg(s)]
7634 +*4r204
7635 l@t16:=th1"l.and"(pos(s),neg(ros),nnotp(s,l)):not(and(pos(s),neg(ros)))
7636 t17:=ore2(and(pos(s),neg(ros)),and(neg(s),pos(ros)),satz196h(s,ros,t13),t16):and(neg(s),pos(ros))
7637 -4r204
7638 l@posovnn:=ande2(neg(s),pos(ov(r,s,n)),t17".4r204"):pos(ov(r,s,n))
7639 @[r0:cut][s0:cut][m:more"rp"(r0,s0)]
7640 morerpep:=morein(pofrp(r0),pofrp(s0),pdofrp(r0),pdofrp(s0),innclass(pdofrp(r0)),innclass(pdofrp(s0)),morerpepd(r0,s0,m)):more(pofrp(r0),pofrp(s0))
7641 s0@[m:more(pofrp(r0),pofrp(s0))]
7642 morerpip:=morerpipd(r0,s0,moreex(pofrp(r0),pofrp(s0),pdofrp(r0),pdofrp(s0),innclass(pdofrp(r0)),innclass(pdofrp(s0)),m)):more"rp"(r0,s0)
7643 s0@[l:less"rp"(r0,s0)]
7644 lessrpep:=lemma1(pofrp(s0),pofrp(r0),morerpep(s0,r0,satz122(r0,s0,l))):less(pofrp(r0),pofrp(s0))
7645 s0@[l:less(pofrp(r0),pofrp(s0))]
7646 lessrpip:=satz121(s0,r0,morerpip(s0,r0,lemma2(pofrp(r0),pofrp(s0),l))):less"rp"(r0,s0)
7647 s@[p:pos(r)][q:pos(s)][m:more(r,s)]
7648 q@[m:more(r,s)]
7649 +ivr5
7650 t1:=ismore12(r,pofrp(rpofp(r,p)),s,pofrp(rpofp(s,q)),isprp1(r,p),isprp1(s,q),m):more(pofrp(rpofp(r,p)),pofrp(rpofp(s,q)))
7651 -ivr5
7652 moreperp:=morerpip(rpofp(r,p),rpofp(s,q),t1".ivr5"):more"rp"(rpofp(r,p),rpofp(s,q))
7653 q@[m:more"rp"(rpofp(r,p),rpofp(s,q))]
7654 +*ivr5
7655 m@t2:=morerpep(rpofp(r,p),rpofp(s,q),m):more(pofrp(rpofp(r,p)),pofrp(rpofp(s,q)))
7656 -ivr5
7657 m@morepirp:=ismore12(pofrp(rpofp(r,p)),r,pofrp(rpofp(s,q)),s,isprp2(r,p),isprp2(s,q),t2".ivr5"):more(r,s)
7658 q@[l:less(r,s)]
7659 lessperp:=satz121(rpofp(s,q),rpofp(r,p),moreperp(s,r,q,p,lemma2(r,s,l))):less"rp"(rpofp(r,p),rpofp(s,q))
7660 q@[l:less"rp"(rpofp(r,p),rpofp(s,q))]
7661 lesspirp:=lemma1(s,r,morepirp(s,r,q,p,satz122(rpofp(r,p),rpofp(s,q),l))):less(r,s)
7662 r@s01:=setof(real,[x:real]lessis(x,r)):set(real)
7663 s02:=setof(real,[x:real]more(x,r)):set(real)
7664 +5r205
7665 s@[n:not(in(s,s01))]
7666 t1:=th3"l.imp"(lessis(s,r),in(s,s01),n,[t:lessis(s,r)]estii(real,[x:real]lessis(x,r),s,t)):not(lessis(s,r))
7667 t2:=estii(real,[x:real]more(x,r),s,satz167k(s,r,t1)):in(s,s02)
7668 -5r205
7669 vb00:=[x:real][t:not(in(x,s01))]t2".5r205"(x,t):all([x:real]or(in(x,s01),in(x,s02)))
7670 +*5r205
7671 r@t3:=estii(real,[x:real]lessis(x,r),r,lessisi2(r,r,refis(real,r))):in(r,s01)
7672 -5r205
7673 r@vb01a:=nonemptyi(real,s01,r,t3".5r205"):nonempty(real,s01)
7674 +*5r205
7675 r@t4:=ismore2(pl(r,0),r,pl(r,1rl),pl02(r,0,refis(real,0)),satz188k(1rl,0,r,satz169a(1rl,pos1))):more(pl(r,1rl),r)
7676 t5:=estii(real,[x:real]more(x,r),pl(r,1rl),t4):in(pl(r,1rl),s02)
7677 -5r205
7678 r@vb01b:=nonemptyi(real,s02,pl(r,1rl),t5".5r205"):nonempty(real,s02)
7679 +*5r205
7680 s@[i:in(s,s01)][t:real][j:in(t,s02)]
7681 t6:=satz172a(s,r,t,estie(real,[x:real]lessis(x,r),s,i),lemma1(t,r,estie(real,[x:real]more(x,r),t,j))):less(s,t)
7682 -5r205
7683 r@vb02:=[x:real][t:in(x,s01)][y:real][u:in(y,s02)]t6".5r205"(x,t,y,u):all([x:real][t:in(x,s01)]all([y:real][u:in(y,s02)]less(x,y)))
7684 s@[l:less(s,r)]
7685 vb03a:=estii(real,[x:real]lessis(x,r),s,lessisi1(s,r,l)):in(s,s01)
7686 s@[m:more(s,r)]
7687 vb03b:=estii(real,[x:real]more(x,r),s,m):in(s,s02)
7688 r@s11:=setof(real,[x:real]less(x,r)):set(real)
7689 s12:=setof(real,[x:real]moreis(x,r)):set(real)
7690 +*5r205
7691 s@[n:not(in(s,s11))]
7692 t7:=th3"l.imp"(less(s,r),in(s,s11),n,[t:less(s,r)]estii(real,[x:real]less(x,r),s,t)):not(less(s,r))
7693 t8:=estii(real,[x:real]moreis(x,r),s,satz167f(s,r,t7)):in(s,s12)
7694 -5r205
7695 r@vb10:=[x:real][t:not(in(x,s11))]t8".5r205"(x,t):all([x:real]or(in(x,s11),in(x,s12)))
7696 +*5r205
7697 r@t9:=isless2(pl(r,0),r,mn(r,1rl),pl02(r,0,refis(real,0)),satz188m(m0(1rl),0,r,satz169c(m0(1rl),satz176a(1rl,pos1)))):less(mn(r,1rl),r)
7698 t10:=estii(real,[x:real]less(x,r),mn(r,1rl),t9):in(mn(r,1rl),s11)
7699 -5r205
7700 r@vb11a:=nonemptyi(real,s11,mn(r,1rl),t10".5r205"):nonempty(real,s11)
7701 +*5r205
7702 r@t11:=estii(real,[x:real]moreis(x,r),r,moreisi2(r,r,refis(real,r))):in(r,s12)
7703 -5r205
7704 r@vb11b:=nonemptyi(real,s12,r,t11".5r205"):nonempty(real,s12)
7705 +*5r205
7706 s@[i:in(s,s11)][t:real][j:in(t,s12)]
7707 t12:=satz172b(s,r,t,estie(real,[x:real]less(x,r),s,i),satz168a(t,r,estie(real,[x:real]moreis(x,r),t,j))):less(s,t)
7708 -5r205
7709 r@vb12:=[x:real][t:in(x,s11)][y:real][u:in(y,s12)]t12".5r205"(x,t,y,u):all([x:real][t:in(x,s11)]all([y:real][u:in(y,s12)]less(x,y)))
7710 s@[l:less(s,r)]
7711 vb13a:=estii(real,[x:real]less(x,r),s,l):in(s,s11)
7712 s@[m:more(s,r)]
7713 vb13b:=estii(real,[x:real]moreis(x,r),s,moreisi1(s,r,m)):in(s,s12)
7714 @2rl:=pl(1rl,1rl):real
7715 pos2:=pospl(1rl,1rl,pos1,pos1):pos(2rl)
7716 half:=ov(1rl,2rl,pnot0(2rl,pos2)):real
7717 poshalf:=posovpp(1rl,2rl,pnot0(2rl,pos2),pos1,pos2):pos(half)
7718 +*ivr5
7719 r@t3:=tris(real,pl(r,r),pl(ts(1rl,r),ts(1rl,r)),ts(2rl,r),ispl12(r,ts(1rl,r),r,ts(1rl,r),satz195c(r),satz195c(r)),distpt1(1rl,1rl,r)):is(pl(r,r),ts(2rl,r))
7720 t4:=tr4is(real,ts(half,pl(r,r)),ts(half,ts(2rl,r)),ts(ts(half,2rl),r),ts(1rl,r),r,ists2(pl(r,r),ts(2rl,r),half,t3),assts2(half,2rl,r),ists1(ts(half,2rl),1rl,r,satz204e(1rl,2rl,pnot0(2rl,pos2))),satz195b(r)):is(ts(half,pl(r,r)),r)
7721 -ivr5
7722 s@[l:less(r,s)]
7723 +*ivr5
7724 l@t5:=satz203k(pl(r,r),pl(r,s),half,satz188m(r,s,r,l),poshalf):less(ts(half,pl(r,r)),ts(half,pl(r,s)))
7725 -ivr5
7726 l@lemma3:=isless1(ts(half,pl(r,r)),r,ts(half,pl(r,s)),t4".ivr5",t5".ivr5"):less(r,ts(half,pl(r,s)))
7727 +*ivr5
7728 l@t6:=satz203k(pl(r,s),pl(s,s),half,satz188f(r,s,s,l),poshalf):less(ts(half,pl(r,s)),ts(half,pl(s,s)))
7729 -ivr5
7730 l@lemma4:=isless2(ts(half,pl(s,s)),s,ts(half,pl(r,s)),t4".ivr5"(s),t6".ivr5"):less(ts(half,pl(r,s)),s)
7731 [p:pos(r)]
7732 lemma5:=satz169b(s,trmore(s,r,0,lemma2(r,s,l),satz169a(r,p))):pos(s)
7733 @[s1:set(real)][s2:set(real)][p0:all([x:real]or(in(x,s1),in(x,s2)))][p1a:nonempty(real,s1)][p1b:nonempty(real,s2)][p2:all([x:real][t:in(x,s1)]all([y:real][u:in(y,s2)]less(x,y)))]
7734 +*5r205
7735 s2@[r:real]
7736 prop1:=all([x:real][t:less(x,r)]in(x,s1)):'prop'
7737 prop2:=all([x:real][t:more(x,r)]in(x,s2)):'prop'
7738 prop3:=and(prop1,prop2):'prop'
7739 p2@[x:real][y:real][px:prop3(x)][py:prop3(y)][l:less(x,y)]
7740 mxy:=ts(half,pl(x,y)):real
7741 t13:=lemma2(x,mxy,lemma3(x,y,l)):more(mxy,x)
7742 t14:=lemma4(x,y,l):less(mxy,y)
7743 t15:=<t14><mxy>ande1(prop1(y),prop2(y),py):in(mxy,s1)
7744 t16:=<t13><mxy>ande2(prop1(x),prop2(x),px):in(mxy,s2)
7745 t17:=<t16><mxy><t15><mxy>p2:less(mxy,mxy)
7746 t18:=<refis(real,mxy)>ec3e31(is(mxy,mxy),more(mxy,mxy),less(mxy,mxy),satz167b(mxy,mxy),t17):con
7747 py@t19:=[t:less(x,y)]t18(t):not(less(x,y))
7748 t20:=[t:more(x,y)]t18(y,x,py,px,lemma1(x,y,t)):not(more(x,y))
7749 t21:=or3e1(is(x,y),more(x,y),less(x,y),satz167a(x,y),t20,t19):is(x,y)
7750 p2@t22:=[x:real][y:real][t:prop3(x)][u:prop3(y)]t21(x,y,t,u):amone(real,[x:real]prop3(x))
7751 [case1:some([x:real]and(pos(x),in(x,s1)))][r:real][a:and(pos(r),in(r,s1))]
7752 t23:=ande1(pos(r),in(r,s1),a):pos(r)
7753 t24:=ande2(pos(r),in(r,s1),a):in(r,s1)
7754 sc1:=setof(cut,[x:cut]in(pofrp(x),s1)):set(cut)
7755 sc2:=setof(cut,[x:cut]in(pofrp(x),s2)):set(cut)
7756 [r0:cut][i:in(pofrp(r0),s1)]
7757 t25:=estii(cut,[x:cut]in(pofrp(x),s1),r0,i):in"rp"(r0,sc1)
7758 r0@[i:in"rp"(r0,sc1)]
7759 t26:=estie(cut,[x:cut]in(pofrp(x),s1),r0,i):in(pofrp(r0),s1)
7760 r0@[i:in(pofrp(r0),s2)]
7761 t27:=estii(cut,[x:cut]in(pofrp(x),s2),r0,i):in"rp"(r0,sc2)
7762 r0@[i:in"rp"(r0,sc2)]
7763 t28:=estie(cut,[x:cut]in(pofrp(x),s2),r0,i):in(pofrp(r0),s2)
7764 r0@t29:=th9"l.or"(in(pofrp(r0),s1),in(pofrp(r0),s2),in"rp"(r0,sc1),in"rp"(r0,sc2),<pofrp(r0)>p0,[t:in(pofrp(r0),s1)]t25(t),[t:in(pofrp(r0),s2)]t27(t)):or(in"rp"(r0,sc1),in"rp"(r0,sc2))
7765 a@pr1:=rpofp(r,t23):cut
7766 t30:=isp(real,[x:real]in(x,s1),r,pofrp(pr1),t24,isprp1(r,t23)):in(pofrp(pr1),s1)
7767 t31:=nonemptyi(cut,sc1,pr1,t25(pr1,t30)):nonempty(cut,sc1)
7768 [s:real][i:in(s,s2)]
7769 t32:=<i><s><t24><r>p2:less(r,s)
7770 t33:=lemma5(r,s,t32,t23):pos(s)
7771 ps1:=rpofp(s,t33):cut
7772 t34:=isp(real,[x:real]in(x,s2),s,pofrp(ps1),i,isprp1(s,t33)):in(pofrp(ps1),s2)
7773 t35:=nonemptyi(cut,sc2,ps1,t27(ps1,t34)):nonempty(cut,sc2)
7774 a@t36:=nonemptyapp(real,s2,p1b,nonempty(cut,sc2),[x:real][t:in(x,s2)]t35(x,t)):nonempty(cut,sc2)
7775 r0@[i:in"rp"(r0,sc1)][s0:cut][j:in"rp"(s0,sc2)]
7776 t37:=<t28(s0,j)><pofrp(s0)><t26(r0,i)><pofrp(r0)>p2:less(pofrp(r0),pofrp(s0))
7777 t38:=lessrpip(r0,s0,t37):less"rp"(r0,s0)
7778 a@stc:=schnitt(sc1,sc2,[x:cut]t29(x),t31,t36,[x:cut][t:in"rp"(x,sc1)][y:cut][u:in"rp"(y,sc2)]t38(x,t,y,u)):cut
7779 t39:=satzp205a(sc1,sc2,[x:cut]t29(x),t31,t36,[x:cut][t:in"rp"(x,sc1)][y:cut][u:in"rp"(y,sc2)]t38(x,t,y,u)):all"rp"([x:cut][t:less"rp"(x,stc)]in"rp"(x,sc1))
7780 t40:=satzp205b(sc1,sc2,[x:cut]t29(x),t31,t36,[x:cut][t:in"rp"(x,sc1)][y:cut][u:in"rp"(y,sc2)]t38(x,t,y,u)):all"rp"([x:cut][t:more"rp"(x,stc)]in"rp"(x,sc2))
7781 stp:=pofrp(stc):real
7782 t41:=posi(stc):pos(stp)
7783 [s:real][l:less(s,stp)][p:pos(s)]
7784 ps2:=rpofp(s,p):cut
7785 t42:=lessrpip(ps2,stc,isless1(s,pofrp(ps2),stp,isprp1(s,p),l)):less"rp"(ps2,stc)
7786 t43:=<t42><ps2>t39:in"rp"(ps2,sc1)
7787 t44:=isp(real,[x:real]in(x,s1),pofrp(ps2),s,t26(ps2,t43),isprp2(s,p)):in(s,s1)
7788 l@[n:not(pos(s))][i:in(s,s2)]
7789 t45:=<i><s><t24><r>p2:less(r,s)
7790 t46:=<lemma5(r,s,t45,t23)>n:con
7791 n@t47:=ore1(in(s,s1),in(s,s2),<s>p0,[t:in(s,s2)]t46(t)):in(s,s1)
7792 l@t48:=th1"l.imp"(pos(s),in(s,s1),[t:pos(s)]t44(t),[t:not(pos(s))]t47(t)):in(s,s1)
7793 s@[m:more(s,stp)]
7794 t49:=lemma5(stp,s,lemma1(s,stp,m),t41):pos(s)
7795 ps3:=rpofp(s,t49):cut
7796 t50:=morerpip(ps3,stc,ismore1(s,pofrp(ps3),stp,isprp1(s,t49),m)):more"rp"(ps3,stc)
7797 t51:=<t50><ps3>t40:in"rp"(ps3,sc2)
7798 t52:=isp(real,[x:real]in(x,s2),pofrp(ps3),s,t28(ps3,t51),isprp2(s,t49)):in(s,s2)
7799 a@t53:=andi(prop1(stp),prop2(stp),[x:real][t:less(x,stp)]t48(x,t),[x:real][t:more(x,stp)]t52(x,t)):prop3(stp)
7800 t54:=somei(real,[x:real]prop3(x),stp,t53):some([x:real]prop3(x))
7801 case1@t55:=someapp(real,[x:real]and(pos(x),in(x,s1)),case1,some([x:real]prop3(x)),[x:real][t:and(pos(x),in(x,s1))]t54(x,t)):some([x:real]prop3(x))
7802 p2@[case2:some([x:real]and(neg(x),in(x,s2)))]
7803 sp1:=setof(real,[x:real]in(m0(x),s1)):set(real)
7804 sp2:=setof(real,[x:real]in(m0(x),s2)):set(real)
7805 [r:real][i:in(m0(r),s1)]
7806 t56:=estii(real,[x:real]in(m0(x),s1),r,i):in(r,sp1)
7807 r@[i:in(r,sp1)]
7808 t57:=estie(real,[x:real]in(m0(x),s1),r,i):in(m0(r),s1)
7809 r@[i:in(m0(r),s2)]
7810 t58:=estii(real,[x:real]in(m0(x),s2),r,i):in(r,sp2)
7811 r@[i:in(r,sp2)]
7812 t59:=estie(real,[x:real]in(m0(x),s2),r,i):in(m0(r),s2)
7813 r@t60:=comor(in(r,sp1),in(r,sp2),th9"l.or"(in(m0(r),s1),in(m0(r),s2),in(r,sp1),in(r,sp2),<m0(r)>p0,[t:in(m0(r),s1)]t56(t),[t:in(m0(r),s2)]t58(t))):or(in(r,sp2),in(r,sp1))
7814 [i:in(r,s2)]
7815 t61:=t58(m0(r),isp(real,[x:real]in(x,s2),r,m0(m0(r)),i,satz177a(r))):in(m0(r),sp2)
7816 t62:=nonemptyi(real,sp2,m0(r),t61):nonempty(real,sp2)
7817 case2@t63:=nonemptyapp(real,s2,p1b,nonempty(real,sp2),[x:real][t:in(x,s2)]t62(x,t)):nonempty(real,sp2)
7818 r@[i:in(r,s1)]
7819 t64:=t56(m0(r),isp(real,[x:real]in(x,s1),r,m0(m0(r)),i,satz177a(r))):in(m0(r),sp1)
7820 t65:=nonemptyi(real,sp1,m0(r),t64):nonempty(real,sp1)
7821 case2@t66:=nonemptyapp(real,s1,p1a,nonempty(real,sp1),[x:real][t:in(x,s1)]t65(x,t)):nonempty(real,sp1)
7822 r@[i:in(r,sp2)][s:real][j:in(s,sp1)]
7823 t67:=<t59(r,i)><m0(r)><t57(s,j)><m0(s)>p2:less(m0(s),m0(r))
7824 t68:=lemma1(s,r,satz183d(s,r,t67)):less(r,s)
7825 r@[a:and(neg(r),in(r,s2))]
7826 t69:=satz176c(r,ande1(neg(r),in(r,s2),a)):pos(m0(r))
7827 t70:=isp(real,[x:real]in(x,s2),r,m0(m0(r)),ande2(neg(r),in(r,s2),a),satz177a(r)):in(m0(m0(r)),s2)
7828 t71:=andi(pos(m0(r)),in(m0(r),sp2),t69,t58(m0(r),t70)):and(pos(m0(r)),in(m0(r),sp2))
7829 t72:=somei(real,[x:real]and(pos(x),in(x,sp2)),m0(r),t71):some([x:real]and(pos(x),in(x,sp2)))
7830 case2@t73:=someapp(real,[x:real]and(neg(x),in(x,s2)),case2,some([x:real]and(pos(x),in(x,sp2))),[x:real][t:and(neg(x),in(x,s2))]t72(x,t)):some([x:real]and(pos(x),in(x,sp2)))
7831 t74:=t55(sp2,sp1,[x:real]t60(x),t63,t66,[x:real][t:in(x,sp2)][y:real][u:in(y,sp1)]t68(x,t,y,u),t73):some([x:real]prop3(sp2,sp1,x))
7832 [r:real][p:prop3(sp2,sp1,r)][s:real][l:less(s,m0(r))]
7833 t75:=ismore2(m0(m0(r)),r,m0(s),satz177(r),satz183c(s,m0(r),l)):more(m0(s),r)
7834 t76:=<t75><m0(s)>ande2(prop1(sp2,sp1,r),prop2(sp2,sp1,r),p):in(m0(s),sp1)
7835 t77:=isp(real,[x:real]in(x,s1),m0(m0(s)),s,t57(m0(s),t76),satz177(s)):in(s,s1)
7836 s@[m:more(s,m0(r))]
7837 t78:=isless2(m0(m0(r)),r,m0(s),satz177(r),satz183a(s,m0(r),m)):less(m0(s),r)
7838 t79:=<t78><m0(s)>ande1(prop1(sp2,sp1,r),prop2(sp2,sp1,r),p):in(m0(s),sp2)
7839 t80:=isp(real,[x:real]in(x,s2),m0(m0(s)),s,t59(m0(s),t79),satz177(s)):in(s,s2)
7840 p@t81:=andi(prop1(m0(r)),prop2(m0(r)),[x:real][t:less(x,m0(r))]t77(x,t),[x:real][t:more(x,m0(r))]t80(x,t)):prop3(m0(r))
7841 t82:=somei(real,[x:real]prop3(x),m0(r),t81):some([x:real]prop3(x))
7842 case2@t83:=someapp(real,[x:real]prop3(sp2,sp1,x),t74,some([x:real]prop3(x)),[x:real][t:prop3(sp2,sp1,x)]t82(x,t)):some([x:real]prop3(x))
7843 p2@[notcase1:not(some([x:real]and(pos(x),in(x,s1))))][notcase2:not(some([x:real]and(neg(x),in(x,s2))))][r:real][l:less(r,0)]
7844 t84:=th4"l.some"(real,[x:real]and(neg(x),in(x,s2)),notcase2,r):not(and(neg(r),in(r,s2)))
7845 t85:=th3"l.and"(neg(r),in(r,s2),t84,satz169d(r,l)):not(in(r,s2))
7846 t86:=ore1(in(r,s1),in(r,s2),<r>p0,t85):in(r,s1)
7847 r@[m:more(r,0)]
7848 t87:=th4"l.some"(real,[x:real]and(pos(x),in(x,s1)),notcase1,r):not(and(pos(r),in(r,s1)))
7849 t88:=th3"l.and"(pos(r),in(r,s1),t87,satz169b(r,m)):not(in(r,s1))
7850 t89:=ore2(in(r,s1),in(r,s2),<r>p0,t88):in(r,s2)
7851 notcase2@t90:=andi(prop1(0),prop2(0),[x:real][t:less(x,0)]t86(x,t),[x:real][t:more(x,0)]t89(x,t)):prop3(0)
7852 t91:=somei(real,[x:real]prop3(x),0,t90):some([x:real]prop3(x))
7853 notcase1@t92:=th1"l.imp"(some([x:real]and(neg(x),in(x,s2))),some([x:real]prop3(x)),[t:some([x:real]and(neg(x),in(x,s2)))]t83(t),[t:not(some([x:real]and(neg(x),in(x,s2))))]t91(t)):some([x:real]prop3(x))
7854 p2@t93:=th1"l.imp"(some([x:real]and(pos(x),in(x,s1))),some([x:real]prop3(x)),[t:some([x:real]and(pos(x),in(x,s1)))]t55(t),[t:not(some([x:real]and(pos(x),in(x,s1))))]t92(t)):some([x:real]prop3(x))
7855 t94:=onei(real,[x:real]prop3(x),t22,t93):one([x:real]prop3(x))
7856 -5r205
7857 p2@satz205:=t94".5r205":one([x:real]and(all([y:real][t:less(y,x)]in(y,s1)),all([y:real][t:more(y,x)]in(y,s2))))
7858 dedekind:=satz205:one([x:real]and(all([y:real][t:less(y,x)]in(y,s1)),all([y:real][t:more(y,x)]in(y,s2))))
7859 schnitt:=ind(real,[x:real]prop3".5r205"(x),satz205):real
7860 [r:real][l:less(r,schnitt)]
7861 satz205a:=<l><r>ande1(prop1".5r205"(schnitt),prop2".5r205"(schnitt),oneax(real,[x:real]prop3".5r205"(x),satz205)):in(r,s1)
7862 r@[m:more(r,schnitt)]
7863 satz205b:=<m><r>ande2(prop1".5r205"(schnitt),prop2".5r205"(schnitt),oneax(real,[x:real]prop3".5r205"(x),satz205)):in(r,s2)
7864 @[r0:cut][s0:cut]
7865 +iva
7866 dr:=pdofrp(r0):dif
7867 ds:=pdofrp(s0):dif
7868 t1:=lemmaivad1(r0,s0):eq"rp"(pd(dr,ds),pdofrp(pl"rp"(r0,s0)))
7869 -iva
7870 lemmaiva1:=isin(pl(pofrp(r0),pofrp(s0)),pofrp(pl"rp"(r0,s0)),pd(pdofrp(r0),pdofrp(s0)),pdofrp(pl"rp"(r0,s0)),picp(pofrp(r0),pofrp(s0),pdofrp(r0),pdofrp(s0),innclass(pdofrp(r0)),innclass(pdofrp(s0))),innclass(pdofrp(pl"rp"(r0,s0))),t1".iva"):is(pl(pofrp(r0),pofrp(s0)),pofrp(pl"rp"(r0,s0)))
7871 +*iva
7872 s0@t2:=lemmaivad2(r0,s0):eq"rp"(td(dr,ds),pdofrp(ts"rp"(r0,s0)))
7873 -iva
7874 s0@lemmaiva2:=isin(ts(pofrp(r0),pofrp(s0)),pofrp(ts"rp"(r0,s0)),td(pdofrp(r0),pdofrp(s0)),pdofrp(ts"rp"(r0,s0)),tict(pofrp(r0),pofrp(s0),pdofrp(r0),pdofrp(s0),innclass(pdofrp(r0)),innclass(pdofrp(s0))),innclass(pdofrp(ts"rp"(r0,s0))),t2".iva"):is(ts(pofrp(r0),pofrp(s0)),pofrp(ts"rp"(r0,s0)))
7875 [m:more(pofrp(r0),pofrp(s0))]
7876 +*iva
7877 m@t3:=moreex(pofrp(r0),pofrp(s0),dr,ds,innclass(dr),innclass(ds),m):mored(dr,ds)
7878 -iva
7879 m@lemmaiva3:=lemmaivad3(r0,s0,t3".iva"):more"rp"(r0,s0)
7880 s0@[m:more"rp"(r0,s0)]
7881 +*iva
7882 m@[l:less(pofrp(r0),pofrp(s0))]
7883 t4:=satz121(s0,r0,lemmaiva3(s0,r0,lemma2(pofrp(r0),pofrp(s0),l))):less"rp"(r0,s0)
7884 m@t5:=ec3e23(is"rp"(r0,s0),more"rp"(r0,s0),less"rp"(r0,s0),satz123b(r0,s0),m):not(less"rp"(r0,s0))
7885 t6:=th3"l.imp"(less(pofrp(r0),pofrp(s0)),less"rp"(r0,s0),t5,[t:less(pofrp(r0),pofrp(s0))]t4(t)):not(less(pofrp(r0),pofrp(s0)))
7886 t7:=ec3e21(is"rp"(r0,s0),more"rp"(r0,s0),less"rp"(r0,s0),satz123b(r0,s0),m):nis"rp"(r0,s0)
7887 t8:=th3"l.imp"(is(pofrp(r0),pofrp(s0)),is"rp"(r0,s0),t7,[t:is(pofrp(r0),pofrp(s0))]isrpip(r0,s0,t)):nis(pofrp(r0),pofrp(s0))
7888 -iva
7889 m@lemmaiva4:=or3e2(is(pofrp(r0),pofrp(s0)),more(pofrp(r0),pofrp(s0)),less(pofrp(r0),pofrp(s0)),satz167a(pofrp(r0),pofrp(s0)),t6".iva",t8".iva"):more(pofrp(r0),pofrp(s0))
7890 @[x:nat][y:nat][m:more(rlofnt(x),rlofnt(y))]
7891 +*iva
7892 m@t9:=lemmaiva3(rpofnt(x),rpofnt(y),m):more"rp"(rpofnt(x),rpofnt(y))
7893 t10:=satz154d(rtofn(x),rtofn(y),t9):more"rt"(rtofn(x),rtofn(y))
7894 t11:=moree"rt"(rtofn(x),rtofn(y),fr(x,1),fr(y,1),inclass"rt"(fr(x,1)),inclass"rt"(fr(y,1)),t10):moref(fr(x,1),fr(y,1))
7895 -iva
7896 m@lemmaiva5:=satz111a(x,y,t11".iva"):more"n"(x,y)
7897 y@[m:more"n"(x,y)]
7898 +*iva
7899 m@t12:=satz111d(x,y,m):moref(fr(x,1),fr(y,1))
7900 t13:=morei"rt"(rtofn(x),rtofn(y),fr(x,1),fr(y,1),inclass"rt"(fr(x,1)),inclass"rt"(fr(y,1)),t12):more"rt"(rtofn(x),rtofn(y))
7901 t14:=satz154a(rtofn(x),rtofn(y),t13):more"rp"(rpofnt(x),rpofnt(y))
7902 -iva
7903 m@lemmaiva6:=lemmaiva4(rpofnt(x),rpofnt(y),t14".iva"):more(rlofnt(x),rlofnt(y))
7904 @[r:real]
7905 +int
7906 a0ir@[i:intrl(r)]
7907 t1:=intabsd(a0,intrlex(r,a0,a0ir,i)):intd(absd(a0))
7908 t2:=intrlin(abs(r),absd(a0),aica,t1):intrl(abs(r))
7909 -int
7910 [i:intrl(r)]
7911 intabs:=realapp1(r,intrl(abs(r)),[x:dif][t:inn(x,class(r))]t2".int"(r,x,t,i)):intrl(abs(r))
7912 +*int
7913 i@t3:=intm0d(a0,intrlex(r,a0,a0ir,i)):intd(m0d(a0))
7914 t4:=intrlin(m0(r),m0d(a0),micm0,t3):intrl(m0(r))
7915 -int
7916 i@intm0:=realapp1(r,intrl(m0(r)),[x:dif][t:inn(x,class(r))]t4".int"(r,x,t,i)):intrl(m0(r))
7917 +*int
7918 b1is@[i:intrl(r)][j:intrl(s)]
7919 t5:=intpd(a1,b1,intrlex(r,a1,a1ir,i),intrlex(s,b1,b1is,j)):intd(pd(a1,b1))
7920 t6:=intrlin(pl(r,s),pd(a1,b1),picp,t5):intrl(pl(r,s))
7921 -int
7922 i@[s:real][j:intrl(s)]
7923 intpl:=realapp2(r,s,intrl(pl(r,s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t6".int"(r,s,x,y,t,u,i,j)):intrl(pl(r,s))
7924 intmn:=intpl(r,i,m0(s),intm0(s,j)):intrl(mn(r,s))
7925 +*int
7926 j@t7:=inttd(a1,b1,intrlex(r,a1,a1ir,i),intrlex(s,b1,b1is,j)):intd(td(a1,b1))
7927 t8:=intrlin(ts(r,s),td(a1,b1),tict,t7):intrl(ts(r,s))
7928 -int
7929 j@intts:=realapp2(r,s,intrl(ts(r,s)),[x:dif][y:dif][t:inn(x,class(r))][u:inn(y,class(s))]t8".int"(r,s,x,y,t,u,i,j)):intrl(ts(r,s))
7930 r@[n:natrl(r)]
7931 +ivr24
7932 t1:=satz24a(ntofrl(r,n)):lessis"n"(1,ntofrl(r,n))
7933 t2:=th3"l.imp"(more(1rl,r),more"n"(1,ntofrl(r,n)),satz10d(1,ntofrl(r,n),t1),[t:more(1rl,r)]lemmaiva5(1,ntofrl(r,n),ismore2(r,rlofnt(ntofrl(r,n)),1rl,isrlnt1(r,n),t))):not(more(1rl,r))
7934 -ivr24
7935 satzr24:=satz167e(1rl,r,t2".ivr24"):lessis(1rl,r)
7936 r@[i:intrl(r)][s:real][j:intrl(s)][l:less(r,s)]
7937 +ivr25
7938 t1:=satz182d(s,r,lemma2(r,s,l)):pos(mn(s,r))
7939 t2:=intmn(s,j,r,i):intrl(mn(s,r))
7940 t3:=posintnatrl(mn(s,r),t1,t2):natrl(mn(s,r))
7941 t4:=satzr24(mn(s,r),t3):lessis(1rl,mn(s,r))
7942 t5:=th9"l.or"(less(1rl,mn(s,r)),is(1rl,mn(s,r)),less(pl(1rl,r),pl(mn(s,r),r)),is(pl(1rl,r),pl(mn(s,r),r)),t4,[t:less(1rl,mn(s,r))]satz188f(1rl,mn(s,r),r,t),[t:is(1rl,mn(s,r))]ispl1(1rl,mn(s,r),r,t)):lessis(pl(1rl,r),pl(mn(s,r),r))
7943 -ivr25
7944 satzr25:=islessis12(pl(1rl,r),pl(r,1rl),pl(mn(s,r),r),s,compl(1rl,r),plmn(s,r),t5".ivr25"):lessis(pl(r,1rl),s)
7945 @[x:nat][y:nat]
7946 +ivr155
7947 t1:=lemmaiva1(rpofnt(x),rpofnt(y)):is(pl(rlofnt(x),rlofnt(y)),pofrp(pl"rp"(rpofnt(x),rpofnt(y))))
7948 t2:=isrpep(rpofnt(pl"n"(x,y)),pl"rp"(rpofnt(x),rpofnt(y)),satz155e(x,y)):is(rlofnt(pl"n"(x,y)),pofrp(pl"rp"(rpofnt(x),rpofnt(y))))
7949 -ivr155
7950 satzr155a:=tris2(real,rlofnt(pl"n"(x,y)),pl(rlofnt(x),rlofnt(y)),pofrp(pl"rp"(rpofnt(x),rpofnt(y))),t2".ivr155",t1".ivr155"):is(rlofnt(pl"n"(x,y)),pl(rlofnt(x),rlofnt(y)))
7951 satzr155b:=symis(real,rlofnt(pl"n"(x,y)),pl(rlofnt(x),rlofnt(y)),satzr155a):is(pl(rlofnt(x),rlofnt(y)),rlofnt(pl"n"(x,y)))
7952 +*ivr155
7953 y@t3:=lemmaiva2(rpofnt(x),rpofnt(y)):is(ts(rlofnt(x),rlofnt(y)),pofrp(ts"rp"(rpofnt(x),rpofnt(y))))
7954 t4:=isrpep(rpofnt(ts"n"(x,y)),ts"rp"(rpofnt(x),rpofnt(y)),satz155f(x,y)):is(rlofnt(ts"n"(x,y)),pofrp(ts"rp"(rpofnt(x),rpofnt(y))))
7955 -ivr155
7956 y@satzr155c:=tris2(real,rlofnt(ts"n"(x,y)),ts(rlofnt(x),rlofnt(y)),pofrp(ts"rp"(rpofnt(x),rpofnt(y))),t4".ivr155",t3".ivr155"):is(rlofnt(ts"n"(x,y)),ts(rlofnt(x),rlofnt(y)))
7957 satzr155d:=symis(real,rlofnt(ts"n"(x,y)),ts(rlofnt(x),rlofnt(y)),satzr155c):is(ts(rlofnt(x),rlofnt(y)),rlofnt(ts"n"(x,y)))
7958 @[t:real][r:real][s:real][a:and(not(neg(r)),is(ts(r,r),t))][b:and(not(neg(s)),is(ts(s,s),t))]
7959 +7r161
7960 [c0:dif][cit:inn(c0,class(t))][a0:dif][air:inn(a0,class(r))][b0:dif][bis:inn(b0,class(s))]
7961 t1:=th3"l.imp"(negd(a0),neg(r),ande1(not(neg(r)),is(ts(r,r),t),a),[u:negd(a0)]negin(r,a0,air,u)):not(negd(a0))
7962 t2:=th3"l.imp"(negd(b0),neg(s),ande1(not(neg(s)),is(ts(s,s),t),b),[u:negd(b0)]negin(s,b0,bis,u)):not(negd(b0))
7963 t3:=isex(ts(r,r),t,td(a0,a0),c0,tict(r,r,a0,a0,air,air),cit,ande2(not(neg(r)),is(ts(r,r),t),a)):eq"rp"(td(a0,a0),c0)
7964 t4:=isex(ts(s,s),t,td(b0,b0),c0,tict(s,s,b0,b0,bis,bis),cit,ande2(not(neg(s)),is(ts(s,s),t),b)):eq"rp"(td(b0,b0),c0)
7965 t5:=isin(r,s,a0,b0,air,bis,satzd161b(c0,a0,b0,t1,t2,t3,t4)):is(r,s)
7966 -7r161
7967 satzr161b:=realapp3(t,r,s,is(r,s),[x:dif][y:dif][z:dif][u:inn(x,class(t))][v:inn(y,class(r))][w:inn(z,class(s))]t5".7r161"(x,u,y,v,z,w)):is(r,s)
7968 t@[n:not(neg(t))]
7969 +*7r161
7970 n@[c0:dif][cit:inn(c0,class(t))]
7971 t6:=th3"l.imp"(negd(c0),neg(t),n,[u:negd(c0)]negin(t,c0,cit,u)):not(negd(c0))
7972 [a0:dif][a:and(not(negd(a0)),eq"rp"(td(a0,a0),c0))]
7973 ar:=realof(a0):real
7974 t7:=th3"l.imp"(neg(ar),negd(a0),ande1(not(negd(a0)),eq"rp"(td(a0,a0),c0),a),[u:neg(ar)]negex(ar,a0,innclass(a0),u)):not(neg(ar))
7975 t8:=isin(ts(ar,ar),t,td(a0,a0),c0,tict(ar,ar,a0,a0,innclass(a0),innclass(a0)),cit,ande2(not(negd(a0)),eq"rp"(td(a0,a0),c0),a)):is(ts(ar,ar),t)
7976 t9:=andi(not(neg(ar)),is(ts(ar,ar),t),t7,t8):and(not(neg(ar)),is(ts(ar,ar),t))
7977 t10:=somei(real,[u:real]and(not(neg(u)),is(ts(u,u),t)),ar,t9):some([u:real]and(not(neg(u)),is(ts(u,u),t)))
7978 cit@t11:=someapp(dif,[x:dif]and(not(negd(x)),eq"rp"(td(x,x),c0)),satzd161a(c0,t6),some([u:real]and(not(neg(u)),is(ts(u,u),t))),[x:dif][v:and(not(negd(x)),eq"rp"(td(x,x),c0))]t10(x,v)):some([u:real]and(not(neg(u)),is(ts(u,u),t)))
7979 -7r161
7980 n@satzr161a:=realapp1(t,some([u:real]and(not(neg(u)),is(ts(u,u),t))),[x:dif][v:inn(x,class(t))]t11".7r161"(x,v)):some([u:real]and(not(neg(u)),is(ts(u,u),t)))
7981 satzr161:=onei(real,[u:real]and(not(neg(u)),is(ts(u,u),t)),[u:real][v:real][a:and(not(neg(u)),is(ts(u,u),t))][b:and(not(neg(v)),is(ts(v,v),t))]satzr161b(u,v,a,b),satzr161a):one([u:real]and(not(neg(u)),is(ts(u,u),t)))
7982 sqrt:=ind(real,[u:real]and(not(neg(u)),is(ts(u,u),t)),satzr161):real
7983 +*7r161
7984 n@t12:=oneax(real,[u:real]and(not(neg(u)),is(ts(u,u),t)),satzr161):and(not(neg(sqrt)),is(ts(sqrt,sqrt),t))
7985 -7r161
7986 n@thsqrt1a:=ande1(not(neg(sqrt)),is(ts(sqrt,sqrt),t),t12".7r161"):not(neg(sqrt))
7987 thsqrt1b:=ande2(not(neg(sqrt)),is(ts(sqrt,sqrt),t),t12".7r161"):is(ts(sqrt,sqrt),t)
7988 [x:real][o:not(neg(x))][i:is(ts(x,x),t)]
7989 thsqrt2:=satzr161b(x,sqrt,andi(not(neg(x)),is(ts(x,x),t),o,i),t12".7r161"):is(x,sqrt(t,n))
7990 o@[i:is(t,ts(x,x))]
7991 thsqrt3:=symis(real,x,sqrt(t,n),thsqrt2(symis(real,t,ts(x,x),i))):is(sqrt(t,n),x)
7992 @[r:real][s:real][n:not(neg(r))][o:not(neg(s))][i:is(r,s)]
7993 issqrt:=thsqrt2(s,o,sqrt(r,n),thsqrt1a(r,n),tris(real,ts(sqrt(r,n),sqrt(r,n)),r,s,thsqrt1b(r,n),i)):is(sqrt(r,n),sqrt(s,o))
7994 r@[n:not(neg(r))][i:is(r,0)]
7995 sqrt0:=thsqrt3(r,n,0,0notn(0,refis(real,0)),tris2(real,r,ts(0,0),0,i,ts01(0,0,refis(real,0)))):is(sqrt(r,n),0)
7996 n@[o:nis(r,0)]
7997 +sqrt
7998 t1:=th3"l.imp"(is(sqrt(r,n),0),is(r,0),o,[t:is(sqrt(r,n),0)]tris1(real,r,0,ts(sqrt(r,n),sqrt(r,n)),thsqrt1b(r,n),ts01(sqrt(r,n),sqrt(r,n),t))):nis(sqrt(r,n),0)
7999 -sqrt
8000 sqrtnot0:=or3e2(is(sqrt(r,n),0),pos(sqrt(r,n)),neg(sqrt(r,n)),axrlo(sqrt(r,n)),thsqrt1a(r,n),t1".sqrt"):pos(sqrt(r,n))
8001 @[r:real][s:real][t:real][n:nis(t,0)]
8002 +v0
8003 t1:=tr3is(real,ts(t,ts(r,ov(s,t,n))),ts(ts(r,ov(s,t,n)),t),ts(r,ts(ov(s,t,n),t)),ts(r,s),comts(t,ts(r,ov(s,t,n))),assts1(r,ov(s,t,n),t),ists2(ts(ov(s,t,n),t),s,r,satz204e(s,t,n))):is(ts(t,ts(r,ov(s,t,n))),ts(r,s))
8004 -v0
8005 lemma6:=satz204g(ts(r,s),t,ts(r,ov(s,t,n)),n,t1".v0"):is(ts(r,ov(s,t,n)),ov(ts(r,s),t,n))
8006 +*v0
8007 n@t2:=tris(real,ts(t,pl(ov(r,t,n),ov(s,t,n))),pl(ts(t,ov(r,t,n)),ts(t,ov(s,t,n))),pl(r,s),disttp2(t,ov(r,t,n),ov(s,t,n)),ispl12(ts(t,ov(r,t,n)),r,ts(t,ov(s,t,n)),s,satz204c(r,t,n),satz204c(s,t,n))):is(ts(t,pl(ov(r,t,n),ov(s,t,n))),pl(r,s))
8008 -v0
8009 n@lemma7:=satz204g(pl(r,s),t,pl(ov(r,t,n),ov(s,t,n)),n,t2".v0"):is(pl(ov(r,t,n),ov(s,t,n)),ov(pl(r,s),t,n))
8010 r@[n:nis(r,0)]
8011 lemma8:=satz204b(r,r,n,ov(r,r,n),1rl,satz204c(r,r,n),satz195(r)):is(ov(r,r,n),1rl)
8012 lemma9:=ore2(is(r,0),is(ov(0,r,n),0),satz192c(r,ov(0,r,n),satz204c(0,r,n)),n):is(ov(0,r,n),0)
8013 r@[i:is(r,m0(r))]
8014 +*v0
8015 i@[p:pos(m0(r))]
8016 t3:=<isneg(r,m0(r),i,satz176f(r,p))>pnotn(m0(r),p):con
8017 i@[n:neg(m0(r))]
8018 t4:=<ispos(r,m0(r),i,satz176d(r,n))>nnotp(m0(r),n):con
8019 -v0
8020 i@lemma10:=satz176e(r,or3e1(is(m0(r),0),pos(m0(r)),neg(m0(r)),axrlo(m0(r)),[t:pos(m0(r))]t3".v0"(t),[t:neg(m0(r))]t4".v0"(t))):is(r,0)
8021 r@lemma11:=satz167f(ts(r,r),0,th3"l.imp"(less(ts(r,r),0),neg(ts(r,r)),nnegsq(r),[t:less(ts(r,r),0)]satz169d(ts(r,r),t))):moreis(ts(r,r),0)
8022 lemma12:=rapp(r,is(ts(r,r),ts(abs(r),abs(r))),[t:pos(r)]satz196a(r,r,t,t),[t:is(r,0)]tris2(real,ts(r,r),ts(abs(r),abs(r)),0,ts01(r,r,t),ts01(abs(r),abs(r),abs0(r,t))),[t:neg(r)]satz196b(r,r,t,t)):is(ts(r,r),ts(abs(r),abs(r)))
8023 @[x:real][ix:intrl(x)][y:real][iy:intrl(y)][ly:lessis(y,x)]
8024 +shift
8025 t1:=satz190a(x,x,1rl,0,moreisi2(x,x,refis(real,x)),satz169a(1rl,natpos(1rl,natrl1))):more(pl(x,1rl),pl(x,0))
8026 t2:=ismore2(pl(x,0),x,pl(x,1rl),pl02(x,0,refis(real,0)),t1):more(pl(x,1rl),x)
8027 t3:=satz172d(pl(x,1rl),x,y,t2,satz168b(y,x,ly)):more(pl(x,1rl),y)
8028 t4:=satz182d(pl(x,1rl),y,t3):pos(mn(pl(x,1rl),y))
8029 t5:=intmn(pl(x,1rl),intpl(x,ix,1rl,intrl1),y,iy):intrl(mn(pl(x,1rl),y))
8030 t6:=posintnatrl(mn(pl(x,1rl),y),t4,t5):natrl(mn(pl(x,1rl),y))
8031 -shift
8032 shiftl:=ntofrl(mn(pl(x,1rl),y),t6".shift"):nat
8033 [n:1to(shiftl)]
8034 +*shift
8035 n@n1:=inn"n"(shiftl,n):nat
8036 t7:=1top(shiftl,n):lessis"n"(n1,shiftl)
8037 n2:=rlofnt(n1):real
8038 t8:=natintrl(n2,natrli(n1)):intrl(n2)
8039 -shift
8040 n@shiftr:=mn(pl(n2".shift",y),1rl):real
8041 intshiftr:=intmn(pl(n2".shift",y),intpl(n2".shift",t8".shift",y,iy),1rl,intrl1):intrl(shiftr)
8042 [m:1to(shiftl)][i:is(shiftr(n),shiftr(m))]
8043 +*shift
8044 n@t8a:=tris(real,mn(pl(shiftr(n),1rl),y),mn(pl(n2,y),y),n2,ismn1(pl(mn(pl(n2,y),1rl),1rl),pl(n2,y),y,plmn(pl(n2,y),1rl)),mnpl(n2,y)):is(mn(pl(shiftr(n),1rl),y),n2)
8045 i@t9a:=ismn1(pl(shiftr(n),1rl),pl(shiftr(m),1rl),y,ispl1(shiftr(n),shiftr(m),1rl,i)):is(mn(pl(shiftr(n),1rl),y),mn(pl(shiftr(m),1rl),y))
8046 t10a:=tr3is(real,n2(n),mn(pl(shiftr(n),1rl),y),mn(pl(shiftr(m),1rl),y),n2(m),symis(real,mn(pl(shiftr(n),1rl),y),n2,t8a),t9a,t8a(m)):is(n2(n),n2(m))
8047 t11a:=isntirl(n1(n),n1(m),t10a):is"n"(n1(n),n1(m))
8048 -shift
8049 i@iseshiftr:=isinne(shiftl,n,m,t11a".shift"):is"e"(1to(shiftl),n,m)
8050 +*shift
8051 n@[m:more(shiftr,x)]
8052 t9:=satz188d(shiftr,x,1rl,m):more(pl(shiftr,1rl),pl(x,1rl))
8053 t10:=ismore1(pl(shiftr,1rl),pl(n2,y),pl(x,1rl),plmn(pl(n2,y),1rl),t9):more(pl(n2,y),pl(x,1rl))
8054 t11:=satz188d(pl(n2,y),pl(x,1rl),m0(y),t10):more(mn(pl(n2,y),y),mn(pl(x,1rl),y))
8055 t12:=ismore1(mn(pl(n2,y),y),n2,mn(pl(x,1rl),y),mnpl(n2,y),t11):more(n2,mn(pl(x,1rl),y))
8056 t13:=ismore12(n2,rlofnt(ntofrl(n2,natrli(n1))),mn(pl(x,1rl),y),rlofnt(ntofrl(mn(pl(x,1rl),y),t6)),isrlnt1(n2,natrli(n1)),isrlnt1(mn(pl(x,1rl),y),t6),t12):more(rlofnt(ntofrl(n2,natrli(n1))),rlofnt(shiftl))
8057 t14:=lemmaiva5(ntofrl(n2,natrli(n1)),shiftl,t13):more"n"(ntofrl(n2,natrli(n1)),shiftl)
8058 t15:=ismore1"n"(ntofrl(n2,natrli(n1)),n1,shiftl,isntrl2(n1),t14):more"n"(n1,shiftl)
8059 n@t16:=th3"l.imp"(more(shiftr,x),more"n"(n1,shiftl),satz10d(n1,shiftl,t7),[t:more(shiftr,x)]t15(t)):not(more(shiftr,x))
8060 -shift
8061 n@shiftrls:=satz167e(shiftr,x,t16".shift"):lessis(shiftr,x)
8062 +*shift
8063 n@[m:more(y,shiftr)]
8064 t17:=satz188d(y,shiftr,1rl,m):more(pl(y,1rl),pl(shiftr,1rl))
8065 t18:=ismore12(pl(y,1rl),pl(1rl,y),pl(shiftr,1rl),pl(n2,y),compl(y,1rl),plmn(pl(n2,y),1rl),t17):more(pl(1rl,y),pl(n2,y))
8066 t19:=satz188a(1rl,n2,y,t18):more(1rl,n2)
8067 t20:=lemmaiva5(1,n1,t19):more"n"(1,n1)
8068 n@t21:=th3"l.imp"(more(y,shiftr),more"n"(1,n1),satz10d(1,n1,satz24a(n1)),[t:more(y,shiftr)]t20(t)):not(more(y,shiftr))
8069 -shift
8070 n@lsshiftr:=satz167e(y,shiftr,t21".shift"):lessis(y,shiftr)
8071 ly@[u:real][a:and3(intrl(u),lessis(y,u),lessis(u,x))]
8072 +*shift
8073 a@t22:=and3e1(intrl(u),lessis(y,u),lessis(u,x),a):intrl(u)
8074 t23:=and3e2(intrl(u),lessis(y,u),lessis(u,x),a):lessis(y,u)
8075 t24:=and3e3(intrl(u),lessis(y,u),lessis(u,x),a):lessis(u,x)
8076 [l:less(u,x)]
8077 t25:=satz188f(pl(u,1rl),pl(x,1rl),m0"r"(y),satz188f(u,x,1rl,l)):less(mn(pl(u,1rl),y),mn(pl(x,1rl),y))
8078 a@[i:is(u,x)]
8079 t26:=ismn1(pl(u,1rl),pl(x,1rl),y,ispl1(u,x,1rl,i)):is(mn(pl(u,1rl),y),mn(pl(x,1rl),y))
8080 a@t27:=th9"l.or"(less(u,x),is(u,x),less(mn(pl(u,1rl),y),mn(pl(x,1rl),y)),is(mn(pl(u,1rl),y),mn(pl(x,1rl),y)),t24,[t:less(u,x)]t25(t),[t:is(u,x)]t26(t)):lessis(mn(pl(u,1rl),y),mn(pl(x,1rl),y))
8081 ul:=shiftl(u,t22,y,iy,t23):nat
8082 t28:=islessis12(mn(pl(u,1rl),y),rlofnt(ul),mn(pl(x,1rl),y),rlofnt(shiftl),isrlnt1(mn(pl(u,1rl),y),t6(u,t22,y,iy,t23)),isrlnt1(mn(pl(x,1rl),y),t6),t27):lessis(rlofnt(ul),rlofnt(shiftl))
8083 t29:=th3"l.imp"(more"n"(ul,shiftl),more(rlofnt(ul),rlofnt(shiftl)),satz167d(rlofnt(ul),rlofnt(shiftl),t28),[t:more"n"(ul,shiftl)]lemmaiva6(ul,shiftl,t)):not(more"n"(ul,shiftl))
8084 t30:=satz10e(ul,shiftl,t29):lessis"n"(ul,shiftl)
8085 -shift
8086 a@shiftl1:=outn(shiftl,ul".shift",t30".shift"):1to(shiftl)
8087 +*shift
8088 a@t31:=isinoutn(shiftl,ul,t30):is"n"(ul,n1(shiftl1))
8089 t32:=tris(real,mn(pl(u,1rl),y),rlofnt(ul),n2(shiftl1),isrlnt1(mn(pl(u,1rl),y),t6(u,t22,y,iy,t23)),isnterl(ul,n1(shiftl1),t31)):is(mn(pl(u,1rl),y),n2(shiftl1))
8090 t33:=tris(real,mn(pl(mn(pl(u,1rl),y),y),1rl),mn(pl(u,1rl),1rl),u,ismn1(pl(mn(pl(u,1rl),y),y),pl(u,1rl),1rl,plmn(pl(u,1rl),y)),mnpl(u,1rl)):is(mn(pl(mn(pl(u,1rl),y),y),1rl),u)
8091 -shift
8092 a@shiftinv1:=tris1(real,u,shiftr(shiftl1),mn(pl(mn(pl(u,1rl),y),y),1rl),t33".shift",ismn1(pl(mn(pl(u,1rl),y),y),pl(n2".shift"(shiftl1),y),1rl,ispl1(mn(pl(u,1rl),y),n2".shift"(shiftl1),y,t32".shift"))):is(u,shiftr(shiftl1))
8093 shiftinv2:=symis(real,u,shiftr(shiftl1),shiftinv1):is(shiftr(shiftl1),u)
8094 ly@[alpha:'type']
8095 seq:=[t:real][u:intrl(t)][v:lessis(y,t)][w:lessis(t,x)]alpha:'type'
8096 [s:seq]
8097 proofsirrelevant:=[t:real][it:intrl(t)][lt:lessis(y,t)][tl:lessis(t,x)][u:real][iu:intrl(u)][lu:lessis(y,u)][ul:lessis(u,x)][v:is(t,u)]is"e"(alpha,<tl><lt><it><t>s,<ul><lu><iu><u>s):'prop'
8098 alpha@[f:seq]
8099 shiftf:=[t:1to(shiftl)]<shiftrls(t)><lsshiftr(t)><intshiftr(t)><shiftr(t)>f:[t:1to(shiftl)]alpha
8100 ly@[s:seq(real)]
8101 inseq:=[t:real][u:intrl(t)][v:lessis(y,t)][w:lessis(t,x)]and3(intrl(<w><v><u><t>s),lessis(y,<w><v><u><t>s),lessis(<w><v><u><t>s,x)):'prop'
8102 injseq:=[t:real][it:intrl(t)][lt:lessis(y,t)][tl:lessis(t,x)][u:real][iu:intrl(u)][lu:lessis(y,u)][ul:lessis(u,x)][v:is"r"(<tl><lt><it><t>s,<ul><lu><iu><u>s)]is"r"(t,u):'prop'
8103 [u:real][v:real][a:and3(intrl(v),lessis(y,v),lessis(v,x))]
8104 +*shift
8105 a@prop1:=is(u,<t24(v,a)><t23(v,a)><t22(v,a)><v>s):'prop'
8106 -shift
8107 v@improp:=and(and3(intrl(v),lessis(y,v),lessis(v,x)),[t:and3(intrl(v),lessis(y,v),lessis(v,x))]prop1".shift"(t)):'prop'
8108 u@imseq:=some([t:real]improp(u,t)):'prop'
8109 s@surjseq:=[t:real][u:intrl(t)][v:lessis(y,t)][w:lessis(t,x)]imseq(t):'prop'
8110 perm:=and(injseq,surjseq):'prop'
8111 [ins:inseq(s)]
8112 +*shift
8113 ins@[n:1to(shiftl)]
8114 ns:=<shiftrls(n)><lsshiftr(n)><intshiftr(n)><shiftr(n)>s:real
8115 t34:=<shiftrls(n)><lsshiftr(n)><intshiftr(n)><shiftr(n)>ins:and3(intrl(ns),lessis(y,ns),lessis(ns,x))
8116 -shift
8117 ins@shiftseq:=[t:1to(shiftl)]shiftl1(ns".shift"(t),t34".shift"(t)):[t:1to(shiftl)]1to(shiftl)
8118 [js:injseq(s)]
8119 +*shift
8120 js@[n:1to(shiftl)][m:1to(shiftl)][i:is"e"(1to(shiftl),<n>shiftseq,<m>shiftseq)]
8121 t35:=isoutne(shiftl,ul(ns(n),t34(n)),t30(ns(n),t34(n)),ul(ns(m),t34(m)),t30(ns(m),t34(m)),i):is"n"(ul(ns(n),t34(n)),ul(ns(m),t34(m)))
8122 t36:=isrlint(mn(pl(ns(n),1rl),y),t6(ns(n),t22(ns(n),t34(n)),y,iy,t23(ns(n),t34(n))),mn(pl(ns(m),1rl),y),t6(ns(m),t22(ns(m),t34(m)),y,iy,t23(ns(m),t34(m))),t35):is(mn(pl(ns(n),1rl),y),mn(pl(ns(m),1rl),y))
8123 t37:=satz188b(ns(n),ns(m),1rl,satz188b(pl(ns(n),1rl),pl(ns(m),1rl),m0"r"(y),t36)):is(ns(n),ns(m))
8124 t38:=<t37><shiftrls(m)><lsshiftr(m)><intshiftr(m)><shiftr(m)><shiftrls(n)><lsshiftr(n)><intshiftr(n)><shiftr(n)>js:is(shiftr(n),shiftr(m))
8125 t39:=satz188b(n2(n),n2(m),y,satz188b(pl(n2(n),y),pl(n2(m),y),m0(1rl),t38)):is(n2(n),n2(m))
8126 t40:=isntirl(n1(n),n1(m),t39):is"n"(n1(n),n1(m))
8127 t41:=isinne(shiftl,n,m,t40):is"e"(1to(shiftl),n,m)
8128 -shift
8129 js@injshiftseq:=[t:1to(shiftl)][u:1to(shiftl)][v:is"e"(1to(shiftl),<t>shiftseq,<u>shiftseq)]t41".shift"(t,u,v):injective(1to(shiftl),1to(shiftl),shiftseq)
8130 ins@[pri:proofsirrelevant(real,s)][ss:surjseq(s)]
8131 +*shift
8132 ss@[n:1to(shiftl)]
8133 t42:=<shiftrls(n)><lsshiftr(n)><intshiftr(n)><shiftr(n)>ss:imseq(shiftr(n))
8134 [u:real][p:improp(shiftr(n),u)]
8135 t43:=ande1(and3(intrl(u),lessis(y,u),lessis(u,x)),[t:and3(intrl(u),lessis(y,u),lessis(u,x))]prop1(shiftr(n),u,t),p):and3(intrl(u),lessis(y,u),lessis(u,x))
8136 t44:=ande2"l.r"(and3(intrl(u),lessis(y,u),lessis(u,x)),[t:and3(intrl(u),lessis(y,u),lessis(u,x))]prop1(shiftr(n),u,t),p):is(shiftr(n),<t24(u,t43)><t23(u,t43)><t22(u,t43)><u>s)
8137 ul1:=shiftl1(u,t43):1to(shiftl)
8138 t45:=<shiftinv1(u,t43)><shiftrls(ul1)><lsshiftr(ul1)><intshiftr(ul1)><shiftr(ul1)><t24(u,t43)><t23(u,t43)><t22(u,t43)><u>pri:is(<t24(u,t43)><t23(u,t43)><t22(u,t43)><u>s,ns(ul1))
8139 t46:=shiftinv1(ns(ul1),t34(ul1)):is(ns(ul1),shiftr(<ul1>shiftseq))
8140 t47:=tr3is(real,shiftr(n),<t24(u,t43)><t23(u,t43)><t22(u,t43)><u>s,ns(ul1),shiftr(<ul1>shiftseq),t44,t45,t46):is(shiftr(n),shiftr(<ul1>shiftseq))
8141 t48:=iseshiftr(n,<ul1>shiftseq,t47):is"e"(1to(shiftl),n,<ul1>shiftseq)
8142 t49:=somei(1to(shiftl),[t:1to(shiftl)]is"e"(1to(shiftl),n,<t>shiftseq),ul1,t48):image(1to(shiftl),1to(shiftl),shiftseq,n)
8143 n@t50:=someapp(real,[t:real]improp(shiftr(n),t),t42,image(1to(shiftl),1to(shiftl),shiftseq,n),[t:real][u:improp(shiftr(n),t)]t49(t,u)):image(1to(shiftl),1to(shiftl),shiftseq,n)
8144 -shift
8145 ss@surjshiftseq:=[t:1to(shiftl)]t50".shift"(t):surjective(1to(shiftl),1to(shiftl),shiftseq)
8146 pri@[ps:perm(s)]
8147 bijshiftseq:=andi(injective(1to(shiftl),1to(shiftl),shiftseq),surjective(1to(shiftl),1to(shiftl),shiftseq),injshiftseq(ande1(injseq(s),surjseq(s),ps)),surjshiftseq(ande2(injseq(s),surjseq(s),ps))):bijective(1to(shiftl),1to(shiftl),shiftseq)
8148 +c
8149 @complex:=pair1type(real):'type'
8150 cx:=complex:'type'
8151 [x:complex][y:complex]
8152 is:=is"e"(cx,x,y):'prop'
8153 nis:=not(is(x,y)):'prop'
8154 @[p:[t:complex]'prop']
8155 some:=some"l"(cx,p):'prop'
8156 all:=all"l"(cx,p):'prop'
8157 one:=one"e"(cx,p):'prop'
8158 @[a:real][b:real]
8159 pli:=pair1(real,a,b):complex
8160 x@re:=first1(real,x):real
8161 im:=second1(real,x):real
8162 b@reis:=first1is1(real,a,b):is"r"(re(pli(a,b)),a)
8163 isre:=first1is2(real,a,b):is"r"(a,re(pli(a,b)))
8164 imis:=second1is1(real,a,b):is"r"(im(pli(a,b)),b)
8165 isim:=second1is2(real,a,b):is"r"(b,im(pli(a,b)))
8166 x@pliis:=pair1is1(real,x):is(pli(re(x),im(x)),x)
8167 ispli:=pair1is2(real,x):is(x,pli(re(x),im(x)))
8168 y@[i:is(x,y)]
8169 iscere:=isf(cx,real,[t:cx]re(t),x,y,i):is"r"(re(x),re(y))
8170 isceim:=isf(cx,real,[t:cx]im(t),x,y,i):is"r"(im(x),im(y))
8171 b@[c:real][i:is"r"(a,b)]
8172 isrecx1:=isf(real,cx,[t:real]pli(t,c),a,b,i):is(pli(a,c),pli(b,c))
8173 isrecx2:=isf(real,cx,[t:real]pli(c,t),a,b,i):is(pli(c,a),pli(c,b))
8174 c@[d:real][i:is"r"(a,b)][j:is"r"(c,d)]
8175 isrecx12:=tris(cx,pli(a,c),pli(b,c),pli(b,d),isrecx1(a,b,c,i),isrecx2(c,d,b,j)):is(pli(a,c),pli(b,d))
8176 x@satz206:=refis(cx,x):is(x,x)
8177 y@[i:is(x,y)]
8178 satz207:=symis(cx,x,y,i):is(y,x)
8179 y@[z:complex][i:is(x,y)][j:is(y,z)]
8180 satz208:=tris(cx,x,y,z,i,j):is(x,z)
8181 @0c:=pli(0,0):complex
8182 1c:=pli(1rl,0):complex
8183 y@pl:=pli(pl"r"(re(x),re(y)),pl"r"(im(x),im(y))):complex
8184 d@plis12a:=isrecx12(pl"r"(re(pli(a,b)),re(pli(c,d))),pl"r"(a,c),pl"r"(im(pli(a,b)),im(pli(c,d))),pl"r"(b,d),ispl12(re(pli(a,b)),a,re(pli(c,d)),c,reis(a,b),reis(c,d)),ispl12(im(pli(a,b)),b,im(pli(c,d)),d,imis(a,b),imis(c,d))):is(pl(pli(a,b),pli(c,d)),pli(pl"r"(a,c),pl"r"(b,d)))
8185 plis12b:=symis(cx,pl(pli(a,b),pli(c,d)),pli(pl"r"(a,c),pl"r"(b,d)),plis12a):is(pli(pl"r"(a,c),pl"r"(b,d)),pl(pli(a,b),pli(c,d)))
8186 x@[r:real][s:real]
8187 plis1a:=isrecx12(pl"r"(re(pli(r,s)),re(x)),pl"r"(r,re(x)),pl"r"(im(pli(r,s)),im(x)),pl"r"(s,im(x)),ispl1(re(pli(r,s)),r,re(x),reis(r,s)),ispl1(im(pli(r,s)),s,im(x),imis(r,s))):is(pl(pli(r,s),x),pli(pl"r"(r,re(x)),pl"r"(s,im(x))))
8188 plis1b:=symis(cx,pl(pli(r,s),x),pli(pl"r"(r,re(x)),pl"r"(s,im(x))),plis1a):is(pli(pl"r"(r,re(x)),pl"r"(s,im(x))),pl(pli(r,s),x))
8189 plis2a:=isrecx12(pl"r"(re(x),re(pli(r,s))),pl"r"(re(x),r),pl"r"(im(x),im(pli(r,s))),pl"r"(im(x),s),ispl2(re(pli(r,s)),r,re(x),reis(r,s)),ispl2(im(pli(r,s)),s,im(x),imis(r,s))):is(pl(x,pli(r,s)),pli(pl"r"(re(x),r),pl"r"(im(x),s)))
8190 plis2b:=symis(cx,pl(x,pli(r,s)),pli(pl"r"(re(x),r),pl"r"(im(x),s)),plis2a):is(pli(pl"r"(re(x),r),pl"r"(im(x),s)),pl(x,pli(r,s)))
8191 z@[i:is(x,y)]
8192 ispl1:=isf(cx,cx,[t:cx]pl(t,z),x,y,i):is(pl(x,z),pl(y,z))
8193 ispl2:=isf(cx,cx,[t:cx]pl(z,t),x,y,i):is(pl(z,x),pl(z,y))
8194 z@[u:complex][i:is(x,y)][j:is(z,u)]
8195 ispl12:=tris(cx,pl(x,z),pl(y,z),pl(y,u),ispl1(x,y,z,i),ispl2(z,u,y,j)):is(pl(x,z),pl(y,u))
8196 y@satz209:=isrecx12(pl"r"(re(x),re(y)),pl"r"(re(y),re(x)),pl"r"(im(x),im(y)),pl"r"(im(y),im(x)),compl(re(x),re(y)),compl(im(x),im(y))):is(pl(x,y),pl(y,x))
8197 compl:=satz209:is(pl(x,y),pl(y,x))
8198 x@satz210:=tr3is(cx,pl(x,0c),pli(pl"r"(re(x),0),pl"r"(im(x),0)),pli(re(x),im(x)),x,plis2a(x,0,0),isrecx12(pl"r"(re(x),0),re(x),pl"r"(im(x),0),im(x),pl02(re(x),0,refis(real,0)),pl02(im(x),0,refis(real,0))),pliis(x)):is(pl(x,0c),x)
8199 satz210a:=symis(cx,pl(x,0c),x,satz210):is(x,pl(x,0c))
8200 satz210b:=tris(cx,pl(0c,x),pl(x,0c),x,compl(0c,x),satz210):is(pl(0c,x),x)
8201 satz210c:=symis(cx,pl(0c,x),x,satz210b):is(x,pl(0c,x))
8202 z@satz211:=tr3is(cx,pl(pl(x,y),z),pli(pl"r"(pl"r"(re(x),re(y)),re(z)),pl"r"(pl"r"(im(x),im(y)),im(z))),pli(pl"r"(re(x),pl"r"(re(y),re(z))),pl"r"(im(x),pl"r"(im(y),im(z)))),pl(x,pl(y,z)),plis1a(z,pl"r"(re(x),re(y)),pl"r"(im(x),im(y))),isrecx12(pl"r"(pl"r"(re(x),re(y)),re(z)),pl"r"(re(x),pl"r"(re(y),re(z))),pl"r"(pl"r"(im(x),im(y)),im(z)),pl"r"(im(x),pl"r"(im(y),im(z))),asspl1(re(x),re(y),re(z)),asspl1(im(x),im(y),im(z))),plis2b(x,pl"r"(re(y),re(z)),pl"r"(im(y),im(z)))):is(pl(pl(x,y),z),pl(x,pl(y,z)))
8203 asspl1:=satz211:is(pl(pl(x,y),z),pl(x,pl(y,z)))
8204 asspl2:=symis(cx,pl(pl(x,y),z),pl(x,pl(y,z)),asspl1):is(pl(x,pl(y,z)),pl(pl(x,y),z))
8205 y@[u:complex][i:is(pl(y,u),x)]
8206 +2212
8207 t1:=tris(real,pl"r"(re(y),re(u)),re(pl(y,u)),re(x),isre(pl"r"(re(y),re(u)),pl"r"(im(y),im(u))),iscere(pl(y,u),x,i)):is"r"(pl"r"(re(y),re(u)),re(x))
8208 t2:=tris(real,pl"r"(im(y),im(u)),im(pl(y,u)),im(x),isim(pl"r"(re(y),re(u)),pl"r"(im(y),im(u))),isceim(pl(y,u),x,i)):is"r"(pl"r"(im(y),im(u)),im(x))
8209 t3:=satz187d(re(x),re(y),re(u),t1):is"r"(re(u),mn(re(x),re(y)))
8210 t4:=satz187d(im(x),im(y),im(u),t2):is"r"(im(u),mn(im(x),im(y)))
8211 -2212
8212 satz212a:=tris(cx,u,pli(re(u),im(u)),pli(mn(re(x),re(y)),mn(im(x),im(y))),ispli(u),isrecx12(re(u),mn(re(x),re(y)),im(u),mn(im(x),im(y)),t3".2212",t4".2212")):is(u,pli(mn(re(x),re(y)),mn(im(x),im(y))))
8213 y@satz212b:=tr3is(cx,pl(y,pli(mn(re(x),re(y)),mn(im(x),im(y)))),pli(pl"r"(re(y),mn(re(x),re(y))),pl"r"(im(y),mn(im(x),im(y)))),pli(re(x),im(x)),x,plis2a(y,mn(re(x),re(y)),mn(im(x),im(y))),isrecx12(pl"r"(re(y),mn(re(x),re(y))),re(x),pl"r"(im(y),mn(im(x),im(y))),im(x),satz187a(re(x),re(y)),satz187a(im(x),im(y))),pliis(x)):is(pl(y,pli(mn(re(x),re(y)),mn(im(x),im(y)))),x)
8214 satz212c:=somei(cx,[t:cx]is(pl(y,t),x),pli(mn(re(x),re(y)),mn(im(x),im(y))),satz212b):some([t:complex]is(pl(y,t),x))
8215 satz212:=onei(cx,[t:cx]is(pl(y,t),x),[t:cx][u:cx][v:is(pl(y,t),x)][w:is(pl(y,u),x)]tris2(cx,t,u,pli(mn(re(x),re(y)),mn(im(x),im(y))),satz212a(t,v),satz212a(u,w)),satz212c):one([t:complex]is(pl(y,t),x))
8216 %satz212:=onei(cx,[t:cx]is(pl(y,t),x),[t:cx][u:cx][v:is(pl(y,t),x)][w:is(pl(y,u),x)]tris2(cx,t,u,v),satz212a(u,w)),satz212c):one([t:complex]is(pl(y,t),x))
8217 mn:=pli(mn"r"(re(x),re(y)),mn"r"(im(x),im(y))):complex
8218 d@mnis12a:=isrecx12(mn"r"(re(pli(a,b)),re(pli(c,d))),mn"r"(a,c),mn"r"(im(pli(a,b)),im(pli(c,d))),mn"r"(b,d),ismn12(re(pli(a,b)),a,re(pli(c,d)),c,reis(a,b),reis(c,d)),ismn12(im(pli(a,b)),b,im(pli(c,d)),d,imis(a,b),imis(c,d))):is(mn(pli(a,b),pli(c,d)),pli(mn"r"(a,c),mn"r"(b,d)))
8219 mnis12b:=symis(cx,mn(pli(a,b),pli(c,d)),pli(mn"r"(a,c),mn"r"(b,d)),mnis12a):is(pli(mn"r"(a,c),mn"r"(b,d)),mn(pli(a,b),pli(c,d)))
8220 x@[r:real][s:real]
8221 mnis1a:=isrecx12(mn"r"(re(pli(r,s)),re(x)),mn"r"(r,re(x)),mn"r"(im(pli(r,s)),im(x)),mn"r"(s,im(x)),ismn1(re(pli(r,s)),r,re(x),reis(r,s)),ismn1(im(pli(r,s)),s,im(x),imis(r,s))):is(mn(pli(r,s),x),pli(mn"r"(r,re(x)),mn"r"(s,im(x))))
8222 mnis1b:=symis(cx,mn(pli(r,s),x),pli(mn"r"(r,re(x)),mn"r"(s,im(x))),mnis1a):is(pli(mn"r"(r,re(x)),mn"r"(s,im(x))),mn(pli(r,s),x))
8223 mnis2a:=isrecx12(mn"r"(re(x),re(pli(r,s))),mn"r"(re(x),r),mn"r"(im(x),im(pli(r,s))),mn"r"(im(x),s),ismn2(re(pli(r,s)),r,re(x),reis(r,s)),ismn2(im(pli(r,s)),s,im(x),imis(r,s))):is(mn(x,pli(r,s)),pli(mn"r"(re(x),r),mn"r"(im(x),s)))
8224 mnis2b:=symis(cx,mn(x,pli(r,s)),pli(mn"r"(re(x),r),mn"r"(im(x),s)),mnis2a):is(pli(mn"r"(re(x),r),mn"r"(im(x),s)),mn(x,pli(r,s)))
8225 z@[i:is(x,y)]
8226 ismn1:=isf(cx,cx,[t:cx]mn(t,z),x,y,i):is(mn(x,z),mn(y,z))
8227 ismn2:=isf(cx,cx,[t:cx]mn(z,t),x,y,i):is(mn(z,x),mn(z,y))
8228 z@[u:complex][i:is(x,y)][j:is(z,u)]
8229 ismn12:=tris(cx,mn(x,z),mn(y,z),mn(y,u),ismn1(x,y,z,i),ismn2(z,u,y,j)):is(mn(x,z),mn(y,u))
8230 y@[u:complex][i:is(pl(y,u),x)]
8231 satz212d:=satz212a(u,i):is(u,mn(x,y))
8232 satz212e:=symis(cx,u,mn(x,y),satz212d):is(mn(x,y),u)
8233 u@[i:is(pl(u,y),x)]
8234 satz212f:=satz212d(tris(cx,pl(y,u),pl(u,y),x,compl(y,u),i)):is(u,mn(x,y))
8235 satz212g:=symis(cx,u,mn(x,y),satz212f):is(mn(x,y),u)
8236 y@satz212h:=satz212b:is(pl(y,mn(x,y)),x)
8237 [i:is(mn(x,y),0c)]
8238 +2213
8239 t1:=tr3is(real,mn"r"(re(x),re(y)),re(mn(x,y)),re(0c),0,isre(mn"r"(re(x),re(y)),mn"r"(im(x),im(y))),iscere(mn(x,y),0c,i),reis(0,0)):is"r"(mn"r"(re(x),re(y)),0)
8240 t2:=tr3is(real,mn"r"(im(x),im(y)),im(mn(x,y)),im(0c),0,isim(mn"r"(re(x),re(y)),mn"r"(im(x),im(y))),isceim(mn(x,y),0c,i),imis(0,0)):is"r"(mn"r"(im(x),im(y)),0)
8241 t3:=satz182b(re(x),re(y),t1):is"r"(re(x),re(y))
8242 t4:=satz182b(im(x),im(y),t2):is"r"(im(x),im(y))
8243 -2213
8244 satz213a:=tr3is(cx,x,pli(re(x),im(x)),pli(re(y),im(y)),y,ispli(x),isrecx12(re(x),re(y),im(x),im(y),t3".2213",t4".2213"),pliis(y)):is(x,y)
8245 y@[i:is(x,y)]
8246 +*2213
8247 i@t5:=satz182e(re(x),re(y),iscere(x,y,i)):is"r"(mn"r"(re(x),re(y)),0)
8248 t6:=satz182e(im(x),im(y),isceim(x,y,i)):is"r"(mn"r"(im(x),im(y)),0)
8249 -2213
8250 i@satz213b:=isrecx12(mn"r"(re(x),re(y)),0,mn"r"(im(x),im(y)),0,t5".2213",t6".2213"):is(mn(x,y),0c)
8251 x@m0:=mn(0c,x):complex
8252 satz214:=isrecx12(mn"r"(re(0c),re(x)),m0"r"(re(x)),mn"r"(im(0c),im(x)),m0"r"(im(x)),pl01(re(0c),m0"r"(re(x)),reis(0,0)),pl01(im(0c),m0"r"(im(x)),imis(0,0))):is(m0(x),pli(m0"r"(re(x)),m0"r"(im(x))))
8253 satz214a:=symis(cx,m0(x),pli(m0"r"(re(x)),m0"r"(im(x))),satz214):is(pli(m0"r"(re(x)),m0"r"(im(x))),m0(x))
8254 b@m0isa:=tris(cx,m0(pli(a,b)),pli(m0"r"(re(pli(a,b))),m0"r"(im(pli(a,b)))),pli(m0"r"(a),m0"r"(b)),satz214(pli(a,b)),isrecx12(m0"r"(re(pli(a,b))),m0"r"(a),m0"r"(im(pli(a,b))),m0"r"(b),ism0(re(pli(a,b)),a,reis(a,b)),ism0(im(pli(a,b)),b,imis(a,b)))):is(m0(pli(a,b)),pli(m0"r"(a),m0"r"(b)))
8255 m0isb:=symis(cx,m0(pli(a,b)),pli(m0"r"(a),m0"r"(b)),m0isa):is(pli(m0"r"(a),m0"r"(b)),m0(pli(a,b)))
8256 y@[i:is(x,y)]
8257 ism0:=isf(cx,cx,[t:cx]m0(t),x,y,i):is(m0(x),m0(y))
8258 x@satz215:=tr4is(cx,m0(m0(x)),m0(pli(m0"r"(re(x)),m0"r"(im(x)))),pli(m0"r"(m0"r"(re(x))),m0"r"(m0"r"(im(x)))),pli(re(x),im(x)),x,ism0(m0(x),pli(m0"r"(re(x)),m0"r"(im(x))),satz214),m0isa(m0"r"(re(x)),m0"r"(im(x))),isrecx12(m0"r"(m0"r"(re(x))),re(x),m0"r"(m0"r"(im(x))),im(x),satz177(re(x)),satz177(im(x))),pliis(x)):is(m0(m0(x)),x)
8259 satz215a:=symis(cx,m0(m0(x)),x,satz215):is(x,m0(m0(x)))
8260 y@[i:is(x,m0(y))]
8261 satz215b:=tris(cx,m0(x),m0(m0(y)),y,ism0(x,m0(y),i),satz215(y)):is(m0(x),y)
8262 satz215c:=symis(cx,m0(x),y,satz215b):is(y,m0(x))
8263 y@[i:is(m0(x),y)]
8264 satz215d:=satz215c(y,x,symis(cx,m0(x),y,i)):is(x,m0(y))
8265 satz215e:=symis(cx,x,m0(y),satz215d):is(m0(y),x)
8266 x@satz216:=tr3is(cx,pl(x,m0(x)),pl(x,pli(m0"r"(re(x)),m0"r"(im(x)))),pli(pl"r"(re(x),m0"r"(re(x))),pl"r"(im(x),m0"r"(im(x)))),0c,ispl2(m0(x),pli(m0"r"(re(x)),m0"r"(im(x))),x,satz214(x)),plis2a(x,m0"r"(re(x)),m0"r"(im(x))),isrecx12(pl"r"(re(x),m0"r"(re(x))),0,pl"r"(im(x),m0"r"(im(x))),0,satz179(re(x)),satz179(im(x)))):is(pl(x,m0(x)),0c)
8267 +2216
8268 anders:=satz212h(0c,x):is(pl(x,m0(x)),0c)
8269 -2216
8270 satz216a:=tris(cx,pl(m0(x),x),pl(x,m0(x)),0c,compl(m0(x),x),satz216):is(pl(m0(x),x),0c)
8271 y@satz217:=tr4is(cx,m0(pl(x,y)),pli(m0"r"(pl"r"(re(x),re(y))),m0"r"(pl"r"(im(x),im(y)))),pli(pl"r"(m0"r"(re(x)),m0"r"(re(y))),pl"r"(m0"r"(im(x)),m0"r"(im(y)))),pl(pli(m0"r"(re(x)),m0"r"(im(x))),pli(m0"r"(re(y)),m0"r"(im(y)))),pl(m0(x),m0(y)),m0isa(pl"r"(re(x),re(y)),pl"r"(im(x),im(y))),isrecx12(m0"r"(pl"r"(re(x),re(y))),pl"r"(m0"r"(re(x)),m0"r"(re(y))),m0"r"(pl"r"(im(x),im(y))),pl"r"(m0"r"(im(x)),m0"r"(im(y))),satz180(re(x),re(y)),satz180(im(x),im(y))),plis12b(m0"r"(re(x)),m0"r"(im(x)),m0"r"(re(y)),m0"r"(im(y))),ispl12(pli(m0"r"(re(x)),m0"r"(im(x))),m0(x),pli(m0"r"(re(y)),m0"r"(im(y))),m0(y),satz214a(x),satz214a(y))):is(m0(pl(x,y)),pl(m0(x),m0(y)))
8272 satz217a:=symis(cx,m0(pl(x,y)),pl(m0(x),m0(y)),satz217):is(pl(m0(x),m0(y)),m0(pl(x,y)))
8273 satz218:=tris(cx,mn(x,y),pl(x,pli(m0"r"(re(y)),m0"r"(im(y)))),pl(x,m0(y)),plis2b(x,m0"r"(re(y)),m0"r"(im(y))),ispl2(pli(m0"r"(re(y)),m0"r"(im(y))),m0(y),x,satz214a(y))):is(mn(x,y),pl(x,m0(y)))
8274 satz218a:=symis(cx,mn(x,y),pl(x,m0(y)),satz218):is(pl(x,m0(y)),mn(x,y))
8275 +2219
8276 t1:=tr3is(cx,m0(mn(x,y)),m0(pl(x,m0(y))),pl(m0(x),m0(m0(y))),pl(m0(x),y),ism0(mn(x,y),pl(x,m0(y)),satz218),satz217(x,m0(y)),ispl2(m0(m0(y)),y,m0(x),satz215(y))):is(m0(mn(x,y)),pl(m0(x),y))
8277 -2219
8278 satz219:=tr3is(cx,m0(mn(x,y)),pl(m0(x),y),pl(y,m0(x)),mn(y,x),t1".2219",compl(m0(x),y),satz218a(y,x)):is(m0(mn(x,y)),mn(y,x))
8279 satz219a:=symis(cx,m0(mn(y,x)),mn(x,y),satz219(y,x)):is(mn(x,y),m0(mn(y,x)))
8280 ts:=pli(mn"r"(ts"r"(re(x),re(y)),ts"r"(im(x),im(y))),pl"r"(ts"r"(re(x),im(y)),ts"r"(im(x),re(y)))):complex
8281 rets:=mn"r"(ts"r"(re(x),re(y)),ts"r"(im(x),im(y))):real
8282 imts:=pl"r"(ts"r"(re(x),im(y)),ts"r"(im(x),re(y))):real
8283 +v3
8284 d@t1:=ismn12"r"(ts"r"(re(pli(a,b)),re(pli(c,d))),ts"r"(a,c),ts"r"(im(pli(a,b)),im(pli(c,d))),ts"r"(b,d),ists12(re(pli(a,b)),a,re(pli(c,d)),c,reis(a,b),reis(c,d)),ists12(im(pli(a,b)),b,im(pli(c,d)),d,imis(a,b),imis(c,d))):is"r"(rets(pli(a,b),pli(c,d)),mn"r"(ts"r"(a,c),ts"r"(b,d)))
8285 t2:=ispl12"r"(ts"r"(re(pli(a,b)),im(pli(c,d))),ts"r"(a,d),ts"r"(im(pli(a,b)),re(pli(c,d))),ts"r"(b,c),ists12(re(pli(a,b)),a,im(pli(c,d)),d,reis(a,b),imis(c,d)),ists12(im(pli(a,b)),b,re(pli(c,d)),c,imis(a,b),reis(c,d))):is"r"(imts(pli(a,b),pli(c,d)),pl"r"(ts"r"(a,d),ts"r"(b,c)))
8286 -v3
8287 d@tsis12a:=isrecx12(rets(pli(a,b),pli(c,d)),mn"r"(ts"r"(a,c),ts"r"(b,d)),imts(pli(a,b),pli(c,d)),pl"r"(ts"r"(a,d),ts"r"(b,c)),t1".v3",t2".v3"):is(ts(pli(a,b),pli(c,d)),pli(mn"r"(ts"r"(a,c),ts"r"(b,d)),pl"r"(ts"r"(a,d),ts"r"(b,c))))
8288 tsis12b:=symis(cx,ts(pli(a,b),pli(c,d)),pli(mn"r"(ts"r"(a,c),ts"r"(b,d)),pl"r"(ts"r"(a,d),ts"r"(b,c))),tsis12a):is(pli(mn"r"(ts"r"(a,c),ts"r"(b,d)),pl"r"(ts"r"(a,d),ts"r"(b,c))),ts(pli(a,b),pli(c,d)))
8289 x@[r:real][s:real]
8290 +*v3
8291 s@t3:=ismn12"r"(ts"r"(re(pli(r,s)),re(x)),ts"r"(r,re(x)),ts"r"(im(pli(r,s)),im(x)),ts"r"(s,im(x)),ists1(re(pli(r,s)),r,re(x),reis(r,s)),ists1(im(pli(r,s)),s,im(x),imis(r,s))):is"r"(rets(pli(r,s),x),mn"r"(ts"r"(r,re(x)),ts"r"(s,im(x))))
8292 t4:=ispl12"r"(ts"r"(re(pli(r,s)),im(x)),ts"r"(r,im(x)),ts"r"(im(pli(r,s)),re(x)),ts"r"(s,re(x)),ists1(re(pli(r,s)),r,im(x),reis(r,s)),ists1(im(pli(r,s)),s,re(x),imis(r,s))):is"r"(imts(pli(r,s),x),pl"r"(ts"r"(r,im(x)),ts"r"(s,re(x))))
8293 -v3
8294 s@tsis1a:=isrecx12(rets(pli(r,s),x),mn"r"(ts"r"(r,re(x)),ts"r"(s,im(x))),imts(pli(r,s),x),pl"r"(ts"r"(r,im(x)),ts"r"(s,re(x))),t3".v3",t4".v3"):is(ts(pli(r,s),x),pli(mn"r"(ts"r"(r,re(x)),ts"r"(s,im(x))),pl"r"(ts"r"(r,im(x)),ts"r"(s,re(x)))))
8295 tsis1b:=symis(cx,ts(pli(r,s),x),pli(mn"r"(ts"r"(r,re(x)),ts"r"(s,im(x))),pl"r"(ts"r"(r,im(x)),ts"r"(s,re(x)))),tsis1a):is(pli(mn"r"(ts"r"(r,re(x)),ts"r"(s,im(x))),pl"r"(ts"r"(r,im(x)),ts"r"(s,re(x)))),ts(pli(r,s),x))
8296 +*v3
8297 s@t5:=ismn12"r"(ts"r"(re(x),re(pli(r,s))),ts"r"(re(x),r),ts"r"(im(x),im(pli(r,s))),ts"r"(im(x),s),ists2(re(pli(r,s)),r,re(x),reis(r,s)),ists2(im(pli(r,s)),s,im(x),imis(r,s))):is"r"(rets(x,pli(r,s)),mn"r"(ts"r"(re(x),r),ts"r"(im(x),s)))
8298 t6:=ispl12"r"(ts"r"(re(x),im(pli(r,s))),ts"r"(re(x),s),ts"r"(im(x),re(pli(r,s))),ts"r"(im(x),r),ists2(im(pli(r,s)),s,re(x),imis(r,s)),ists2(re(pli(r,s)),r,im(x),reis(r,s))):is"r"(imts(x,pli(r,s)),pl"r"(ts"r"(re(x),s),ts"r"(im(x),r)))
8299 -v3
8300 s@tsis2a:=isrecx12(rets(x,pli(r,s)),mn"r"(ts"r"(re(x),r),ts"r"(im(x),s)),imts(x,pli(r,s)),pl"r"(ts"r"(re(x),s),ts"r"(im(x),r)),t5".v3",t6".v3"):is(ts(x,pli(r,s)),pli(mn"r"(ts"r"(re(x),r),ts"r"(im(x),s)),pl"r"(ts"r"(re(x),s),ts"r"(im(x),r))))
8301 tsis2b:=symis(cx,ts(x,pli(r,s)),pli(mn"r"(ts"r"(re(x),r),ts"r"(im(x),s)),pl"r"(ts"r"(re(x),s),ts"r"(im(x),r))),tsis2a):is(pli(mn"r"(ts"r"(re(x),r),ts"r"(im(x),s)),pl"r"(ts"r"(re(x),s),ts"r"(im(x),r))),ts(x,pli(r,s)))
8302 z@[i:is(x,y)]
8303 ists1:=isf(cx,cx,[t:cx]ts(t,z),x,y,i):is(ts(x,z),ts(y,z))
8304 ists2:=isf(cx,cx,[t:cx]ts(z,t),x,y,i):is(ts(z,x),ts(z,y))
8305 z@[u:complex][i:is(x,y)][j:is(z,u)]
8306 ists12:=tris(cx,ts(x,z),ts(y,z),ts(y,u),ists1(x,y,z,i),ists2(z,u,y,j)):is(ts(x,z),ts(y,u))
8307 +3220
8308 y@t1:=ismn12"r"(ts"r"(re(x),re(y)),ts"r"(re(y),re(x)),ts"r"(im(x),im(y)),ts"r"(im(y),im(x)),comts(re(x),re(y)),comts(im(x),im(y))):is"r"(rets(x,y),rets(y,x))
8309 t2:=tris(real,imts(x,y),pl"r"(ts"r"(im(x),re(y)),ts"r"(re(x),im(y))),imts(y,x),compl"r"(ts"r"(re(x),im(y)),ts"r"(im(x),re(y))),ispl12"r"(ts"r"(im(x),re(y)),ts"r"(re(y),im(x)),ts"r"(re(x),im(y)),ts"r"(im(y),re(x)),comts(im(x),re(y)),comts(re(x),im(y)))):is"r"(imts(x,y),imts(y,x))
8310 -3220
8311 y@satz220:=isrecx12(rets(x,y),rets(y,x),imts(x,y),imts(y,x),t1".3220",t2".3220"):is(ts(x,y),ts(y,x))
8312 comts:=satz220:is(ts(x,y),ts(y,x))
8313 x@[i:is(x,0c)]
8314 lemma1:=tris(real,re(x),re(0c),0,iscere(x,0c,i),reis(0,0)):is"r"(re(x),0)
8315 lemma2:=tris(real,im(x),im(0c),0,isceim(x,0c,i),imis(0,0)):is"r"(im(x),0)
8316 x@mod2:=pl"r"(ts"r"(re(x),re(x)),ts"r"(im(x),im(x))):real
8317 i@lemma3:=tris(real,mod2,pl"r"(0,0),0,ispl12"r"(ts"r"(re(x),re(x)),0,ts"r"(im(x),im(x)),0,ts01(re(x),re(x),lemma1),ts01(im(x),im(x),lemma2)),pl01(0,0,refis(real,0))):is"r"(mod2(x),0)
8318 x@[n:nis(x,0c)]
8319 +*v3
8320 x@re2:=ts"r"(re(x),re(x)):real
8321 im2:=ts"r"(im(x),im(x)):real
8322 n@[i:is"r"(re(x),0)]
8323 t7:=th3"l.imp"(is"r"(im(x),0),is(x,0c),n,[t:is"r"(im(x),0)]tris(cx,x,pli(re(x),im(x)),0c,ispli(x),isrecx12(re(x),0,im(x),0,i,t))):nis"r"(im(x),0)
8324 t8:=ispos(im2,mod2,symis(real,mod2,im2,pl01(re2,im2,ts01(re(x),re(x),i))),possq(im(x),t7)):pos(mod2)
8325 n@[o:nis"r"(re(x),0)][i:is"r"(im(x),0)]
8326 t9:=ispos(re2,mod2,symis(real,mod2,re2,pl02(re2,im2,ts01(im(x),im(x),i))),possq(re(x),o)):pos(mod2)
8327 o@[p:nis"r"(im(x),0)]
8328 t10:=pospl(re2,im2,possq(re(x),o),possq(im(x),p)):pos(mod2)
8329 o@t11:=th1"l.imp"(is"r"(im(x),0),pos(mod2),[t:is"r"(im(x),0)]t9(t),[t:nis"r"(im(x),0)]t10(t)):pos(mod2)
8330 -v3
8331 n@lemma4:=th1"l.imp"(is"r"(re(x),0),pos(mod2),[t:is"r"(re(x),0)]t8".v3"(t),[t:nis"r"(re(x),0)]t11".v3"(t)):pos(mod2(x))
8332 x@lemma5:=th1"l.imp"(is(x,0c),not(neg(mod2)),[t:is(x,0c)]0notn(mod2,lemma3(t)),[t:nis(x,0c)]pnotn(mod2,lemma4(t))):not(neg(mod2(x)))
8333 y@[i:is(x,0c)]
8334 +3221
8335 t1:=lemma1(x,i):is"r"(re(x),0)
8336 t2:=lemma2(x,i):is"r"(im(x),0)
8337 t3:=tris(real,rets(x,y),mn"r"(0,0),0,ismn12"r"(ts"r"(re(x),re(y)),0,ts"r"(im(x),im(y)),0,ts01(re(x),re(y),t1),ts01(im(x),im(y),t2)),satz187c(0,0,0,pl02(0,0,refis(real,0)))):is"r"(rets(x,y),0)
8338 t4:=tris(real,imts(x,y),pl"r"(0,0),0,ispl12"r"(ts"r"(re(x),im(y)),0,ts"r"(im(x),re(y)),0,ts01(re(x),im(y),t1),ts01(im(x),re(y),t2)),pl02(0,0,refis(real,0))):is"r"(imts(x,y),0)
8339 -3221
8340 satz221a:=isrecx12(rets(x,y),0,imts(x,y),0,t3".3221",t4".3221"):is(ts(x,y),0c)
8341 y@[i:is(y,0c)]
8342 satz221b:=tris(cx,ts(x,y),ts(y,x),0c,comts(x,y),satz221a(y,x,i)):is(ts(x,y),0c)
8343 y@[i:is(ts(x,y),0c)]
8344 +*3221
8345 i@[n:nis(y,0c)]
8346 t5:=lemma4(y,n):pos(mod2(y))
8347 t6:=tris1(real,rets(x,y),0,re(ts(x,y)),reis(rets(x,y),imts(x,y)),lemma1(ts(x,y),i)):is"r"(rets(x,y),0)
8348 t7:=tris1(real,imts(x,y),0,im(ts(x,y)),imis(rets(x,y),imts(x,y)),lemma2(ts(x,y),i)):is"r"(imts(x,y),0)
8349 t8:=tris(real,pl"r"(ts"r"(rets(x,y),re(y)),ts"r"(imts(x,y),im(y))),pl"r"(0,0),0,ispl12"r"(ts"r"(rets(x,y),re(y)),0,ts"r"(imts(x,y),im(y)),0,ts01(rets(x,y),re(y),t6),ts01(imts(x,y),im(y),t7)),pl02(0,0,refis(real,0))):is"r"(pl"r"(ts"r"(rets(x,y),re(y)),ts"r"(imts(x,y),im(y))),0)
8350 y@ii1r:=ts"r"(ts"r"(im(x),im(y)),re(y)):real
8351 i1ir:=ts"r"(im(x),ts"r"(im(y),re(y))):real
8352 ir1i:=ts"r"(ts"r"(im(x),re(y)),im(y)):real
8353 i1ri:=ts"r"(im(x),ts"r"(re(y),im(y))):real
8354 rr1r:=ts"r"(ts"r"(re(x),re(y)),re(y)):real
8355 r1rr:=ts"r"(re(x),ts"r"(re(y),re(y))):real
8356 ri1r:=ts"r"(ts"r"(re(x),im(y)),re(y)):real
8357 r1ir:=ts"r"(re(x),ts"r"(im(y),re(y))):real
8358 ri1i:=ts"r"(ts"r"(re(x),im(y)),im(y)):real
8359 r1ii:=ts"r"(re(x),ts"r"(im(y),im(y))):real
8360 n@t9:=tris(real,ii1r,i1ir,i1ri,assts1(im(x),im(y),re(y)),ists2"r"(ts"r"(im(y),re(y)),ts"r"(re(y),im(y)),im(x),comts"r"(im(y),re(y)))):is"r"(ii1r,i1ri)
8361 t10:=tris(real,ts"r"(rets(x,y),re(y)),mn"r"(rr1r,ii1r),mn"r"(r1rr,i1ri),disttm1(ts"r"(re(x),re(y)),ts"r"(im(x),im(y)),re(y)),ismn12"r"(rr1r,r1rr,ii1r,i1ri,assts1(re(x),re(y),re(y)),t9)):is"r"(ts"r"(rets(x,y),re(y)),mn"r"(r1rr,i1ri))
8362 t11:=tr3is(real,ts"r"(imts(x,y),im(y)),pl"r"(ri1i,ir1i),pl"r"(r1ii,i1ri),pl"r"(i1ri,r1ii),disttp1(ts"r"(re(x),im(y)),ts"r"(im(x),re(y)),im(y)),ispl12"r"(ri1i,r1ii,ir1i,i1ri,assts1(re(x),im(y),im(y)),assts1(im(x),re(y),im(y))),compl"r"(r1ii,i1ri)):is"r"(ts"r"(imts(x,y),im(y)),pl"r"(i1ri,r1ii))
8363 t12:=tr4is(real,pl"r"(ts"r"(rets(x,y),re(y)),ts"r"(imts(x,y),im(y))),pl"r"(mn"r"(r1rr,i1ri),pl"r"(i1ri,r1ii)),pl"r"(pl"r"(mn"r"(r1rr,i1ri),i1ri),r1ii),pl"r"(r1rr,r1ii),ts"r"(re(x),mod2(y)),ispl12"r"(ts"r"(rets(x,y),re(y)),mn"r"(r1rr,i1ri),ts"r"(imts(x,y),im(y)),pl"r"(i1ri,r1ii),t10,t11),asspl2"r"(mn"r"(r1rr,i1ri),i1ri,r1ii),ispl1"r"(pl"r"(mn"r"(r1rr,i1ri),i1ri),r1rr,r1ii,plmn(r1rr,i1ri)),distpt2(re(x),ts"r"(re(y),re(y)),ts"r"(im(y),im(y)))):is"r"(pl"r"(ts"r"(rets(x,y),re(y)),ts"r"(imts(x,y),im(y))),ts"r"(re(x),mod2(y)))
8364 t13:=tris1(real,ts"r"(re(x),mod2(y)),0,pl"r"(ts"r"(rets(x,y),re(y)),ts"r"(imts(x,y),im(y))),t12,t8):is"r"(ts"r"(re(x),mod2(y)),0)
8365 t14:=ore1(is"r"(re(x),0),is"r"(mod2(y),0),satz192c(re(x),mod2(y),t13),pnot0(mod2(y),t5)):is"r"(re(x),0)
8366 t15:=tris1(real,ts"r"(im(x),im(y)),0,ts"r"(re(x),re(y)),satz182b(ts"r"(re(x),re(y)),ts"r"(im(x),im(y)),t6),ts01(re(x),re(y),t14)):is"r"(ts"r"(im(x),im(y)),0)
8367 t16:=tris1(real,ts"r"(im(x),re(y)),0,imts(x,y),pl01(ts"r"(re(x),im(y)),ts"r"(im(x),re(y)),ts01(re(x),im(y),t14)),t7):is"r"(ts"r"(im(x),re(y)),0)
8368 [j:is"r"(re(y),0)]
8369 t17:=t7"c.v3"(y,n,j):nis"r"(im(y),0)
8370 t18:=ore1(is"r"(im(x),0),is"r"(im(y),0),satz192c(im(x),im(y),t15),t17):is"r"(im(x),0)
8371 n@[o:nis"r"(re(y),0)]
8372 t19:=ore1(is"r"(im(x),0),is"r"(re(y),0),satz192c(im(x),re(y),t16),o):is"r"(im(x),0)
8373 n@t20:=th1"l.imp"(is"r"(re(y),0),is"r"(im(x),0),[t:is"r"(re(y),0)]t18(t),[t:nis"r"(re(y),0)]t19(t)):is"r"(im(x),0)
8374 t21:=tris(cx,x,pli(re(x),im(x)),0c,ispli(x),isrecx12(re(x),0,im(x),0,t14,t20)):is(x,0c)
8375 -3221
8376 i@satz221c:=th2"l.or"(is(x,0c),is(y,0c),[t:nis(y,0c)]t21".3221"(t)):or(is(x,0c),is(y,0c))
8377 y@[n:nis(x,0c)][o:nis(y,0c)]
8378 satz221d:=th3"l.imp"(is(ts(x,y),0c),or(is(x,0c),is(y,0c)),th3"l.or"(is(x,0c),is(y,0c),n,o),[t:is(ts(x,y),0c)]satz221c(t)):nis(ts(x,y),0c)
8379 +3222
8380 x@t1:=tris(real,mn"r"(ts"r"(re(x),1rl),ts"r"(im(x),0)),mn"r"(re(x),0),re(x),ismn12"r"(ts"r"(re(x),1rl),re(x),ts"r"(im(x),0),0,satz195(re(x)),ts02(im(x),0,refis(real,0))),pl02(re(x),m0"r"(0),satz176b(0,refis(real,0)))):is"r"(mn"r"(ts"r"(re(x),1rl),ts"r"(im(x),0)),re(x))
8381 t2:=tris(real,pl"r"(ts"r"(re(x),0),ts"r"(im(x),1rl)),pl"r"(0,im(x)),im(x),ispl12"r"(ts"r"(re(x),0),0,ts"r"(im(x),1rl),im(x),ts02(re(x),0,refis(real,0)),satz195(im(x))),pl01(0,im(x),refis(real,0))):is"r"(pl"r"(ts"r"(re(x),0),ts"r"(im(x),1rl)),im(x))
8382 -3222
8383 x@satz222:=tr3is(cx,ts(x,1c),pli(mn"r"(ts"r"(re(x),1rl),ts"r"(im(x),0)),pl"r"(ts"r"(re(x),0),ts"r"(im(x),1rl))),pli(re(x),im(x)),x,tsis2a(x,1rl,0),isrecx12(mn"r"(ts"r"(re(x),1rl),ts"r"(im(x),0)),re(x),pl"r"(ts"r"(re(x),0),ts"r"(im(x),1rl)),im(x),t1".3222",t2".3222"),pliis(x)):is(ts(x,1c),x)
8384 satz222a:=symis(cx,ts(x,1c),x,satz222):is(x,ts(x,1c))
8385 satz222b:=tris(cx,ts(1c,x),ts(x,1c),x,comts(1c,x),satz222):is(ts(1c,x),x)
8386 satz222c:=symis(cx,ts(1c,x),x,satz222b):is(x,ts(1c,x))
8387 +3223
8388 t1:=tris(real,mn"r"(ts"r"(re(x),m0"r"(1rl)),ts"r"(im(x),m0"r"(0))),mn"r"(m0"r"(re(x)),0),m0"r"(re(x)),ismn12"r"(ts"r"(re(x),m0"r"(1rl)),m0"r"(re(x)),ts"r"(im(x),m0"r"(0)),0,tris(real,ts"r"(re(x),m0"r"(1rl)),m0"r"(ts"r"(re(x),1rl)),m0"r"(re(x)),satz197b(re(x),1rl),ism0"r"(ts"r"(re(x),1rl),re(x),satz195(re(x)))),ts02(im(x),m0"r"(0),satz176b(0,refis(real,0)))),pl02(m0"r"(re(x)),m0"r"(0),satz176b(0,refis(real,0)))):is"r"(mn"r"(ts"r"(re(x),m0"r"(1rl)),ts"r"(im(x),m0"r"(0))),m0"r"(re(x)))
8389 t2:=tris(real,pl"r"(ts"r"(re(x),m0"r"(0)),ts"r"(im(x),m0"r"(1rl))),pl"r"(0,m0"r"(im(x))),m0"r"(im(x)),ispl12"r"(ts"r"(re(x),m0"r"(0)),0,ts"r"(im(x),m0"r"(1rl)),m0"r"(im(x)),ts02(re(x),m0"r"(0),satz176b(0,refis(real,0))),tris(real,ts"r"(im(x),m0"r"(1rl)),m0"r"(ts"r"(im(x),1rl)),m0"r"(im(x)),satz197b(im(x),1rl),ism0"r"(ts"r"(im(x),1rl),im(x),satz195(im(x))))),pl01(0,m0"r"(im(x)),refis(real,0))):is"r"(pl"r"(ts"r"(re(x),m0"r"(0)),ts"r"(im(x),m0"r"(1rl))),m0"r"(im(x)))
8390 -3223
8391 satz223:=tr4is(cx,ts(x,m0(1c)),ts(x,pli(m0"r"(1rl),m0"r"(0))),pli(mn"r"(ts"r"(re(x),m0"r"(1rl)),ts"r"(im(x),m0"r"(0))),pl"r"(ts"r"(re(x),m0"r"(0)),ts"r"(im(x),m0"r"(1rl)))),pli(m0"r"(re(x)),m0"r"(im(x))),m0(x),ists2(m0(1c),pli(m0"r"(1rl),m0"r"(0)),x,m0isa(1rl,0)),tsis2a(x,m0"r"(1rl),m0"r"(0)),isrecx12(mn"r"(ts"r"(re(x),m0"r"(1rl)),ts"r"(im(x),m0"r"(0))),m0"r"(re(x)),pl"r"(ts"r"(re(x),m0"r"(0)),ts"r"(im(x),m0"r"(1rl))),m0"r"(im(x)),t1".3223",t2".3223"),satz214a(x)):is(ts(x,m0(1c)),m0(x))
8392 satz223a:=symis(cx,ts(x,m0(1c)),m0(x),satz223):is(m0(x),ts(x,m0(1c)))
8393 satz223b:=tris(cx,ts(m0(1c),x),ts(x,m0(1c)),m0(x),comts(m0(1c),x),satz223):is(ts(m0(1c),x),m0(x))
8394 satz223c:=symis(cx,ts(m0(1c),x),m0(x),satz223b):is(m0(x),ts(m0(1c),x))
8395 +3224
8396 y@rxry:=ts"r"(re(x),re(y)):real
8397 ixiy:=ts"r"(im(x),im(y)):real
8398 rxiy:=ts"r"(re(x),im(y)):real
8399 ixry:=ts"r"(im(x),re(y)):real
8400 t1:=tr4is(real,mn"r"(ts"r"(m0"r"(re(x)),re(y)),ts"r"(m0"r"(im(x)),im(y))),mn"r"(m0"r"(rxry),m0"r"(ixiy)),pl"r"(m0"r"(rxry),ixiy),mn"r"(ixiy,rxry),m0"r"(mn"r"(rxry,ixiy)),ismn12"r"(ts"r"(m0"r"(re(x)),re(y)),m0"r"(rxry),ts"r"(m0"r"(im(x)),im(y)),m0"r"(ixiy),satz197a(re(x),re(y)),satz197a(im(x),im(y))),ispl2"r"(m0"r"(m0"r"(ixiy)),ixiy,m0"r"(rxry),satz177(ixiy)),compl"r"(m0"r"(rxry),ixiy),satz181a(ixiy,rxry)):is"r"(mn"r"(ts"r"(m0"r"(re(x)),re(y)),ts"r"(m0"r"(im(x)),im(y))),m0"r"(mn"r"(rxry,ixiy)))
8401 t2:=tris(real,pl"r"(ts"r"(m0"r"(re(x)),im(y)),ts"r"(m0"r"(im(x)),re(y))),mn"r"(m0"r"(rxiy),ixry),m0"r"(pl"r"(rxiy,ixry)),ispl12"r"(ts"r"(m0"r"(re(x)),im(y)),m0"r"(rxiy),ts"r"(m0"r"(im(x)),re(y)),m0"r"(ixry),satz197a(re(x),im(y)),satz197a(im(x),re(y))),satz180a(rxiy,ixry)):is"r"(pl"r"(ts"r"(m0"r"(re(x)),im(y)),ts"r"(m0"r"(im(x)),re(y))),m0"r"(pl"r"(rxiy,ixry)))
8402 -3224
8403 y@satz224a:=tr4is(cx,ts(m0(x),y),ts(pli(m0"r"(re(x)),m0"r"(im(x))),y),pli(mn"r"(ts"r"(m0"r"(re(x)),re(y)),ts"r"(m0"r"(im(x)),im(y))),pl"r"(ts"r"(m0"r"(re(x)),im(y)),ts"r"(m0"r"(im(x)),re(y)))),pli(m0"r"(mn"r"(ts"r"(re(x),re(y)),ts"r"(im(x),im(y)))),m0"r"(pl"r"(ts"r"(re(x),im(y)),ts"r"(im(x),re(y))))),m0(ts(x,y)),ists1(m0(x),pli(m0"r"(re(x)),m0"r"(im(x))),y,satz214(x)),tsis1a(y,m0"r"(re(x)),m0"r"(im(x))),isrecx12(mn"r"(ts"r"(m0"r"(re(x)),re(y)),ts"r"(m0"r"(im(x)),im(y))),m0"r"(mn"r"(ts"r"(re(x),re(y)),ts"r"(im(x),im(y)))),pl"r"(ts"r"(m0"r"(re(x)),im(y)),ts"r"(m0"r"(im(x)),re(y))),m0"r"(pl"r"(ts"r"(re(x),im(y)),ts"r"(im(x),re(y)))),t1".3224",t2".3224"),m0isb(mn"r"(ts"r"(re(x),re(y)),ts"r"(im(x),im(y))),pl"r"(ts"r"(re(x),im(y)),ts"r"(im(x),re(y))))):is(ts(m0(x),y),m0(ts(x,y)))
8404 satz224b:=tr3is(cx,ts(x,m0(y)),ts(m0(y),x),m0(ts(y,x)),m0(ts(x,y)),comts(x,m0(y)),satz224a(y,x),ism0(ts(y,x),ts(x,y),comts(y,x))):is(ts(x,m0(y)),m0(ts(x,y)))
8405 satz224c:=tris2(cx,ts(m0(x),y),ts(x,m0(y)),m0(ts(x,y)),satz224a,satz224b):is(ts(m0(x),y),ts(x,m0(y)))
8406 satz224d:=symis(cx,ts(m0(x),y),ts(x,m0(y)),satz224c):is(ts(x,m0(y)),ts(m0(x),y))
8407 satz224e:=symis(cx,ts(m0(x),y),m0(ts(x,y)),satz224a):is(m0(ts(x,y)),ts(m0(x),y))
8408 satz224f:=symis(cx,ts(x,m0(y)),m0(ts(x,y)),satz224b):is(m0(ts(x,y)),ts(x,m0(y)))
8409 satz225:=tris(cx,ts(m0(x),m0(y)),ts(x,m0(m0(y))),ts(x,y),satz224c(x,m0(y)),ists2(m0(m0(y)),y,x,satz215(y))):is(ts(m0(x),m0(y)),ts(x,y))
8410 satz225a:=symis(cx,ts(m0(x),m0(y)),ts(x,y),satz225):is(ts(x,y),ts(m0(x),m0(y)))
8411 +3226
8412 z@rrr:=ts"r"(ts"r"(re(x),re(y)),re(z)):real
8413 iir:=ts"r"(ts"r"(im(x),im(y)),re(z)):real
8414 rii:=ts"r"(ts"r"(re(x),im(y)),im(z)):real
8415 iri:=ts"r"(ts"r"(im(x),re(y)),im(z)):real
8416 rri:=ts"r"(ts"r"(re(x),re(y)),im(z)):real
8417 iii:=ts"r"(ts"r"(im(x),im(y)),im(z)):real
8418 rir:=ts"r"(ts"r"(re(x),im(y)),re(z)):real
8419 irr:=ts"r"(ts"r"(im(x),re(y)),re(z)):real
8420 t1:=tr3is(real,mn"r"(ts"r"(rets(x,y),re(z)),ts"r"(imts(x,y),im(z))),mn"r"(mn"r"(rrr,iir),pl"r"(rii,iri)),pl"r"(rrr,pl"r"(m0"r"(iir),m0"r"(pl"r"(rii,iri)))),mn"r"(rrr,pl"r"(iir,pl"r"(rii,iri))),ismn12"r"(ts"r"(rets(x,y),re(z)),mn"r"(rrr,iir),ts"r"(imts(x,y),im(z)),pl"r"(rii,iri),disttm1(ts"r"(re(x),re(y)),ts"r"(im(x),im(y)),re(z)),disttp1(ts"r"(re(x),im(y)),ts"r"(im(x),re(y)),im(z))),asspl1"r"(rrr,m0"r"(iir),m0"r"(pl"r"(rii,iri))),ispl2"r"(pl"r"(m0"r"(iir),m0"r"(pl"r"(rii,iri))),m0"r"(pl"r"(iir,pl"r"(rii,iri))),rrr,satz180a(iir,pl"r"(rii,iri)))):is"r"(mn"r"(ts"r"(rets(x,y),re(z)),ts"r"(imts(x,y),im(z))),mn"r"(rrr,pl"r"(iir,pl"r"(rii,iri))))
8421 t2:=tr3is(real,pl"r"(ts"r"(rets(x,y),im(z)),ts"r"(imts(x,y),re(z))),pl"r"(mn"r"(rri,iii),pl"r"(rir,irr)),pl"r"(pl"r"(rir,irr),mn"r"(rri,iii)),mn"r"(pl"r"(pl"r"(rir,irr),rri),iii),ispl12"r"(ts"r"(rets(x,y),im(z)),mn"r"(rri,iii),ts"r"(imts(x,y),re(z)),pl"r"(rir,irr),disttm1(ts"r"(re(x),re(y)),ts"r"(im(x),im(y)),im(z)),disttp1(ts"r"(re(x),im(y)),ts"r"(im(x),re(y)),re(z))),compl"r"(mn"r"(rri,iii),pl"r"(rir,irr)),asspl2"r"(pl"r"(rir,irr),rri,m0"r"(iii))):is"r"(pl"r"(ts"r"(rets(x,y),im(z)),ts"r"(imts(x,y),re(z))),mn"r"(pl"r"(pl"r"(rir,irr),rri),iii))
8422 t3:=tris(cx,ts(ts(x,y),z),pli(mn"r"(ts"r"(rets(x,y),re(z)),ts"r"(imts(x,y),im(z))),pl"r"(ts"r"(rets(x,y),im(z)),ts"r"(imts(x,y),re(z)))),pli(mn"r"(rrr,pl"r"(iir,pl"r"(rii,iri))),mn"r"(pl"r"(pl"r"(rir,irr),rri),iii)),tsis1a(z,rets(x,y),imts(x,y)),isrecx12(mn"r"(ts"r"(rets(x,y),re(z)),ts"r"(imts(x,y),im(z))),mn"r"(rrr,pl"r"(iir,pl"r"(rii,iri))),pl"r"(ts"r"(rets(x,y),im(z)),ts"r"(imts(x,y),re(z))),mn"r"(pl"r"(pl"r"(rir,irr),rri),iii),t1,t2)):is(ts(ts(x,y),z),pli(mn"r"(rrr,pl"r"(iir,pl"r"(rii,iri))),mn"r"(pl"r"(pl"r"(rir,irr),rri),iii)))
8423 c@t4:=tris(real,ts"r"(ts"r"(a,b),c),ts"r"(a,ts"r"(b,c)),ts"r"(ts"r"(b,c),a),assts1(a,b,c),comts"r"(a,ts"r"(b,c))):is"r"(ts"r"(ts"r"(a,b),c),ts"r"(ts"r"(b,c),a))
8424 t5:=tris(real,pl"r"(pl"r"(a,b),c),pl"r"(c,pl"r"(a,b)),pl"r"(pl"r"(c,a),b),compl"r"(pl"r"(a,b),c),asspl2"r"(c,a,b)):is"r"(pl"r"(pl"r"(a,b),c),pl"r"(pl"r"(c,a),b))
8425 t6:=tris(real,pl"r"(a,pl"r"(b,c)),pl"r"(pl"r"(b,c),a),pl"r"(b,pl"r"(c,a)),compl"r"(a,pl"r"(b,c)),asspl1"r"(b,c,a)):is"r"(pl"r"(a,pl"r"(b,c)),pl"r"(b,pl"r"(c,a)))
8426 z@rrr1:=rrr(y,z,x):real
8427 iir1:=iir(y,z,x):real
8428 rii1:=rii(y,z,x):real
8429 iri1:=iri(y,z,x):real
8430 rri1:=rri(y,z,x):real
8431 iii1:=iii(y,z,x):real
8432 rir1:=rir(y,z,x):real
8433 irr1:=irr(y,z,x):real
8434 t7:=tris(real,pl"r"(iir,pl"r"(rii,iri)),pl"r"(rii,pl"r"(iri,iir)),pl"r"(iir1,pl"r"(rii1,iri1)),t6(iir,rii,iri),ispl12"r"(rii,iir1,pl"r"(iri,iir),pl"r"(rii1,iri1),t4(re(x),im(y),im(z)),ispl12"r"(iri,rii1,iir,iri1,t4(im(x),re(y),im(z)),t4(im(x),im(y),re(z))))):is"r"(pl"r"(iir,pl"r"(rii,iri)),pl"r"(iir1,pl"r"(rii1,iri1)))
8435 t8:=tris(real,pl"r"(pl"r"(rir,irr),rri),pl"r"(pl"r"(rri,rir),irr),pl"r"(pl"r"(rir1,irr1),rri1),t5(rir,irr,rri),ispl12"r"(pl"r"(rri,rir),pl"r"(rir1,irr1),irr,rri1,ispl12"r"(rri,rir1,rir,irr1,t4(re(x),re(y),im(z)),t4(re(x),im(y),re(z))),t4(im(x),re(y),re(z)))):is"r"(pl"r"(pl"r"(rir,irr),rri),pl"r"(pl"r"(rir1,irr1),rri1))
8436 t9:=ismn12"r"(rrr,rrr1,pl"r"(iir,pl"r"(rii,iri)),pl"r"(iir1,pl"r"(rii1,iri1)),t4(re(x),re(y),re(z)),t7):is"r"(mn"r"(rrr,pl"r"(iir,pl"r"(rii,iri))),mn"r"(rrr1,pl"r"(iir1,pl"r"(rii1,iri1))))
8437 t10:=ismn12"r"(pl"r"(pl"r"(rir,irr),rri),pl"r"(pl"r"(rir1,irr1),rri1),iii,iii1,t8,t4(im(x),im(y),im(z))):is"r"(mn"r"(pl"r"(pl"r"(rir,irr),rri),iii),mn"r"(pl"r"(pl"r"(rir1,irr1),rri1),iii1))
8438 t11:=tris(cx,ts(ts(x,y),z),pli(mn"r"(rrr,pl"r"(iir,pl"r"(rii,iri))),mn"r"(pl"r"(pl"r"(rir,irr),rri),iii)),pli(mn"r"(rrr1,pl"r"(iir1,pl"r"(rii1,iri1))),mn"r"(pl"r"(pl"r"(rir1,irr1),rri1),iii1)),t3,isrecx12(mn"r"(rrr,pl"r"(iir,pl"r"(rii,iri))),mn"r"(rrr1,pl"r"(iir1,pl"r"(rii1,iri1))),mn"r"(pl"r"(pl"r"(rir,irr),rri),iii),mn"r"(pl"r"(pl"r"(rir1,irr1),rri1),iii1),t9,t10)):is(ts(ts(x,y),z),pli(mn"r"(rrr1,pl"r"(iir1,pl"r"(rii1,iri1))),mn"r"(pl"r"(pl"r"(rir1,irr1),rri1),iii1)))
8439 t12:=tris(cx,ts(x,ts(y,z)),ts(ts(y,z),x),pli(mn"r"(rrr1,pl"r"(iir1,pl"r"(rii1,iri1))),mn"r"(pl"r"(pl"r"(rir1,irr1),rri1),iii1)),comts(x,ts(y,z)),t3(y,z,x)):is(ts(x,ts(y,z)),pli(mn"r"(rrr1,pl"r"(iir1,pl"r"(rii1,iri1))),mn"r"(pl"r"(pl"r"(rir1,irr1),rri1),iii1)))
8440 t13:=tris2(cx,ts(ts(x,y),z),ts(x,ts(y,z)),pli(mn"r"(rrr1,pl"r"(iir1,pl"r"(rii1,iri1))),mn"r"(pl"r"(pl"r"(rir1,irr1),rri1),iii1)),t11,t12):is(ts(ts(x,y),z),ts(x,ts(y,z)))
8441 -3226
8442 z@satz226:=t13".3226":is(ts(ts(x,y),z),ts(x,ts(y,z)))
8443 assts1:=satz226:is(ts(ts(x,y),z),ts(x,ts(y,z)))
8444 assts2:=symis(cx,ts(ts(x,y),z),ts(x,ts(y,z)),assts1):is(ts(x,ts(y,z)),ts(ts(x,y),z))
8445 +3227
8446 c@t1:=tr3is(real,pl"r"(pl"r"(a,b),c),pl"r"(a,pl"r"(b,c)),pl"r"(a,pl"r"(c,b)),pl"r"(pl"r"(a,c),b),asspl1"r"(a,b,c),ispl2"r"(pl"r"(b,c),pl"r"(c,b),a,compl"r"(b,c)),asspl2"r"(a,c,b)):is"r"(pl"r"(pl"r"(a,b),c),pl"r"(pl"r"(a,c),b))
8447 d@t2:=tr3is(real,pl"r"(pl"r"(a,b),pl"r"(c,d)),pl"r"(pl"r"(pl"r"(a,b),c),d),pl"r"(pl"r"(pl"r"(a,c),b),d),pl"r"(pl"r"(a,c),pl"r"(b,d)),asspl2"r"(pl"r"(a,b),c,d),ispl1"r"(pl"r"(pl"r"(a,b),c),pl"r"(pl"r"(a,c),b),d,t1),asspl1"r"(pl"r"(a,c),b,d)):is"r"(pl"r"(pl"r"(a,b),pl"r"(c,d)),pl"r"(pl"r"(a,c),pl"r"(b,d)))
8448 t3:=tris(real,mn"r"(pl"r"(a,b),pl"r"(c,d)),pl"r"(pl"r"(a,b),pl"r"(m0"r"(c),m0"r"(d))),pl"r"(mn"r"(a,c),mn"r"(b,d)),ispl2"r"(m0"r"(pl"r"(c,d)),pl"r"(m0"r"(c),m0"r"(d)),pl"r"(a,b),satz180(c,d)),t2(a,b,m0"r"(c),m0"r"(d))):is"r"(mn"r"(pl"r"(a,b),pl"r"(c,d)),pl"r"(mn"r"(a,c),mn"r"(b,d)))
8449 z@t4:=tris(real,mn"r"(ts"r"(re(x),pl"r"(re(y),re(z))),ts"r"(im(x),pl"r"(im(y),im(z)))),mn"r"(pl"r"(ts"r"(re(x),re(y)),ts"r"(re(x),re(z))),pl"r"(ts"r"(im(x),im(y)),ts"r"(im(x),im(z)))),pl"r"(rets(x,y),rets(x,z)),ismn12"r"(ts"r"(re(x),pl"r"(re(y),re(z))),pl"r"(ts"r"(re(x),re(y)),ts"r"(re(x),re(z))),ts"r"(im(x),pl"r"(im(y),im(z))),pl"r"(ts"r"(im(x),im(y)),ts"r"(im(x),im(z))),disttp2(re(x),re(y),re(z)),disttp2(im(x),im(y),im(z))),t3(ts"r"(re(x),re(y)),ts"r"(re(x),re(z)),ts"r"(im(x),im(y)),ts"r"(im(x),im(z)))):is"r"(mn"r"(ts"r"(re(x),pl"r"(re(y),re(z))),ts"r"(im(x),pl"r"(im(y),im(z)))),pl"r"(rets(x,y),rets(x,z)))
8450 t5:=tris(real,pl"r"(ts"r"(re(x),pl"r"(im(y),im(z))),ts"r"(im(x),pl"r"(re(y),re(z)))),pl"r"(pl"r"(ts"r"(re(x),im(y)),ts"r"(re(x),im(z))),pl"r"(ts"r"(im(x),re(y)),ts"r"(im(x),re(z)))),pl"r"(imts(x,y),imts(x,z)),ispl12"r"(ts"r"(re(x),pl"r"(im(y),im(z))),pl"r"(ts"r"(re(x),im(y)),ts"r"(re(x),im(z))),ts"r"(im(x),pl"r"(re(y),re(z))),pl"r"(ts"r"(im(x),re(y)),ts"r"(im(x),re(z))),disttp2(re(x),im(y),im(z)),disttp2(im(x),re(y),re(z))),t2(ts"r"(re(x),im(y)),ts"r"(re(x),im(z)),ts"r"(im(x),re(y)),ts"r"(im(x),re(z)))):is"r"(pl"r"(ts"r"(re(x),pl"r"(im(y),im(z))),ts"r"(im(x),pl"r"(re(y),re(z)))),pl"r"(imts(x,y),imts(x,z)))
8451 t6:=tr3is(cx,ts(x,pl(y,z)),pli(mn"r"(ts"r"(re(x),pl"r"(re(y),re(z))),ts"r"(im(x),pl"r"(im(y),im(z)))),pl"r"(ts"r"(re(x),pl"r"(im(y),im(z))),ts"r"(im(x),pl"r"(re(y),re(z))))),pli(pl"r"(rets(x,y),rets(x,z)),pl"r"(imts(x,y),imts(x,z))),pl(ts(x,y),ts(x,z)),tsis2a(x,pl"r"(re(y),re(z)),pl"r"(im(y),im(z))),isrecx12(mn"r"(ts"r"(re(x),pl"r"(re(y),re(z))),ts"r"(im(x),pl"r"(im(y),im(z)))),pl"r"(rets(x,y),rets(x,z)),pl"r"(ts"r"(re(x),pl"r"(im(y),im(z))),ts"r"(im(x),pl"r"(re(y),re(z)))),pl"r"(imts(x,y),imts(x,z)),t4,t5),plis12b(rets(x,y),imts(x,y),rets(x,z),imts(x,z))):is(ts(x,pl(y,z)),pl(ts(x,y),ts(x,z)))
8452 -3227
8453 satz227:=t6".3227":is(ts(x,pl(y,z)),pl(ts(x,y),ts(x,z)))
8454 disttp1:=tr3is(cx,ts(pl(x,y),z),ts(z,pl(x,y)),pl(ts(z,x),ts(z,y)),pl(ts(x,z),ts(y,z)),comts(pl(x,y),z),satz227(z,x,y),ispl12(ts(z,x),ts(x,z),ts(z,y),ts(y,z),comts(z,x),comts(z,y))):is(ts(pl(x,y),z),pl(ts(x,z),ts(y,z)))
8455 disttp2:=satz227:is(ts(x,pl(y,z)),pl(ts(x,y),ts(x,z)))
8456 distpt1:=symis(cx,ts(pl(x,y),z),pl(ts(x,z),ts(y,z)),disttp1):is(pl(ts(x,z),ts(y,z)),ts(pl(x,y),z))
8457 distpt2:=symis(cx,ts(x,pl(y,z)),pl(ts(x,y),ts(x,z)),disttp2):is(pl(ts(x,y),ts(x,z)),ts(x,pl(y,z)))
8458 satz228:=tr4is(cx,ts(x,mn(y,z)),ts(x,pl(y,m0(z))),pl(ts(x,y),ts(x,m0(z))),pl(ts(x,y),m0(ts(x,z))),mn(ts(x,y),ts(x,z)),ists2(mn(y,z),pl(y,m0(z)),x,satz218(y,z)),disttp2(x,y,m0(z)),ispl2(ts(x,m0(z)),m0(ts(x,z)),ts(x,y),satz224b(x,z)),satz218a(ts(x,y),ts(x,z))):is(ts(x,mn(y,z)),mn(ts(x,y),ts(x,z)))
8459 disttm1:=tr3is(cx,ts(mn(x,y),z),ts(z,mn(x,y)),mn(ts(z,x),ts(z,y)),mn(ts(x,z),ts(y,z)),comts(mn(x,y),z),satz228(z,x,y),ismn12(ts(z,x),ts(x,z),ts(z,y),ts(y,z),comts(z,x),comts(z,y))):is(ts(mn(x,y),z),mn(ts(x,z),ts(y,z)))
8460 disttm2:=satz228:is(ts(x,mn(y,z)),mn(ts(x,y),ts(x,z)))
8461 distmt1:=symis(cx,ts(mn(x,y),z),mn(ts(x,z),ts(y,z)),disttm1):is(mn(ts(x,z),ts(y,z)),ts(mn(x,y),z))
8462 distmt2:=symis(cx,ts(x,mn(y,z)),mn(ts(x,y),ts(x,z)),disttm2):is(mn(ts(x,y),ts(x,z)),ts(x,mn(y,z)))
8463 y@[n:nis(y,0c)][u1:complex][u2:complex][i:is(ts(y,u1),x)][j:is(ts(y,u2),x)]
8464 +3229
8465 t1:=tris2(cx,ts(y,u1),ts(y,u2),x,i,j):is(ts(y,u1),ts(y,u2))
8466 t2:=tris(cx,ts(y,mn(u1,u2)),mn(ts(y,u1),ts(y,u2)),0c,disttm2(y,u1,u2),satz213b(ts(y,u1),ts(y,u2),t1)):is(ts(y,mn(u1,u2)),0c)
8467 t3:=ore2(is(y,0c),is(mn(u1,u2),0c),satz221c(y,mn(u1,u2),t2),n):is(mn(u1,u2),0c)
8468 -3229
8469 satz229b:=satz213a(u1,u2,t3".3229"):is(u1,u2)
8470 +*3229
8471 n@t4:=pnot0(mod2(y),lemma4(y,n)):nis"r"(mod2(y),0)
8472 u:=ts(pli(ov(re(y),mod2(y),t4),m0"r"(ov(im(y),mod2(y),t4))),x):complex
8473 [v:real]
8474 dd:=ov(v,mod2(y),t4):real
8475 n@t5:=tr3is(real,m0"r"(ts"r"(im(y),m0"r"(dd(im(y))))),m0"r"(m0"r"(ts"r"(im(y),dd(im(y))))),ts"r"(im(y),dd(im(y))),dd(ts"r"(im(y),im(y))),ism0"r"(ts"r"(im(y),m0"r"(dd(im(y)))),m0"r"(ts"r"(im(y),dd(im(y)))),satz197b(im(y),dd(im(y)))),satz177(ts"r"(im(y),dd(im(y)))),lemma6(im(y),im(y),mod2(y),t4)):is"r"(m0"r"(ts"r"(im(y),m0"r"(dd(im(y))))),dd(ts"r"(im(y),im(y))))
8476 t6:=tr3is(real,mn"r"(ts"r"(re(y),dd(re(y))),ts"r"(im(y),m0"r"(dd(im(y))))),pl"r"(dd(ts"r"(re(y),re(y))),dd(ts"r"(im(y),im(y)))),dd(mod2(y)),1rl,ispl12"r"(ts"r"(re(y),dd(re(y))),dd(ts"r"(re(y),re(y))),m0"r"(ts"r"(im(y),m0"r"(dd(im(y))))),dd(ts"r"(im(y),im(y))),lemma6(re(y),re(y),mod2(y),t4),t5),lemma7(ts"r"(re(y),re(y)),ts"r"(im(y),im(y)),mod2(y),t4),lemma8(mod2(y),t4)):is"r"(mn"r"(ts"r"(re(y),dd(re(y))),ts"r"(im(y),m0"r"(dd(im(y))))),1rl)
8477 t7:=tris(real,ts"r"(re(y),m0"r"(dd(im(y)))),ts"r"(m0"r"(re(y)),dd(im(y))),dd(ts"r"(m0"r"(re(y)),im(y))),satz197d(re(y),dd(im(y))),lemma6(m0"r"(re(y)),im(y),mod2(y),t4)):is"r"(ts"r"(re(y),m0"r"(dd(im(y)))),dd(ts"r"(m0"r"(re(y)),im(y))))
8478 t8:=tris(real,pl"r"(ts"r"(re(y),m0"r"(dd(im(y)))),ts"r"(im(y),dd(re(y)))),pl"r"(dd(ts"r"(m0"r"(re(y)),im(y))),dd(ts"r"(im(y),re(y)))),dd(pl"r"(ts"r"(m0"r"(re(y)),im(y)),ts"r"(im(y),re(y)))),ispl12"r"(ts"r"(re(y),m0"r"(dd(im(y)))),dd(ts"r"(m0"r"(re(y)),im(y))),ts"r"(im(y),dd(re(y))),dd(ts"r"(im(y),re(y))),t7,lemma6(im(y),re(y),mod2(y),t4)),lemma7(ts"r"(m0"r"(re(y)),im(y)),ts"r"(im(y),re(y)),mod2(y),t4)):is"r"(pl"r"(ts"r"(re(y),m0"r"(dd(im(y)))),ts"r"(im(y),dd(re(y)))),dd(pl"r"(ts"r"(m0"r"(re(y)),im(y)),ts"r"(im(y),re(y)))))
8479 t9:=tr3is(real,pl"r"(ts"r"(m0"r"(re(y)),im(y)),ts"r"(im(y),re(y))),pl"r"(m0"r"(ts"r"(re(y),im(y))),ts"r"(im(y),re(y))),mn"r"(ts"r"(im(y),re(y)),ts"r"(re(y),im(y))),0,ispl1"r"(ts"r"(m0"r"(re(y)),im(y)),m0"r"(ts"r"(re(y),im(y))),ts"r"(im(y),re(y)),satz197a(re(y),im(y))),compl"r"(m0"r"(ts"r"(re(y),im(y))),ts"r"(im(y),re(y))),satz182e(ts"r"(im(y),re(y)),ts"r"(re(y),im(y)),comts"r"(im(y),re(y)))):is"r"(pl"r"(ts"r"(m0"r"(re(y)),im(y)),ts"r"(im(y),re(y))),0)
8480 t10:=tr3is(real,pl"r"(ts"r"(re(y),m0"r"(dd(im(y)))),ts"r"(im(y),dd(re(y)))),dd(pl"r"(ts"r"(m0"r"(re(y)),im(y)),ts"r"(im(y),re(y)))),dd(0),0,t8,isf(real,real,[t:real]dd(t),pl"r"(ts"r"(m0"r"(re(y)),im(y)),ts"r"(im(y),re(y))),0,t9),lemma9(mod2(y),t4)):is"r"(pl"r"(ts"r"(re(y),m0"r"(dd(im(y)))),ts"r"(im(y),dd(re(y)))),0)
8481 t11:=tris(cx,ts(y,pli(dd(re(y)),m0"r"(dd(im(y))))),pli(mn"r"(ts"r"(re(y),dd(re(y))),ts"r"(im(y),m0"r"(dd(im(y))))),pl"r"(ts"r"(re(y),m0"r"(dd(im(y)))),ts"r"(im(y),dd(re(y))))),1c,tsis2a(y,dd(re(y)),m0"r"(dd(im(y)))),isrecx12(mn"r"(ts"r"(re(y),dd(re(y))),ts"r"(im(y),m0"r"(dd(im(y))))),1rl,pl"r"(ts"r"(re(y),m0"r"(dd(im(y)))),ts"r"(im(y),dd(re(y)))),0,t6,t10)):is(ts(y,pli(dd(re(y)),m0"r"(dd(im(y))))),1c)
8482 t12:=tr3is(cx,ts(y,u),ts(ts(y,pli(dd(re(y)),m0"r"(dd(im(y))))),x),ts(1c,x),x,assts2(y,pli(dd(re(y)),m0"r"(dd(im(y)))),x),ists1(ts(y,pli(dd(re(y)),m0"r"(dd(im(y))))),1c,x,t11),satz222b(x)):is(ts(y,u),x)
8483 -3229
8484 n@satz229a:=somei(cx,[t:cx]is(ts(y,t),x),u".3229",t12".3229"):some([t:cx]is(ts(y,t),x))
8485 satz229:=onei(cx,[t:cx]is(ts(y,t),x),[t:cx][u:cx][v:is(ts(y,t),x)][w:is(ts(y,u),x)]satz229b(t,u,v,w),satz229a):one([t:cx]is(ts(y,t),x))
8486 ov:=ind(cx,[t:cx]is(ts(y,t),x),satz229):complex
8487 satz229c:=oneax(cx,[t:cx]is(ts(y,t),x),satz229):is(ts(y,ov(x,y,n)),x)
8488 satz229d:=symis(cx,ts(y,ov(x,y,n)),x,satz229c):is(x,ts(y,ov(x,y,n)))
8489 satz229e:=tris(cx,ts(ov(x,y,n),y),ts(y,ov(x,y,n)),x,comts(ov(x,y,n),y),satz229c):is(ts(ov(x,y,n),y),x)
8490 satz229f:=symis(cx,ts(ov(x,y,n),y),x,satz229e):is(x,ts(ov(x,y,n),y))
8491 y@[u:complex][n:nis(y,0c)][i:is(ts(y,u),x)]
8492 satz229g:=satz229b(n,u,ov(x,y,n),i,satz229c(n)):is(u,ov(x,y,n))
8493 satz229h:=symis(cx,u,ov(x,y,n),satz229g):is(ov(x,y,n),u)
8494 n@[i:is(ts(u,y),x)]
8495 satz229j:=satz229g(tris(cx,ts(y,u),ts(u,y),x,comts(y,u),i)):is(u,ov(x,y,n))
8496 satz229k:=symis(cx,u,ov(x,y,n),satz229j):is(ov(x,y,n),u)
8497 z@[i:is(x,y)][n:nis(z,0c)]
8498 isov1:=isf(cx,cx,[t:cx]ov(t,z,n),x,y,i):is(ov(x,z,n),ov(y,z,n))
8499 i@[n:nis(x,0c)][o:nis(y,0c)]
8500 isov2:=satz229h(z,x,ov(z,y,o),n,tris(cx,ts(x,ov(z,y,o)),ts(y,ov(z,y,o)),z,ists1(x,y,ov(z,y,o),i),satz229c(z,y,o))):is(ov(z,x,n),ov(z,y,o))
8501 z@[u:complex][i:is(x,y)][j:is(z,u)][n:nis(z,0c)][o:nis(u,0c)]
8502 isov12:=tris(cx,ov(x,z,n),ov(y,z,n),ov(y,u,o),isov1(x,y,z,i,n),isov2(z,u,y,j,n,o)):is(ov(x,z,n),ov(y,u,o))
8503 y@satz230:=tris(cx,pl(mn(x,y),y),pl(y,mn(x,y)),x,compl(mn(x,y),y),satz212h(x,y)):is(pl(mn(x,y),y),x)
8504 satz231:=satz212e(pl(x,y),y,x,compl(y,x)):is(mn(pl(x,y),y),x)
8505 satz232:=satz212e(x,mn(x,y),y,satz230):is(mn(x,mn(x,y)),y)
8506 +4233
8507 z@t1:=tr3is(cx,pl(pl(y,z),mn(mn(x,y),z)),pl(mn(mn(x,y),z),pl(y,z)),pl(mn(mn(x,y),z),pl(z,y)),pl(pl(mn(mn(x,y),z),z),y),compl(pl(y,z),mn(mn(x,y),z)),ispl2(pl(y,z),pl(z,y),mn(mn(x,y),z),compl(y,z)),asspl2(mn(mn(x,y),z),z,y)):is(pl(pl(y,z),mn(mn(x,y),z)),pl(pl(mn(mn(x,y),z),z),y))
8508 t2:=tr3is(cx,pl(pl(y,z),mn(mn(x,y),z)),pl(pl(mn(mn(x,y),z),z),y),pl(mn(x,y),y),x,t1,ispl1(pl(mn(mn(x,y),z),z),mn(x,y),y,satz230(mn(x,y),z)),satz230(x,y)):is(pl(pl(y,z),mn(mn(x,y),z)),x)
8509 -4233
8510 z@satz233:=satz212d(x,pl(y,z),mn(mn(x,y),z),t2".4233"):is(mn(mn(x,y),z),mn(x,pl(y,z)))
8511 satz234:=satz212g(pl(x,y),z,pl(x,mn(y,z)),tris(cx,pl(pl(x,mn(y,z)),z),pl(x,pl(mn(y,z),z)),pl(x,y),asspl1(x,mn(y,z),z),ispl2(pl(mn(y,z),z),y,x,satz230(y,z)))):is(mn(pl(x,y),z),pl(x,mn(y,z)))
8512 satz234a:=symis(cx,mn(pl(x,y),z),pl(x,mn(y,z)),satz234):is(pl(x,mn(y,z)),mn(pl(x,y),z))
8513 satz234b:=tr3is(cx,mn(pl(x,y),z),mn(pl(y,x),z),pl(y,mn(x,z)),pl(mn(x,z),y),ismn1(pl(x,y),pl(y,x),z,compl(x,y)),satz234(y,x,z),compl(y,mn(x,z))):is(mn(pl(x,y),z),pl(mn(x,z),y))
8514 satz234c:=symis(cx,mn(pl(x,y),z),pl(mn(x,z),y),satz234b):is(pl(mn(x,z),y),mn(pl(x,y),z))
8515 satz235:=satz212f(x,mn(y,z),pl(mn(x,y),z),tr3is(cx,pl(pl(mn(x,y),z),mn(y,z)),pl(mn(x,y),pl(z,mn(y,z))),pl(mn(x,y),y),x,asspl1(mn(x,y),z,mn(y,z)),ispl2(pl(z,mn(y,z)),y,mn(x,y),satz212h(y,z)),satz230(x,y))):is(pl(mn(x,y),z),mn(x,mn(y,z)))
8516 satz235a:=symis(cx,pl(mn(x,y),z),mn(x,mn(y,z)),satz235):is(mn(x,mn(y,z)),pl(mn(x,y),z))
8517 satz235b:=tris(cx,mn(x,mn(y,z)),pl(mn(x,y),z),mn(pl(x,z),y),satz235a,satz234c(x,z,y)):is(mn(x,mn(y,z)),mn(pl(x,z),y))
8518 satz235c:=tris(cx,mn(x,mn(y,z)),mn(pl(x,z),y),mn(pl(z,x),y),satz235b,ismn1(pl(x,z),pl(z,x),y,compl(x,z))):is(mn(x,mn(y,z)),mn(pl(z,x),y))
8519 satz236:=satz212g(pl(x,z),pl(y,z),mn(x,y),tris(cx,pl(mn(x,y),pl(y,z)),pl(pl(mn(x,y),y),z),pl(x,z),asspl2(mn(x,y),y,z),ispl1(pl(mn(x,y),y),x,z,satz230(x,y)))):is(mn(pl(x,z),pl(y,z)),mn(x,y))
8520 satz236a:=tris(cx,mn(pl(z,x),pl(z,y)),mn(pl(x,z),pl(y,z)),mn(x,y),ismn12(pl(z,x),pl(x,z),pl(z,y),pl(y,z),compl(z,x),compl(z,y)),satz236):is(mn(pl(z,x),pl(z,y)),mn(x,y))
8521 [u:complex]
8522 +4237
8523 t1:=tr3is(cx,pl(mn(z,u),pl(u,y)),pl(pl(mn(z,u),u),y),pl(z,y),pl(y,z),asspl2(mn(z,u),u,y),ispl1(pl(mn(z,u),u),z,y,satz230(z,u)),compl(z,y)):is(pl(mn(z,u),pl(u,y)),pl(y,z))
8524 t2:=tr3is(cx,pl(pl(mn(x,y),mn(z,u)),pl(y,u)),pl(pl(mn(x,y),mn(z,u)),pl(u,y)),pl(mn(x,y),pl(mn(z,u),pl(u,y))),pl(mn(x,y),pl(y,z)),ispl2(pl(y,u),pl(u,y),pl(mn(x,y),mn(z,u)),compl(y,u)),asspl1(mn(x,y),mn(z,u),pl(u,y)),ispl2(pl(mn(z,u),pl(u,y)),pl(y,z),mn(x,y),t1)):is(pl(pl(mn(x,y),mn(z,u)),pl(y,u)),pl(mn(x,y),pl(y,z)))
8525 t3:=tr3is(cx,pl(pl(mn(x,y),mn(z,u)),pl(y,u)),pl(mn(x,y),pl(y,z)),pl(pl(mn(x,y),y),z),pl(x,z),t2,asspl2(mn(x,y),y,z),ispl1(pl(mn(x,y),y),x,z,satz230(x,y))):is(pl(pl(mn(x,y),mn(z,u)),pl(y,u)),pl(x,z))
8526 -4237
8527 satz237:=satz212f(pl(x,z),pl(y,u),pl(mn(x,y),mn(z,u)),t3".4237"):is(pl(mn(x,y),mn(z,u)),mn(pl(x,z),pl(y,u)))
8528 +4238
8529 t1:=tris(cx,pl(pl(x,u),z),pl(x,pl(u,z)),pl(x,pl(z,u)),asspl1(x,u,z),ispl2(pl(u,z),pl(z,u),x,compl(u,z))):is(pl(pl(x,u),z),pl(x,pl(z,u)))
8530 t2:=tr3is(cx,pl(mn(pl(x,u),pl(y,z)),mn(z,u)),mn(pl(pl(x,u),z),pl(pl(y,z),u)),mn(pl(x,pl(z,u)),pl(y,pl(z,u))),mn(x,y),satz237(pl(x,u),pl(y,z),z,u),ismn12(pl(pl(x,u),z),pl(x,pl(z,u)),pl(pl(y,z),u),pl(y,pl(z,u)),t1,asspl1(y,z,u)),satz236(x,y,pl(z,u))):is(pl(mn(pl(x,u),pl(y,z)),mn(z,u)),mn(x,y))
8531 -4238
8532 satz238:=satz212g(mn(x,y),mn(z,u),mn(pl(x,u),pl(y,z)),t2".4238"):is(mn(mn(x,y),mn(z,u)),mn(pl(x,u),pl(y,z)))
8533 [i:is(mn(x,y),mn(z,u))]
8534 +4239
8535 t1:=tris1(cx,mn(pl(x,u),pl(y,z)),0c,mn(mn(x,y),mn(z,u)),satz238,satz213b(mn(x,y),mn(z,u),i)):is(mn(pl(x,u),pl(y,z)),0c)
8536 -4239
8537 satz239a:=satz213a(pl(x,u),pl(y,z),t1".4239"):is(pl(x,u),pl(y,z))
8538 u@[i:is(pl(x,u),pl(y,z))]
8539 +*4239
8540 i@t2:=tris(cx,mn(mn(x,y),mn(z,u)),mn(pl(x,u),pl(y,z)),0c,satz238,satz213b(pl(x,u),pl(y,z),i)):is(mn(mn(x,y),mn(z,u)),0c)
8541 -4239
8542 i@satz239b:=satz213a(mn(x,y),mn(z,u),t2".4239"):is(mn(x,y),mn(z,u))
8543 y@[n:nis(y,0c)]
8544 satz240:=tris(cx,ts(ov(x,y,n),y),ts(y,ov(x,y,n)),x,comts(ov(x,y,n),y),satz229c(x,y,n)):is(ts(ov(x,y,n),y),x)
8545 satz241:=satz229h(ts(x,y),y,x,n,comts(y,x)):is(ov(ts(x,y),y,n),x)
8546 y@[n:nis(x,0c)][o:nis(y,0c)]
8547 lemma6:=th3"l.imp"(is(ov(x,y,o),0c),is(x,0c),n,[t:is(ov(x,y,o),0c)]tris1(cx,x,0c,ts(y,ov(x,y,o)),satz229c(x,y,o),satz221b(y,ov(x,y,o),t))):nis(ov(x,y,o),0c)
8548 satz242:=satz229h(x,ov(x,y,o),y,lemma6,satz240(o)):is(ov(x,ov(x,y,o),lemma6),y)
8549 z@[n:nis(y,0c)][o:nis(z,0c)]
8550 +5243
8551 t1:=tr3is(cx,ts(ts(y,z),ov(ov(x,y,n),z,o)),ts(ov(ov(x,y,n),z,o),ts(y,z)),ts(ov(ov(x,y,n),z,o),ts(z,y)),ts(ts(ov(ov(x,y,n),z,o),z),y),comts(ts(y,z),ov(ov(x,y,n),z,o)),ists2(ts(y,z),ts(z,y),ov(ov(x,y,n),z,o),comts(y,z)),assts2(ov(ov(x,y,n),z,o),z,y)):is(ts(ts(y,z),ov(ov(x,y,n),z,o)),ts(ts(ov(ov(x,y,n),z,o),z),y))
8552 t2:=tr3is(cx,ts(ts(y,z),ov(ov(x,y,n),z,o)),ts(ts(ov(ov(x,y,n),z,o),z),y),ts(ov(x,y,n),y),x,t1,ists1(ts(ov(ov(x,y,n),z,o),z),ov(x,y,n),y,satz240(ov(x,y,n),z,o)),satz240(x,y,n)):is(ts(ts(y,z),ov(ov(x,y,n),z,o)),x)
8553 -5243
8554 satz243:=satz229g(x,ts(y,z),ov(ov(x,y,n),z,o),satz221d(y,z,n,o),t2".5243"):is(ov(ov(x,y,n),z,o),ov(x,ts(y,z),satz221d(y,z,n,o)))
8555 z@[n:nis(z,0c)]
8556 satz244:=satz229k(ts(x,y),z,ts(x,ov(y,z,n)),n,tris(cx,ts(ts(x,ov(y,z,n)),z),ts(x,ts(ov(y,z,n),z)),ts(x,y),assts1(x,ov(y,z,n),z),ists2(ts(ov(y,z,n),z),y,x,satz240(y,z,n)))):is(ov(ts(x,y),z,n),ts(x,ov(y,z,n)))
8557 satz244a:=symis(cx,ov(ts(x,y),z,n),ts(x,ov(y,z,n)),satz244):is(ts(x,ov(y,z,n)),ov(ts(x,y),z,n))
8558 satz244b:=tr3is(cx,ov(ts(x,y),z,n),ov(ts(y,x),z,n),ts(y,ov(x,z,n)),ts(ov(x,z,n),y),isov1(ts(x,y),ts(y,x),z,comts(x,y),n),satz244(y,x,z,n),comts(y,ov(x,z,n))):is(ov(ts(x,y),z,n),ts(ov(x,z,n),y))
8559 satz244c:=symis(cx,ov(ts(x,y),z,n),ts(ov(x,z,n),y),satz244b):is(ts(ov(x,z,n),y),ov(ts(x,y),z,n))
8560 z@[n:nis(y,0c)][o:nis(z,0c)]
8561 satz245:=satz229j(x,ov(y,z,o),ts(ov(x,y,n),z),lemma6(y,z,n,o),tr3is(cx,ts(ts(ov(x,y,n),z),ov(y,z,o)),ts(ov(x,y,n),ts(z,ov(y,z,o))),ts(ov(x,y,n),y),x,assts1(ov(x,y,n),z,ov(y,z,o)),ists2(ts(z,ov(y,z,o)),y,ov(x,y,n),satz229c(y,z,o)),satz240(x,y,n))):is(ts(ov(x,y,n),z),ov(x,ov(y,z,o),lemma6(y,z,n,o)))
8562 satz245a:=symis(cx,ts(ov(x,y,n),z),ov(x,ov(y,z,o),lemma6(y,z,n,o)),satz245):is(ov(x,ov(y,z,o),lemma6(y,z,n,o)),ts(ov(x,y,n),z))
8563 satz245b:=tris(cx,ov(x,ov(y,z,o),lemma6(y,z,n,o)),ts(ov(x,y,n),z),ov(ts(x,z),y,n),satz245a,satz244c(x,z,y,n)):is(ov(x,ov(y,z,o),lemma6(y,z,n,o)),ov(ts(x,z),y,n))
8564 satz245c:=tris(cx,ov(x,ov(y,z,o),lemma6(y,z,n,o)),ov(ts(x,z),y,n),ov(ts(z,x),y,n),satz245b,isov1(ts(x,z),ts(z,x),y,comts(x,z),n)):is(ov(x,ov(y,z,o),lemma6(y,z,n,o)),ov(ts(z,x),y,n))
8565 satz246:=satz229k(ts(x,z),ts(y,z),ov(x,y,n),satz221d(y,z,n,o),tris(cx,ts(ov(x,y,n),ts(y,z)),ts(ts(ov(x,y,n),y),z),ts(x,z),assts2(ov(x,y,n),y,z),ists1(ts(ov(x,y,n),y),x,z,satz240(x,y,n)))):is(ov(ts(x,z),ts(y,z),satz221d(y,z,n,o)),ov(x,y,n))
8566 satz246a:=tris(cx,ov(ts(z,x),ts(z,y),satz221d(z,y,o,n)),ov(ts(x,z),ts(y,z),satz221d(y,z,n,o)),ov(x,y,n),isov12(ts(z,x),ts(x,z),ts(z,y),ts(y,z),comts(z,x),comts(z,y),satz221d(z,y,o,n),satz221d(y,z,n,o)),satz246):is(ov(ts(z,x),ts(z,y),satz221d(z,y,o,n)),ov(x,y,n))
8567 z@[u:complex][n:nis(y,0c)][o:nis(u,0c)]
8568 +5247
8569 t1:=tr3is(cx,ts(ov(z,u,o),ts(u,y)),ts(ts(ov(z,u,o),u),y),ts(z,y),ts(y,z),assts2(ov(z,u,o),u,y),ists1(ts(ov(z,u,o),u),z,y,satz240(z,u,o)),comts(z,y)):is(ts(ov(z,u,o),ts(u,y)),ts(y,z))
8570 t2:=tr3is(cx,ts(ts(ov(x,y,n),ov(z,u,o)),ts(y,u)),ts(ts(ov(x,y,n),ov(z,u,o)),ts(u,y)),ts(ov(x,y,n),ts(ov(z,u,o),ts(u,y))),ts(ov(x,y,n),ts(y,z)),ists2(ts(y,u),ts(u,y),ts(ov(x,y,n),ov(z,u,o)),comts(y,u)),assts1(ov(x,y,n),ov(z,u,o),ts(u,y)),ists2(ts(ov(z,u,o),ts(u,y)),ts(y,z),ov(x,y,n),t1)):is(ts(ts(ov(x,y,n),ov(z,u,o)),ts(y,u)),ts(ov(x,y,n),ts(y,z)))
8571 t3:=tr3is(cx,ts(ts(ov(x,y,n),ov(z,u,o)),ts(y,u)),ts(ov(x,y,n),ts(y,z)),ts(ts(ov(x,y,n),y),z),ts(x,z),t2,assts2(ov(x,y,n),y,z),ists1(ts(ov(x,y,n),y),x,z,satz240(x,y,n))):is(ts(ts(ov(x,y,n),ov(z,u,o)),ts(y,u)),ts(x,z))
8572 -5247
8573 satz247:=satz229j(ts(x,z),ts(y,u),ts(ov(x,y,n),ov(z,u,o)),satz221d(y,u,n,o),t3".5247"):is(ts(ov(x,y,n),ov(z,u,o)),ov(ts(x,z),ts(y,u),satz221d(y,u,n,o)))
8574 u@[n:nis(y,0c)][o:nis(z,0c)][p:nis(u,0c)]
8575 +5248
8576 t1:=tris(cx,ts(ts(x,u),z),ts(x,ts(u,z)),ts(x,ts(z,u)),assts1(x,u,z),ists2(ts(u,z),ts(z,u),x,comts(u,z))):is(ts(ts(x,u),z),ts(x,ts(z,u)))
8577 t2:=tr3is(cx,ts(ov(ts(x,u),ts(y,z),satz221d(y,z,n,o)),ov(z,u,p)),ov(ts(ts(x,u),z),ts(ts(y,z),u),satz221d(ts(y,z),u,satz221d(y,z,n,o),p)),ov(ts(x,ts(z,u)),ts(y,ts(z,u)),satz221d(y,ts(z,u),n,satz221d(z,u,o,p))),ov(x,y,n),satz247(ts(x,u),ts(y,z),z,u,satz221d(y,z,n,o),p),isov12(ts(ts(x,u),z),ts(x,ts(z,u)),ts(ts(y,z),u),ts(y,ts(z,u)),t1,assts1(y,z,u),satz221d(ts(y,z),u,satz221d(y,z,n,o),p),satz221d(y,ts(z,u),n,satz221d(z,u,o,p))),satz246(x,y,ts(z,u),n,satz221d(z,u,o,p))):is(ts(ov(ts(x,u),ts(y,z),satz221d(y,z,n,o)),ov(z,u,p)),ov(x,y,n))
8578 -5248
8579 satz248:=satz229k(ov(x,y,n),ov(z,u,p),ov(ts(x,u),ts(y,z),satz221d(y,z,n,o)),lemma6(z,u,o,p),t2".5248"):is(ov(ov(x,y,n),ov(z,u,p),lemma6(z,u,o,p)),ov(ts(x,u),ts(y,z),satz221d(y,z,n,o)))
8580 x@[n:nis(x,0c)]
8581 satz249:=satz229h(0c,x,0c,n,satz221b(x,0c,refis(cx,0c))):is(ov(0c,x,n),0c)
8582 satz250:=satz229h(x,x,1c,n,satz222(x)):is(ov(x,x,n),1c)
8583 y@[n:nis(y,0c)][i:is(x,y)]
8584 satz251a:=tris(cx,ov(x,y,n),ov(x,x,th2"e.notis"(cx,y,0c,x,n,i)),1c,isov2(y,x,x,symis(cx,x,y,i),n,th2"e.notis"(cx,y,0c,x,n,i)),satz250(x,th2"e.notis"(cx,y,0c,x,n,i))):is(ov(x,y,n),1c)
8585 n@[i:is(ov(x,y,n),1c)]
8586 satz251b:=tr3is(cx,x,ts(y,ov(x,y,n)),ts(y,1c),y,satz229d(x,y,n),ists2(ov(x,y,n),1c,y,i),satz222(y)):is(x,y)
8587 u@[n:nis(y,0c)][o:nis(u,0c)][i:is(ov(x,y,n),ov(z,u,o))]
8588 +5252
8589 [j:is(z,0c)]
8590 t1:=tr3is(cx,ov(x,y,n),ov(z,u,o),ov(0c,u,o),0c,i,isov1(z,0c,u,j,o),satz249(u,o)):is(ov(x,y,n),0c)
8591 t2:=tris(cx,x,ts(y,ov(x,y,n)),0c,satz229d(x,y,n),satz221b(y,ov(x,y,n),t1)):is(x,0c)
8592 t3:=tris2(cx,ts(x,u),ts(y,z),0c,satz221a(x,u,t2),satz221b(y,z,j)):is(ts(x,u),ts(y,z))
8593 i@[p:nis(z,0c)]
8594 t4:=tris1(cx,ov(ts(x,u),ts(y,z),satz221d(y,z,n,p)),1c,ov(ov(x,y,n),ov(z,u,o),lemma6(z,u,p,o)),satz248(x,y,z,u,n,p,o),satz251a(ov(x,y,n),ov(z,u,o),lemma6(z,u,p,o),i)):is(ov(ts(x,u),ts(y,z),satz221d(y,z,n,p)),1c)
8595 t5:=satz251b(ts(x,u),ts(y,z),satz221d(y,z,n,p),t4):is(ts(x,u),ts(y,z))
8596 -5252
8597 satz252a:=th1"l.imp"(is(z,0c),is(ts(x,u),ts(y,z)),[t:is(z,0c)]t3".5252"(t),[t:nis(z,0c)]t5".5252"(t)):is(ts(x,u),ts(y,z))
8598 o@[i:is(ts(x,u),ts(y,z))]
8599 +*5252
8600 i@[j:is(z,0c)]
8601 t6:=tris(cx,ts(x,u),ts(y,z),0c,i,satz221b(y,z,j)):is(ts(x,u),0c)
8602 t7:=ore1(is(x,0c),is(u,0c),satz221c(x,u,t6),o):is(x,0c)
8603 t8:=tris2(cx,ov(x,y,n),ov(z,u,o),0c,tris(cx,ov(x,y,n),ov(0c,y,n),0c,isov1(x,0c,y,t7,n),satz249(y,n)),tris(cx,ov(z,u,o),ov(0c,u,o),0c,isov1(z,0c,u,j,o),satz249(u,o))):is(ov(x,y,n),ov(z,u,o))
8604 i@[p:nis(z,0c)]
8605 t9:=tris(cx,ov(ov(x,y,n),ov(z,u,o),lemma6(z,u,p,o)),ov(ts(x,u),ts(y,z),satz221d(y,z,n,p)),1c,satz248(x,y,z,u,n,p,o),satz251a(ts(x,u),ts(y,z),satz221d(y,z,n,p),i)):is(ov(ov(x,y,n),ov(z,u,o),lemma6(z,u,p,o)),1c)
8606 t10:=satz251b(ov(x,y,n),ov(z,u,o),lemma6(z,u,p,o),t9):is(ov(x,y,n),ov(z,u,o))
8607 -5252
8608 i@satz252b:=th1"l.imp"(is(z,0c),is(ov(x,y,n),ov(z,u,o)),[t:is(z,0c)]t8".5252"(t),[t:nis(z,0c)]t10".5252"(t)):is(ov(x,y,n),ov(z,u,o))
8609 z@[n:nis(y,0c)]
8610 satz253:=satz229g(pl(x,z),y,pl(ov(x,y,n),ov(z,y,n)),n,tris(cx,ts(y,pl(ov(x,y,n),ov(z,y,n))),pl(ts(y,ov(x,y,n)),ts(y,ov(z,y,n))),pl(x,z),disttp2(y,ov(x,y,n),ov(z,y,n)),ispl12(ts(y,ov(x,y,n)),x,ts(y,ov(z,y,n)),z,satz229c(x,y,n),satz229c(z,y,n)))):is(pl(ov(x,y,n),ov(z,y,n)),ov(pl(x,z),y,n))
8611 z@[n:nis(z,0c)]
8612 distop:=symis(cx,pl(ov(x,z,n),ov(y,z,n)),ov(pl(x,y),z,n),satz253(x,z,y,n)):is(ov(pl(x,y),z,n),pl(ov(x,z,n),ov(y,z,n)))
8613 distpo:=satz253(x,z,y,n):is(pl(ov(x,z,n),ov(y,z,n)),ov(pl(x,y),z,n))
8614 u@[n:nis(y,0c)][o:nis(u,0c)]
8615 satz254:=tris1(cx,pl(ov(x,y,n),ov(z,u,o)),ov(pl(ts(x,u),ts(y,z)),ts(y,u),satz221d(y,u,n,o)),pl(ov(ts(x,u),ts(y,u),satz221d(y,u,n,o)),ov(ts(y,z),ts(y,u),satz221d(y,u,n,o))),ispl12(ov(ts(x,u),ts(y,u),satz221d(y,u,n,o)),ov(x,y,n),ov(ts(y,z),ts(y,u),satz221d(y,u,n,o)),ov(z,u,o),satz246(x,y,u,n,o),satz246a(z,u,y,o,n)),satz253(ts(x,u),ts(y,u),ts(y,z),satz221d(y,u,n,o))):is(pl(ov(x,y,n),ov(z,u,o)),ov(pl(ts(x,u),ts(y,z)),ts(y,u),satz221d(y,u,n,o)))
8616 z@[n:nis(y,0c)]
8617 satz255:=satz229g(mn(x,z),y,mn(ov(x,y,n),ov(z,y,n)),n,tris(cx,ts(y,mn(ov(x,y,n),ov(z,y,n))),mn(ts(y,ov(x,y,n)),ts(y,ov(z,y,n))),mn(x,z),disttm2(y,ov(x,y,n),ov(z,y,n)),ismn12(ts(y,ov(x,y,n)),x,ts(y,ov(z,y,n)),z,satz229c(x,y,n),satz229c(z,y,n)))):is(mn(ov(x,y,n),ov(z,y,n)),ov(mn(x,z),y,n))
8618 z@[n:nis(z,0c)]
8619 distom:=symis(cx,mn(ov(x,z,n),ov(y,z,n)),ov(mn(x,y),z,n),satz255(x,z,y,n)):is(ov(mn(x,y),z,n),mn(ov(x,z,n),ov(y,z,n)))
8620 distmo:=satz255(x,z,y,n):is(mn(ov(x,z,n),ov(y,z,n)),ov(mn(x,y),z,n))
8621 u@[n:nis(y,0c)][o:nis(u,0c)]
8622 satz256:=tris1(cx,mn(ov(x,y,n),ov(z,u,o)),ov(mn(ts(x,u),ts(y,z)),ts(y,u),satz221d(y,u,n,o)),mn(ov(ts(x,u),ts(y,u),satz221d(y,u,n,o)),ov(ts(y,z),ts(y,u),satz221d(y,u,n,o))),ismn12(ov(ts(x,u),ts(y,u),satz221d(y,u,n,o)),ov(x,y,n),ov(ts(y,z),ts(y,u),satz221d(y,u,n,o)),ov(z,u,o),satz246(x,y,u,n,o),satz246a(z,u,y,o,n)),satz255(ts(x,u),ts(y,u),ts(y,z),satz221d(y,u,n,o))):is(mn(ov(x,y,n),ov(z,u,o)),ov(mn(ts(x,u),ts(y,z)),ts(y,u),satz221d(y,u,n,o)))
8623 x@conj:=pli(re(x),m0"r"(im(x))):complex
8624 b@conjisa:=isrecx12(re(pli(a,b)),a,m0"r"(im(pli(a,b))),m0"r"(b),reis(a,b),ism0"r"(im(pli(a,b)),b,imis(a,b))):is(conj(pli(a,b)),pli(a,m0"r"(b)))
8625 conjisb:=symis(cx,conj(pli(a,b)),pli(a,m0"r"(b)),conjisa):is(pli(a,m0"r"(b)),conj(pli(a,b)))
8626 y@[i:is(x,y)]
8627 isconj:=isf(cx,cx,[t:cx]conj(t),x,y,i):is(conj(x),conj(y))
8628 x@satz257:=tr3is(cx,conj(conj(x)),pli(re(x),m0"r"(m0"r"(im(x)))),pli(re(x),im(x)),x,conjisa(re(x),m0"r"(im(x))),isrecx2(m0"r"(m0"r"(im(x))),im(x),re(x),satz177(im(x))),pliis(x)):is(conj(conj(x)),x)
8629 [i:is(x,0c)]
8630 satz258a:=tr3is(cx,conj(x),conj(0c),pli(0,m0"r"(0)),0c,isconj(x,0c,i),conjisa(0,0),isrecx2(m0"r"(0),0,0,satz176b(0,refis(real,0)))):is(conj(x),0c)
8631 x@[i:is(conj(x),0c)]
8632 +6258
8633 t1:=tris(real,re(x),re(conj(x)),0,isre(re(x),m0"r"(im(x))),lemma1(conj(x),i)):is"r"(re(x),0)
8634 t2:=satz176e(im(x),tris(real,m0"r"(im(x)),im(conj(x)),0,isim(re(x),m0"r"(im(x))),lemma2(conj(x),i))):is"r"(im(x),0)
8635 -6258
8636 satz258b:=tris(cx,x,pli(re(x),im(x)),0c,ispli(x),isrecx12(re(x),0,im(x),0,t1".6258",t2".6258")):is(x,0c)
8637 +*6258
8638 i@anders:=tris1(cx,x,0c,conj(conj(x)),satz257,satz258a(conj(x),i)):is(x,0c)
8639 -6258
8640 x@[n:nis(x,0c)]
8641 satz258c:=th3"l.imp"(is(conj(x),0c),is(x,0c),n,[t:is(conj(x),0c)]satz258b(t)):nis(conj(x),0c)
8642 x@[i:is(conj(x),x)]
8643 +6259
8644 t1:=tris(real,m0"r"(im(x)),im(conj(x)),im(x),isim(re(x),m0"r"(im(x))),isceim(conj(x),x,i)):is"r"(m0"r"(im(x)),im(x))
8645 -6259
8646 satz259a:=lemma10(im(x),symis(real,m0"r"(im(x)),im(x),t1".6259")):is"r"(im(x),0)
8647 x@[i:is"r"(im(x),0)]
8648 satz259b:=tris(cx,conj(x),pli(re(x),im(x)),x,isrecx2(m0"r"(im(x)),im(x),re(x),tris2(real,m0"r"(im(x)),im(x),0,satz176b(im(x),i),i)),pliis(x)):is(conj(x),x)
8649 x@[i:is(x,conj(x))]
8650 satz269c:=satz259a(x,symis(cx,x,conj(x),i)):is"r"(im(x),0)
8651 x@[i:is"r"(im(x),0)]
8652 satz269d:=symis(cx,conj(x),x,satz259b(i)):is(x,conj(x))
8653 y@satz260:=tr3is(cx,conj(pl(x,y)),pli(pl"r"(re(x),re(y)),m0"r"(pl"r"(im(x),im(y)))),pli(pl"r"(re(x),re(y)),pl"r"(m0"r"(im(x)),m0"r"(im(y)))),pl(conj(x),conj(y)),conjisa(pl"r"(re(x),re(y)),pl"r"(im(x),im(y))),isrecx2(m0"r"(pl"r"(im(x),im(y))),pl"r"(m0"r"(im(x)),m0"r"(im(y))),pl"r"(re(x),re(y)),satz180(im(x),im(y))),plis12b(re(x),m0"r"(im(x)),re(y),m0"r"(im(y)))):is(conj(pl(x,y)),pl(conj(x),conj(y)))
8654 satz260a:=symis(cx,conj(pl(x,y)),pl(conj(x),conj(y)),satz260):is(pl(conj(x),conj(y)),conj(pl(x,y)))
8655 +6261
8656 t1:=tris(real,m0"r"(imts(x,y)),pl"r"(m0"r"(ts"r"(re(x),im(y))),m0"r"(ts"r"(im(x),re(y)))),pl"r"(ts"r"(re(x),m0"r"(im(y))),ts"r"(m0"r"(im(x)),re(y))),satz180(ts"r"(re(x),im(y)),ts"r"(im(x),re(y))),ispl12"r"(m0"r"(ts"r"(re(x),im(y))),ts"r"(re(x),m0"r"(im(y))),m0"r"(ts"r"(im(x),re(y))),ts"r"(m0"r"(im(x)),re(y)),satz197f(re(x),im(y)),satz197e(im(x),re(y)))):is"r"(m0"r"(imts(x,y)),pl"r"(ts"r"(re(x),m0"r"(im(y))),ts"r"(m0"r"(im(x)),re(y))))
8657 -6261
8658 satz261:=tr3is(cx,conj(ts(x,y)),pli(rets(x,y),m0"r"(imts(x,y))),pli(mn"r"(ts"r"(re(x),re(y)),ts"r"(m0"r"(im(x)),m0"r"(im(y)))),pl"r"(ts"r"(re(x),m0"r"(im(y))),ts"r"(m0"r"(im(x)),re(y)))),ts(conj(x),conj(y)),conjisa(rets(x,y),imts(x,y)),isrecx12(rets(x,y),mn"r"(ts"r"(re(x),re(y)),ts"r"(m0"r"(im(x)),m0"r"(im(y)))),m0"r"(imts(x,y)),pl"r"(ts"r"(re(x),m0"r"(im(y))),ts"r"(m0"r"(im(x)),re(y))),ismn2"r"(ts"r"(im(x),im(y)),ts"r"(m0"r"(im(x)),m0"r"(im(y))),ts"r"(re(x),re(y)),satz198a(im(x),im(y))),t1".6261"),tsis12b(re(x),m0"r"(im(x)),re(y),m0"r"(im(y)))):is(conj(ts(x,y)),ts(conj(x),conj(y)))
8659 satz261a:=symis(cx,conj(ts(x,y)),ts(conj(x),conj(y)),satz261):is(ts(conj(x),conj(y)),conj(ts(x,y)))
8660 +6262
8661 t1:=symis(cx,pl(mn(x,y),y),x,satz230(x,y)):is(x,pl(mn(x,y),y))
8662 t2:=tris(cx,conj(x),conj(pl(mn(x,y),y)),pl(conj(mn(x,y)),conj(y)),isconj(x,pl(mn(x,y),y),t1),satz260(mn(x,y),y)):is(conj(x),pl(conj(mn(x,y)),conj(y)))
8663 -6262
8664 satz262:=satz212f(conj(x),conj(y),conj(mn(x,y)),symis(cx,conj(x),pl(conj(mn(x,y)),conj(y)),t2".6262")):is(conj(mn(x,y)),mn(conj(x),conj(y)))
8665 satz262a:=symis(cx,conj(mn(x,y)),mn(conj(x),conj(y)),satz262):is(mn(conj(x),conj(y)),conj(mn(x,y)))
8666 [n:nis(y,0c)]
8667 +6263
8668 t1:=satz229f(x,y,n):is(x,ts(ov(x,y,n),y))
8669 t2:=isconj(x,ts(ov(x,y,n),y),t1):is(conj(x),conj(ts(ov(x,y,n),y)))
8670 t3:=satz261(ov(x,y,n),y):is(conj(ts(ov(x,y,n),y)),ts(conj(ov(x,y,n)),conj(y)))
8671 t4:=tris(cx,conj(x),conj(ts(ov(x,y,n),y)),ts(conj(ov(x,y,n)),conj(y)),t2,t3):is(conj(x),ts(conj(ov(x,y,n)),conj(y)))
8672 t5:=satz258c(y,n):nis(conj(y),0c)
8673 -6263
8674 satz263:=satz229j(conj(x),conj(y),conj(ov(x,y,n)),t5".6263",symis(cx,conj(x),ts(conj(ov(x,y,n)),conj(y)),t4".6263")):is(conj(ov(x,y,n)),ov(conj(x),conj(y),satz258c(y,n)))
8675 satz263a:=symis(cx,conj(ov(x,y,n)),ov(conj(x),conj(y),satz258c(y,n)),satz263):is(ov(conj(x),conj(y),satz258c(y,n)),conj(ov(x,y,n)))
8676 x@mod:=sqrt(mod2(x),lemma5(x)):real
8677 y@[i:is(x,y)]
8678 ismod:=isf(cx,real,[t:cx]mod(t),x,y,i):is"r"(mod(x),mod(y))
8679 x@[n:nis(x,0c)]
8680 satz264a:=sqrtnot0(mod2(x),lemma5(x),pnot0(mod2(x),lemma4(x,n))):pos(mod(x))
8681 x@[i:is(x,0c)]
8682 satz264b:=sqrt0(mod2(x),lemma5(x),lemma3(x,i)):is"r"(mod(x),0)
8683 x@satz264c:=thsqrt1a(mod2(x),lemma5(x)):not(neg(mod(x)))
8684 satz264d:=satz167f(mod(x),0,th3"l.imp"(less(mod(x),0),neg(mod(x)),satz264c,[t:less(mod(x),0)]satz169d(mod(x),t))):moreis(mod(x),0)
8685 +7265
8686 t1:=symis(real,ts"r"(mod(x),mod(x)),mod2(x),thsqrt1b(mod2(x),lemma5(x))):is"r"(mod2(x),ts"r"(mod(x),mod(x)))
8687 t2:=tris(real,pl"r"(ts"r"(re(x),re(x)),0),ts"r"(re(x),re(x)),ts"r"(abs(re(x)),abs(re(x))),pl02(ts"r"(re(x),re(x)),0,refis(real,0)),lemma12"r"(re(x))):is"r"(pl"r"(ts"r"(re(x),re(x)),0),ts"r"(abs(re(x)),abs(re(x))))
8688 t3:=satz191(ts"r"(re(x),re(x)),ts"r"(re(x),re(x)),ts"r"(im(x),im(x)),0,moreisi2(ts"r"(re(x),re(x)),ts"r"(re(x),re(x)),refis(real,ts"r"(re(x),re(x)))),lemma11"r"(im(x))):moreis(mod2(x),pl"r"(ts"r"(re(x),re(x)),0))
8689 t4:=ismoreis12(mod2(x),ts"r"(mod(x),mod(x)),pl"r"(ts"r"(re(x),re(x)),0),ts"r"(abs(re(x)),abs(re(x))),t1,t2,t3):moreis(ts"r"(mod(x),mod(x)),ts"r"(abs(re(x)),abs(re(x))))
8690 t5:=tris(real,pl"r"(0,ts"r"(im(x),im(x))),ts"r"(im(x),im(x)),ts"r"(abs(im(x)),abs(im(x))),pl01(0,ts"r"(im(x),im(x)),refis(real,0)),lemma12"r"(im(x))):is"r"(pl"r"(0,ts"r"(im(x),im(x))),ts"r"(abs(im(x)),abs(im(x))))
8691 t6:=satz191(ts"r"(re(x),re(x)),0,ts"r"(im(x),im(x)),ts"r"(im(x),im(x)),lemma11"r"(re(x)),moreisi2(ts"r"(im(x),im(x)),ts"r"(im(x),im(x)),refis(real,ts"r"(im(x),im(x))))):moreis(mod2(x),pl"r"(0,ts"r"(im(x),im(x))))
8692 t7:=ismoreis12(mod2(x),ts"r"(mod(x),mod(x)),pl"r"(0,ts"r"(im(x),im(x))),ts"r"(abs(im(x)),abs(im(x))),t1,t5,t6):moreis(ts"r"(mod(x),mod(x)),ts"r"(abs(im(x)),abs(im(x))))
8693 @[r:real][s:real][m:moreis(ts"r"(r,r),ts"r"(s,s))][n:moreis(r,0)][l:less(r,s)]
8694 t8:=lemma2"r"(r,s,l):more(s,r)
8695 t9:=satz169b(s,satz172d(s,r,0,t8,n)):pos(s)
8696 [o:more(r,0)]
8697 t10:=trmore(ts"r"(s,s),ts"r"(r,s),ts"r"(r,r),satz203a(s,r,s,t8,t9),satz203d(s,r,r,t8,satz169b(r,o))):more(ts"r"(s,s),ts"r"(r,r))
8698 l@[i:is"r"(r,0)]
8699 t11:=ismore2(0,ts"r"(r,r),ts"r"(s,s),symis(real,ts"r"(r,r),0,ts01(r,r,i)),satz169a(ts"r"(s,s),possq(s,pnot0(s,t9)))):more(ts"r"(s,s),ts"r"(r,r))
8700 l@t12:=lemma1"r"(ts"r"(s,s),ts"r"(r,r),orapp(more(r,0),is"r"(r,0),more(ts"r"(s,s),ts"r"(r,r)),n,[t:more(r,0)]t10(t),[t:is"r"(r,0)]t11(t))):less(ts"r"(r,r),ts"r"(s,s))
8701 n@t13:=satz167f(r,s,[t:less(r,s)]<t12(t)>satz167c(ts"r"(r,r),ts"r"(s,s),m)):moreis(r,s)
8702 -7265
8703 satz265a:=t13".7265"(mod(x),abs(re(x)),t4".7265",satz264d(x)):moreis(mod(x),abs(re(x)))
8704 satz265b:=t13".7265"(mod(x),abs(im(x)),t7".7265",satz264d(x)):moreis(mod(x),abs(im(x)))
8705 @[r:real][s:real][i:is(ts(pli(r,0),pli(r,0)),ts(pli(s,0),pli(s,0)))][n:not(neg(r))][o:not(neg(s))]
8706 +7266
8707 @[t:real]
8708 t1:=pl02(ts"r"(t,t),m0"r"(ts"r"(0,0)),satz176b(ts"r"(0,0),ts01(0,0,refis(real,0)))):is"r"(mn"r"(ts"r"(t,t),ts"r"(0,0)),ts"r"(t,t))
8709 t2:=tris(real,pl"r"(ts"r"(t,0),ts"r"(0,t)),ts"r"(t,0),0,pl02(ts"r"(t,0),ts"r"(0,t),ts01(0,t,refis(real,0))),ts02(t,0,refis(real,0))):is"r"(pl"r"(ts"r"(t,0),ts"r"(0,t)),0)
8710 t3:=tris(cx,ts(pli(t,0),pli(t,0)),pli(mn"r"(ts"r"(t,t),ts"r"(0,0)),pl"r"(ts"r"(t,0),ts"r"(0,t))),pli(ts"r"(t,t),0),tsis12a(t,0,t,0),isrecx12(mn"r"(ts"r"(t,t),ts"r"(0,0)),ts"r"(t,t),pl"r"(ts"r"(t,0),ts"r"(0,t)),0,t1,t2)):is(ts(pli(t,0),pli(t,0)),pli(ts"r"(t,t),0))
8711 o@t4:=tr3is(cx,pli(ts"r"(r,r),0),ts(pli(r,0),pli(r,0)),ts(pli(s,0),pli(s,0)),pli(ts"r"(s,s),0),symis(cx,ts(pli(r,0),pli(r,0)),pli(ts"r"(r,r),0),t3(r)),i,t3(s)):is(pli(ts"r"(r,r),0),pli(ts"r"(s,s),0))
8712 t5:=tr3is(real,ts"r"(r,r),re(pli(ts"r"(r,r),0)),re(pli(ts"r"(s,s),0)),ts"r"(s,s),isre(ts"r"(r,r),0),iscere(pli(ts"r"(r,r),0),pli(ts"r"(s,s),0),t4),reis(ts"r"(s,s),0)):is"r"(ts"r"(r,r),ts"r"(s,s))
8713 t6:=andi(not(neg(r)),is"r"(ts"r"(r,r),ts"r"(s,s)),n,t5):and(not(neg(r)),is"r"(ts"r"(r,r),ts"r"(s,s)))
8714 t7:=andi(not(neg(s)),is"r"(ts"r"(s,s),ts"r"(s,s)),o,refis(real,ts"r"(s,s))):and(not(neg(s)),is"r"(ts"r"(s,s),ts"r"(s,s)))
8715 -7266
8716 satz266:=satzr161b(ts"r"(s,s),r,s,t6".7266",t7".7266"):is"r"(r,s)
8717 +7267
8718 x@t1:=tris(cx,ts(pli(mod(x),0),pli(mod(x),0)),pli(ts"r"(mod(x),mod(x)),0),pli(mod2(x),0),t3"c.7266"(mod(x)),isrecx1(ts"r"(mod(x),mod(x)),mod2(x),0,thsqrt1b(mod2(x),lemma5(x)))):is(ts(pli(mod(x),0),pli(mod(x),0)),pli(mod2(x),0))
8719 t2:=ispl2"r"(m0"r"(ts"r"(im(x),m0"r"(im(x)))),ts"r"(im(x),im(x)),ts"r"(re(x),re(x)),tris(real,m0"r"(ts"r"(im(x),m0"r"(im(x)))),ts"r"(m0"r"(im(x)),m0"r"(im(x))),ts"r"(im(x),im(x)),satz197e(im(x),m0"r"(im(x))),satz198(im(x),im(x)))):is"r"(mn"r"(ts"r"(re(x),re(x)),ts"r"(im(x),m0"r"(im(x)))),mod2(x))
8720 t3:=tris(real,pl"r"(ts"r"(re(x),m0"r"(im(x))),ts"r"(im(x),re(x))),pl"r"(m0"r"(ts"r"(re(x),im(x))),ts"r"(re(x),im(x))),0,ispl12"r"(ts"r"(re(x),m0"r"(im(x))),m0"r"(ts"r"(re(x),im(x))),ts"r"(im(x),re(x)),ts"r"(re(x),im(x)),satz197b(re(x),im(x)),comts"r"(im(x),re(x))),satz179a(ts"r"(re(x),im(x)))):is"r"(pl"r"(ts"r"(re(x),m0"r"(im(x))),ts"r"(im(x),re(x))),0)
8721 t4:=tris(cx,ts(x,conj(x)),pli(mn"r"(ts"r"(re(x),re(x)),ts"r"(im(x),m0"r"(im(x)))),pl"r"(ts"r"(re(x),m0"r"(im(x))),ts"r"(im(x),re(x)))),pli(mod2(x),0),tsis2a(x,re(x),m0"r"(im(x))),isrecx12(mn"r"(ts"r"(re(x),re(x)),ts"r"(im(x),m0"r"(im(x)))),mod2(x),pl"r"(ts"r"(re(x),m0"r"(im(x))),ts"r"(im(x),re(x))),0,t2,t3)):is(ts(x,conj(x)),pli(mod2(x),0))
8722 -7267
8723 x@satz267:=tris2(cx,ts(pli(mod(x),0),pli(mod(x),0)),ts(x,conj(x)),pli(mod2(x),0),t1".7267",t4".7267"):is(ts(pli(mod(x),0),pli(mod(x),0)),ts(x,conj(x)))
8724 satz267a:=symis(cx,ts(pli(mod(x),0),pli(mod(x),0)),ts(x,conj(x)),satz267):is(ts(x,conj(x)),ts(pli(mod(x),0),pli(mod(x),0)))
8725 +7268
8726 z@t1:=tr3is(cx,ts(x,ts(y,z)),ts(ts(x,y),z),ts(ts(y,x),z),ts(y,ts(x,z)),assts2(x,y,z),ists1(ts(x,y),ts(y,x),z,comts(x,y)),assts1(y,x,z)):is(ts(x,ts(y,z)),ts(y,ts(x,z)))
8727 [u:complex]
8728 t2:=tr3is(cx,ts(ts(x,y),ts(z,u)),ts(x,ts(y,ts(z,u))),ts(x,ts(z,ts(y,u))),ts(ts(x,z),ts(y,u)),assts1(x,y,ts(z,u)),ists2(ts(y,ts(z,u)),ts(z,ts(y,u)),x,t1(y,z,u)),assts2(x,z,ts(y,u))):is(ts(ts(x,y),ts(z,u)),ts(ts(x,z),ts(y,u)))
8729 y@t3:=tr3is(cx,ts(pli(mod(ts(x,y)),0),pli(mod(ts(x,y)),0)),ts(ts(x,y),conj(ts(x,y))),ts(ts(x,y),ts(conj(x),conj(y))),ts(ts(x,conj(x)),ts(y,conj(y))),satz267(ts(x,y)),ists2(conj(ts(x,y)),ts(conj(x),conj(y)),ts(x,y),satz261(x,y)),t2(x,y,conj(x),conj(y))):is(ts(pli(mod(ts(x,y)),0),pli(mod(ts(x,y)),0)),ts(ts(x,conj(x)),ts(y,conj(y))))
8730 t4:=tr3is(cx,ts(pli(mod(ts(x,y)),0),pli(mod(ts(x,y)),0)),ts(ts(x,conj(x)),ts(y,conj(y))),ts(ts(pli(mod(x),0),pli(mod(x),0)),ts(pli(mod(y),0),pli(mod(y),0))),ts(ts(pli(mod(x),0),pli(mod(y),0)),ts(pli(mod(x),0),pli(mod(y),0))),t3,ists12(ts(x,conj(x)),ts(pli(mod(x),0),pli(mod(x),0)),ts(y,conj(y)),ts(pli(mod(y),0),pli(mod(y),0)),satz267a(x),satz267a(y)),t2(pli(mod(x),0),pli(mod(x),0),pli(mod(y),0),pli(mod(y),0))):is(ts(pli(mod(ts(x,y)),0),pli(mod(ts(x,y)),0)),ts(ts(pli(mod(x),0),pli(mod(y),0)),ts(pli(mod(x),0),pli(mod(y),0))))
8731 @[r:real][s:real]
8732 t5:=pl02(ts"r"(r,s),m0"r"(ts"r"(0,0)),satz176b(ts"r"(0,0),ts01(0,0,refis(real,0)))):is"r"(mn"r"(ts"r"(r,s),ts"r"(0,0)),ts"r"(r,s))
8733 t6:=tris(real,pl"r"(ts"r"(r,0),ts"r"(0,s)),ts"r"(r,0),0,pl02(ts"r"(r,0),ts"r"(0,s),ts01(0,s,refis(real,0))),ts02(r,0,refis(real,0))):is"r"(pl"r"(ts"r"(r,0),ts"r"(0,s)),0)
8734 t7:=tris(cx,ts(pli(r,0),pli(s,0)),pli(mn"r"(ts"r"(r,s),ts"r"(0,0)),pl"r"(ts"r"(r,0),ts"r"(0,s))),pli(ts"r"(r,s),0),tsis12a(r,0,s,0),isrecx12(mn"r"(ts"r"(r,s),ts"r"(0,0)),ts"r"(r,s),pl"r"(ts"r"(r,0),ts"r"(0,s)),0,t5,t6)):is(ts(pli(r,0),pli(s,0)),pli(ts"r"(r,s),0))
8735 y@t8:=tris(cx,ts(pli(mod(ts(x,y)),0),pli(mod(ts(x,y)),0)),ts(ts(pli(mod(x),0),pli(mod(y),0)),ts(pli(mod(x),0),pli(mod(y),0))),ts(pli(ts"r"(mod(x),mod(y)),0),pli(ts"r"(mod(x),mod(y)),0)),t4,ists12(ts(pli(mod(x),0),pli(mod(y),0)),pli(ts"r"(mod(x),mod(y)),0),ts(pli(mod(x),0),pli(mod(y),0)),pli(ts"r"(mod(x),mod(y)),0),t7(mod(x),mod(y)),t7(mod(x),mod(y)))):is(ts(pli(mod(ts(x,y)),0),pli(mod(ts(x,y)),0)),ts(pli(ts"r"(mod(x),mod(y)),0),pli(ts"r"(mod(x),mod(y)),0)))
8736 [n:neg(ts"r"(mod(x),mod(y)))]
8737 t9:=orapp(and(pos(mod(x)),neg(mod(y))),and(neg(mod(x)),pos(mod(y))),con,satz196h(mod(x),mod(y),n),[t:and(pos(mod(x)),neg(mod(y)))]<ande2(pos(mod(x)),neg(mod(y)),t)>satz264c(y),[t:and(neg(mod(x)),pos(mod(y)))]<ande1(neg(mod(x)),pos(mod(y)),t)>satz264c(x)):con
8738 -7268
8739 y@satz268:=satz266(mod(ts(x,y)),ts"r"(mod(x),mod(y)),t8".7268",satz264c(ts(x,y)),[t:neg(ts"r"(mod(x),mod(y)))]t9".7268"(t)):is"r"(mod(ts(x,y)),ts"r"(mod(x),mod(y)))
8740 satz268a:=symis(real,mod(ts(x,y)),ts"r"(mod(x),mod(y)),satz268):is"r"(ts"r"(mod(x),mod(y)),mod(ts(x,y)))
8741 [n:nis(y,0c)]
8742 +7269
8743 t1:=pnot0(mod(y),satz264a(y,n)):nis"r"(mod(y),0)
8744 t2:=tris1(real,ts"r"(mod(ov(x,y,n)),mod(y)),mod(x),mod(ts(ov(x,y,n),y)),satz268(ov(x,y,n),y),ismod(ts(ov(x,y,n),y),x,satz240(x,y,n))):is"r"(ts"r"(mod(ov(x,y,n)),mod(y)),mod(x))
8745 t3:=satz204g(mod(x),mod(y),mod(ov(x,y,n)),t1,tris(real,ts"r"(mod(y),mod(ov(x,y,n))),ts"r"(mod(ov(x,y,n)),mod(y)),mod(x),comts"r"(mod(y),mod(ov(x,y,n))),t2)):is"r"(mod(ov(x,y,n)),ov"r"(mod(x),mod(y),t1))
8746 -7269
8747 satz269:=t3".7269":is"r"(mod(ov(x,y,n)),ov"r"(mod(x),mod(y),pnot0(mod(y),satz264a(y,n))))
8748 y@[i:is(pl(x,y),1c)]
8749 +7270
8750 @[r:real]
8751 t1:=th1"l.imp"(neg(r),moreis(abs(r),r),[t:neg(r)]moreisi1(abs(r),r,trmore(abs(r),0,r,satz169a(abs(r),satz166b(r,t)),lemma2"r"(r,0,satz169c(r,t)))),[t:not(neg(r))]moreisi2(abs(r),r,absnn(r,t))):moreis(abs(r),r)
8752 x@t2:=trmoreis(mod(x),abs(re(x)),re(x),satz265a(x),t1(re(x))):moreis(mod(x),re(x))
8753 i@t3:=tr3is(real,pl"r"(re(x),re(y)),re(pl(x,y)),re(1c),1rl,isre(pl"r"(re(x),re(y)),pl"r"(im(x),im(y))),iscere(pl(x,y),1c,i),reis(1rl,0)):is"r"(pl"r"(re(x),re(y)),1rl)
8754 -7270
8755 satz270:=ismoreis2(pl"r"(re(x),re(y)),1rl,pl"r"(mod(x),mod(y)),t3".7270",satz191(mod(x),re(x),mod(y),re(y),t2".7270",t2".7270"(y))):moreis(pl"r"(mod(x),mod(y)),1rl)
8756 +7271
8757 y@[i:is(pl(x,y),0c)]
8758 t1:=satz264b(pl(x,y),i):is"r"(mod(pl(x,y)),0)
8759 t2:=ismoreis2(pl"r"(0,0),mod(pl(x,y)),pl"r"(mod(x),mod(y)),tris2(real,pl"r"(0,0),mod(pl(x,y)),0,pl01(0,0,refis(real,0)),t1),satz191(mod(x),0,mod(y),0,satz264d(x),satz264d(y))):moreis(pl"r"(mod(x),mod(y)),mod(pl(x,y)))
8760 y@[n:nis(pl(x,y),0c)]
8761 t3:=pnot0(mod(pl(x,y)),satz264a(pl(x,y),n)):nis"r"(mod(pl(x,y)),0)
8762 t4:=tris(cx,pl(ov(x,pl(x,y),n),ov(y,pl(x,y),n)),ov(pl(x,y),pl(x,y),n),1c,satz253(x,pl(x,y),y,n),satz250(pl(x,y),n)):is(pl(ov(x,pl(x,y),n),ov(y,pl(x,y),n)),1c)
8763 t5:=satz270(ov(x,pl(x,y),n),ov(y,pl(x,y),n),t4):moreis(pl"r"(mod(ov(x,pl(x,y),n)),mod(ov(y,pl(x,y),n))),1rl)
8764 fx:=ov"r"(mod(x),mod(pl(x,y)),t3):real
8765 fy:=ov"r"(mod(y),mod(pl(x,y)),t3):real
8766 t6:=ismoreis1(pl"r"(mod(ov(x,pl(x,y),n)),mod(ov(y,pl(x,y),n))),pl"r"(fx,fy),1rl,ispl12"r"(mod(ov(x,pl(x,y),n)),fx,mod(ov(y,pl(x,y),n)),fy,satz269(x,pl(x,y),n),satz269(y,pl(x,y),n)),t5):moreis(pl"r"(fx,fy),1rl)
8767 prl:=ts"r"(pl"r"(fx,fy),mod(pl(x,y))):real
8768 prr:=ts"r"(1rl,mod(pl(x,y))):real
8769 t7:=orapp(more(pl"r"(fx,fy),1rl),is"r"(pl"r"(fx,fy),1rl),moreis(prl,prr),t6,[t:more(pl"r"(fx,fy),1rl)]moreisi1(prl,prr,satz203a(pl"r"(fx,fy),1rl,mod(pl(x,y)),t,satz264a(pl(x,y),n))),[t:is"r"(pl"r"(fx,fy),1rl)]moreisi2(prl,prr,ists1"r"(pl"r"(fx,fy),1rl,mod(pl(x,y)),t))):moreis(prl,prr)
8770 t8:=tris(real,prl,pl"r"(ts"r"(fx,mod(pl(x,y))),ts"r"(fy,mod(pl(x,y)))),pl"r"(mod(x),mod(y)),disttp1"r"(fx,fy,mod(pl(x,y))),ispl12"r"(ts"r"(fx,mod(pl(x,y))),mod(x),ts"r"(fy,mod(pl(x,y))),mod(y),satz204e(mod(x),mod(pl(x,y)),t3),satz204e(mod(y),mod(pl(x,y)),t3))):is"r"(prl,pl"r"(mod(x),mod(y)))
8771 t9:=satz195b(mod(pl(x,y))):is"r"(prr,mod(pl(x,y)))
8772 t10:=ismoreis12(prl,pl"r"(mod(x),mod(y)),prr,mod(pl(x,y)),t8,t9,t7):moreis(pl"r"(mod(x),mod(y)),mod(pl(x,y)))
8773 y@t11:=th1"l.imp"(is(pl(x,y),0c),moreis(pl"r"(mod(x),mod(y)),mod(pl(x,y))),[t:is(pl(x,y),0c)]t2(t),[t:nis(pl(x,y),0c)]t10(t)):moreis(pl"r"(mod(x),mod(y)),mod(pl(x,y)))
8774 -7271
8775 y@satz271:=satz168a(pl"r"(mod(x),mod(y)),mod(pl(x,y)),t11".7271"):lessis(mod(pl(x,y)),pl"r"(mod(x),mod(y)))
8776 satz271a:=t11".7271":moreis(pl"r"(mod(x),mod(y)),mod(pl(x,y)))
8777 +7272
8778 x@t1:=tris(real,re(m0(x)),re(pli(m0"r"(re(x)),m0"r"(im(x)))),m0"r"(re(x)),iscere(m0(x),pli(m0"r"(re(x)),m0"r"(im(x))),satz214(x)),reis(m0"r"(re(x)),m0"r"(im(x)))):is"r"(re(m0(x)),m0"r"(re(x)))
8779 t2:=tris(real,ts"r"(re(m0(x)),re(m0(x))),ts"r"(m0"r"(re(x)),m0"r"(re(x))),ts"r"(re(x),re(x)),ists12"r"(re(m0(x)),m0"r"(re(x)),re(m0(x)),m0"r"(re(x)),t1,t1),satz198(re(x),re(x))):is"r"(ts"r"(re(m0(x)),re(m0(x))),ts"r"(re(x),re(x)))
8780 t3:=tris(real,im(m0(x)),im(pli(m0"r"(re(x)),m0"r"(im(x)))),m0"r"(im(x)),isceim(m0(x),pli(m0"r"(re(x)),m0"r"(im(x))),satz214(x)),imis(m0"r"(re(x)),m0"r"(im(x)))):is"r"(im(m0(x)),m0"r"(im(x)))
8781 t4:=tris(real,ts"r"(im(m0(x)),im(m0(x))),ts"r"(m0"r"(im(x)),m0"r"(im(x))),ts"r"(im(x),im(x)),ists12"r"(im(m0(x)),m0"r"(im(x)),im(m0(x)),m0"r"(im(x)),t3,t3),satz198(im(x),im(x))):is"r"(ts"r"(im(m0(x)),im(m0(x))),ts"r"(im(x),im(x)))
8782 t5:=ispl12"r"(ts"r"(re(m0(x)),re(m0(x))),ts"r"(re(x),re(x)),ts"r"(im(m0(x)),im(m0(x))),ts"r"(im(x),im(x)),t2,t4):is"r"(mod2(m0(x)),mod2(x))
8783 -7272
8784 x@satz272:=issqrt(mod2(m0(x)),mod2(x),lemma5(m0(x)),lemma5(x),t5".7272"):is"r"(mod(m0(x)),mod(x))
8785 satz272a:=symis(real,mod(m0(x)),mod(x),satz272):is"r"(mod(x),mod(m0(x)))
8786 +7273
8787 y@sum:=pl"r"(mod(y),mod(mn(x,y))):real
8788 t1:=islessis1(mod(pl(y,mn(x,y))),mod(x),sum,ismod(pl(y,mn(x,y)),x,satz212h(x,y)),satz271(y,mn(x,y))):lessis(mod(x),sum)
8789 t2:=th9"l.or"(less(mod(x),sum),is"r"(mod(x),sum),less(mn"r"(mod(x),mod(y)),mn"r"(sum,mod(y))),is"r"(mn"r"(mod(x),mod(y)),mn"r"(sum,mod(y))),t1,[t:less(mod(x),sum)]satz188f(mod(x),sum,m0"r"(mod(y)),t),[t:is"r"(mod(x),sum)]ismn1"r"(mod(x),sum,mod(y),t)):lessis(mn"r"(mod(x),mod(y)),mn"r"(sum,mod(y)))
8790 t3:=tris(real,mn"r"(sum,mod(y)),mn"r"(pl"r"(mod(mn(x,y)),mod(y)),mod(y)),mod(mn(x,y)),ismn1"r"(sum,pl"r"(mod(mn(x,y)),mod(y)),mod(y),compl"r"(mod(y),mod(mn(x,y)))),mnpl(mod(mn(x,y)),mod(y))):is"r"(mn"r"(sum,mod(y)),mod(mn(x,y)))
8791 t4:=satz168b(mn"r"(mod(x),mod(y)),mod(mn(x,y)),islessis2(mn"r"(sum,mod(y)),mod(mn(x,y)),mn"r"(mod(x),mod(y)),t3,t2)):moreis(mod(mn(x,y)),mn"r"(mod(x),mod(y)))
8792 t5:=ismoreis12(mod(mn(y,x)),mod(mn(x,y)),mn"r"(mod(y),mod(x)),m0"r"(mn"r"(mod(x),mod(y))),tris1(real,mod(mn(y,x)),mod(mn(x,y)),mod(m0(mn(x,y))),ismod(m0(mn(x,y)),mn(y,x),satz219(x,y)),satz272(mn(x,y))),satz181a(mod(y),mod(x)),t4(y,x)):moreis(mod(mn(x,y)),m0"r"(mn"r"(mod(x),mod(y))))
8793 @[r:real][s:real][m:moreis(r,s)][n:moreis(r,m0"r"(s))]
8794 r@t6:=th9"l.or"(neg(r),not(neg(r)),is"r"(abs(r),m0"r"(r)),is"r"(abs(r),r),th6"l.or"(neg(r)),[t:neg(r)]absn(r,t),[t:not(neg(r))]absnn(r,t)):or(is"r"(abs(r),m0"r"(r)),is"r"(abs(r),r))
8795 n@t7:=orapp(is"r"(abs(s),m0"r"(s)),is"r"(abs(s),s),moreis(r,abs(s)),t6(s),[t:is"r"(abs(s),m0"r"(s))]ismoreis2(m0"r"(s),abs(s),r,symis(real,abs(s),m0"r"(s),t),n),[t:is"r"(abs(s),s)]ismoreis2(s,abs(s),r,symis(real,abs(s),s,t),m)):moreis(r,abs(s))
8796 -7273
8797 y@satz273:=t7".7273"(mod(mn(x,y)),mn"r"(mod(x),mod(y)),t4".7273",t5".7273"):moreis(mod(mn(x,y)),abs(mn"r"(mod(x),mod(y))))
8798 -c
8799 -r
8800 -rp
8801 -rt
8802 @[x:nat][y:nat]
8803 +8274
8804 prop1:=some"l"([t:1to(x)]1to(y),[f:[t:1to(x)]1to(y)]bijective(1to(x),1to(y),f)):'prop'
8805 x@prop2:=all([y:nat]imp(less(x,y),not(prop1(y)))):'prop'
8806 @[y:nat][l:less(1,y)][f:[t:1to(1)]1to(y)]
8807 1y:=1out(y):1to(y)
8808 yy:=xout(y):1to(y)
8809 [i:is"e"(1to(y),1y,yy)]
8810 t1:=isoutne(y,1,satz24a(y),y,lessisi3(y),i):is(1,y)
8811 f@t2:=ec3e31(is(1,y),more(1,y),less(1,y),satz10b(1,y),l):nis(1,y)
8812 t3:=th3"l.imp"(is"e"(1to(y),1y,yy),is(1,y),t2,[t:is"e"(1to(y),1y,yy)]t1(t)):not(is"e"(1to(y),1y,yy))
8813 [u:1to(1)]
8814 t4:=isf(1to(1),1to(y),f,u,1o,th1"n.singlet"(u)):is"e"(1to(y),<u>f,<1o>f)
8815 f@[i:is"e"(1to(y),<1o>f,1y)][u:1to(1)]
8816 t5:=th2"e.notis"(1to(y),1y,yy,<u>f,t3,tris(1to(y),<u>f,<1o>f,1y,t4(u),i)):not(is"e"(1to(y),<u>f,yy))
8817 i@t6:=th5"l.some"(1to(1),[u:1to(1)]is"e"(1to(y),yy,<u>f),[u:1to(1)]symnotis(1to(y),<u>f,yy,t5(u))):not(image(1to(1),1to(y),f,yy))
8818 t7:=th1"l.all"(1to(y),[u:1to(y)]image(1to(1),1to(y),f,u),yy,t6):not(surjective(1to(1),1to(y),f))
8819 f@[n:not(is"e"(1to(y),<1o>f,1y))][u:1to(1)]
8820 t8:=th2"e.notis"(1to(y),<1o>f,1y,<u>f,n,t4(u)):not(is"e"(1to(y),<u>f,1y))
8821 n@t9:=th5"l.some"(1to(1),[u:1to(1)]is"e"(1to(y),1y,<u>f),[u:1to(1)]symnotis(1to(y),<u>f,1y,t8(u))):not(image(1to(1),1to(y),f,1y))
8822 t10:=th1"l.all"(1to(y),[u:1to(y)]image(1to(1),1to(y),f,u),1y,t9):not(surjective(1to(1),1to(y),f))
8823 f@t11:=th1"l.imp"(is"e"(1to(y),<1o>f,1y),not(surjective(1to(1),1to(y),f)),[t:is"e"(1to(y),<1o>f,1y)]t7(t),[t:not(is"e"(1to(y),<1o>f,1y))]t10(t)):not(surjective(1to(1),1to(y),f))
8824 t12:=th2"l.and"(injective(1to(1),1to(y),f),surjective(1to(1),1to(y),f),t11):not(bijective(1to(1),1to(y),f))
8825 l@t13:=th5"l.some"([t:1to(1)]1to(y),[f:[t:1to(1)]1to(y)]bijective(1to(1),1to(y),f),[f:[t:1to(1)]1to(y)]t12(f)):not(prop1(1,y))
8826 @t14:=[y:nat][t:less(1,y)]t13(y,t):prop2(1)
8827 x@[p:prop2(x)][y:nat][l:less(<x>suc,y)]
8828 x@xs:=<x>suc:nat
8829 l@xxs:=xout(xs):1to(<x>suc)
8830 yy1:=xout(y):1to(y)
8831 t15:=trless(1,<x>suc,y,satz24c(x),l):less(1,y)
8832 ym1:=mn(y,1,t15):nat
8833 t16:=isless12(<x>suc,pl(x,1),y,pl(ym1,1),satz4e(x),th1c"n.mn"(y,1,t15),l):less(pl(x,1),pl(ym1,1))
8834 t17:=satz20c(x,ym1,1,t16):less(x,ym1)
8835 t18:=isless2(pl(ym1,1),y,ym1,th1d"n.mn"(y,1,t15),satz18a(ym1,1)):less(ym1,y)
8836 [f:[t:1to(xs)]1to(y)][b:bijective(1to(xs),1to(y),f)]
8837 t19:=ande1(injective(1to(xs),1to(y),f),surjective(1to(xs),1to(y),f),b):injective(1to(xs),1to(y),f)
8838 t20:=ande2(injective(1to(xs),1to(y),f),surjective(1to(xs),1to(y),f),b):surjective(1to(xs),1to(y),f)
8839 [i:is"e"(1to(y),<xxs>f,yy1)][u:1to(x)]
8840 u1:=inn(x,u):nat
8841 t21:=satz16a(u1,x,xs,1top(x,u),satz18c(x)):less(u1,xs)
8842 t22:=ec3e31(is(u1,xs),more(u1,xs),less(u1,xs),satz10b(u1,xs),t21):nis(u1,xs)
8843 t23:=lessisi1(u1,xs,t21):lessis(u1,xs)
8844 u2:=outn(xs,u1,t23):1to(xs)
8845 t24:=th3"l.imp"(is"e"(1to(xs),u2,xxs),is(u1,xs),t22,[t:is"e"(1to(xs),u2,xxs)]isoutne(xs,u1,t23,xs,lessisi3(xs),t)):not(is"e"(1to(xs),u2,xxs))
8846 [j:is"e"(1to(y),<u2>f,yy1)]
8847 t25:=tris2(1to(y),<u2>f,<xxs>f,yy1,j,i):is"e"(1to(y),<u2>f,<xxs>f)
8848 t26:=isfe(1to(xs),1to(y),f,t19,u2,xxs,t25):is"e"(1to(xs),u2,xxs)
8849 u@t27:=th3"l.imp"(is"e"(1to(y),<u2>f,yy1),is"e"(1to(xs),u2,xxs),t24,[t:is"e"(1to(y),<u2>f,yy1)]t26(t)):not(is"e"(1to(y),<u2>f,yy1))
8850 w1:=inn(y,<u2>f):nat
8851 [j:is(w1,y)]
8852 t28:=tris(1to(y),<u2>f,outn(y,w1,1top(y,<u2>f)),yy1,isoutinn(y,<u2>f),isoutni(y,w1,1top(y,<u2>f),y,lessisi3(y),j)):is"e"(1to(y),<u2>f,yy1)
8853 u@t29:=th3"l.imp"(is(w1,y),is"e"(1to(y),<u2>f,yy1),t27,[t:is(w1,y)]t28(t)):nis(w1,y)
8854 t30:=ore1(less(w1,y),is(w1,y),1top(y,<u2>f),t29):less(w1,y)
8855 t31:=islessis2(y,pl(ym1,1),pl(w1,1),th1c"n.mn"(y,1,t15),satz25b(y,w1,t30)):lessis(pl(w1,1),pl(ym1,1))
8856 t32:=th9"l.or"(less(pl(w1,1),pl(ym1,1)),is(pl(w1,1),pl(ym1,1)),less(w1,ym1),is(w1,ym1),t31,[t:less(pl(w1,1),pl(ym1,1))]satz20c(w1,ym1,1,t),[t:is(pl(w1,1),pl(ym1,1))]satz20b(w1,ym1,1,t)):lessis(w1,ym1)
8857 w2:=outn(ym1,w1,t32):1to(ym1)
8858 i@f1:=[t:1to(x)]w2(t):[t:1to(x)]1to(ym1)
8859 u@[v:1to(x)][j:is"e"(1to(ym1),<u>f1,<v>f1)]
8860 t33:=isoutne(ym1,w1(u),t32(u),w1(v),t32(v),j):is(w1(u),w1(v))
8861 t34:=isinne(y,<u2(u)>f,<u2(v)>f,t33):is"e"(1to(y),<u2(u)>f,<u2(v)>f)
8862 t35:=<t34><u2(v)><u2(u)>t19:is"e"(1to(xs),u2(u),u2(v))
8863 t36:=isoutne(xs,u1(u),t23(u),u1(v),t23(v),t35):is(u1(u),u1(v))
8864 t37:=isinne(x,u,v,t36):is"e"(1to(x),u,v)
8865 i@[v:1to(ym1)]
8866 v1:=inn(ym1,v):nat
8867 t38:=satz16a(v1,ym1,y,1top(ym1,v),t18):less(v1,y)
8868 t39:=ec3e31(is(v1,y),more(v1,y),less(v1,y),satz10b(v1,y),t38):nis(v1,y)
8869 t40:=lessisi1(v1,y,t38):lessis(v1,y)
8870 v2:=outn(y,v1,t40):1to(y)
8871 w3:=<v2>invf(1to(xs),1to(y),f,b):1to(xs)
8872 t41:=thinvf2(1to(xs),1to(y),f,b,v2):is"e"(1to(y),v2,<w3>f)
8873 [j:is"e"(1to(xs),w3,xxs)]
8874 t42:=isf(1to(xs),1to(y),f,w3,xxs,j):is"e"(1to(y),<w3>f,<xxs>f)
8875 t43:=tr3is(1to(y),v2,<w3>f,<xxs>f,yy1,t41,t42,i):is"e"(1to(y),v2,yy1)
8876 t44:=isoutne(y,v1,t40,y,lessisi3(y),t43):is(v1,y)
8877 v@t45:=th3"l.imp"(is"e"(1to(xs),w3,xxs),is(v1,y),t39,[t:is"e"(1to(xs),w3,xxs)]t44(t)):not(is"e"(1to(xs),w3,xxs))
8878 w4:=inn(xs,w3):nat
8879 [j:is(w4,xs)]
8880 t46:=tris(1to(xs),w3,outn(xs,w4,1top(xs,w3)),xxs,isoutinn(xs,w3),isoutni(xs,w4,1top(xs,w3),xs,lessisi3(xs),j)):is"e"(1to(xs),w3,xxs)
8881 v@t47:=th3"l.imp"(is(w4,xs),is"e"(1to(xs),w3,xxs),t45,[t:is(w4,xs)]t46(t)):nis(w4,xs)
8882 t48:=ore1(less(w4,xs),is(w4,xs),1top(xs,w3),t47):less(w4,xs)
8883 t49:=satz26a(x,w4,t48):lessis(w4,x)
8884 w5:=outn(x,w4,t49):1to(x)
8885 t50:=isinoutn(x,w4,t49):is(w4,u1(w5))
8886 t51:=tris(1to(xs),w3,outn(xs,w4,1top(xs,w3)),u2(w5),isoutinn(xs,w3),isoutni(xs,w4,1top(xs,w3),u1(w5),t23(w5),t50)):is"e"(1to(xs),w3,u2(w5))
8887 t52:=isf(1to(xs),1to(y),f,w3,u2(w5),t51):is"e"(1to(y),<w3>f,<u2(w5)>f)
8888 t53:=tris(1to(y),v2,<w3>f,<u2(w5)>f,t41,t52):is"e"(1to(y),v2,<u2(w5)>f)
8889 t54:=tris(nat,v1,inn(y,v2),w1(w5),isinoutn(y,v1,t40),isinni(y,v2,<u2(w5)>f,t53)):is(v1,w1(w5))
8890 t55:=tris(1to(ym1),v,outn(ym1,v1,1top(ym1,v)),w2(w5),isoutinn(ym1,v),isoutni(ym1,v1,1top(ym1,v),w1(w5),t32(w5),t54)):is"e"(1to(ym1),v,<w5>f1)
8891 t56:=somei(1to(x),[t:1to(x)]is"e"(1to(ym1),v,<t>f1),w5,t55):image(1to(x),1to(ym1),f1,v)
8892 i@t57:=andi(injective(1to(x),1to(ym1),f1),surjective(1to(x),1to(ym1),f1),[u:1to(x)][v:1to(x)][t:is"e"(1to(ym1),<u>f1,<v>f1)]t37(u,v,t),[u:1to(ym1)]t56(u)):bijective(1to(x),1to(ym1),f1)
8893 t58:=somei([t:1to(x)]1to(ym1),[g:[t:1to(x)]1to(ym1)]bijective(1to(x),1to(ym1),g),f1,t57):prop1(ym1)
8894 t59:=<t58><t17><ym1>p:con
8895 b@[n:not(is"e"(1to(y),<xxs>f,yy1))]
8896 m0:=<yy1>invf(1to(xs),1to(y),f,b):1to(xs)
8897 t60:=thinvf2(1to(xs),1to(y),f,b,yy1):is"e"(1to(y),yy1,<m0>f)
8898 f2:=changef(1to(xs),1to(y),f,m0,xxs):[t:1to(xs)]1to(y)
8899 t61:=changef2(1to(xs),1to(y),f,m0,xxs,xxs,refis(1to(xs),xxs)):is"e"(1to(y),<xxs>f2,<m0>f)
8900 t62:=tris2(1to(y),<xxs>f2,yy1,<m0>f,t61,t60):is"e"(1to(y),<xxs>f2,yy1)
8901 t63:=th6"e.wissel"(1to(xs),1to(y),f,m0,xxs,b):bijective(1to(xs),1to(y),f2)
8902 t64:=t59(f2,t63,t62):con
8903 b@t65:=th1"l.imp"(is"e"(1to(y),<xxs>f,yy1),con,[t:is"e"(1to(y),<xxs>f,yy1)]t59(t),[t:not(is"e"(1to(y),<xxs>f,yy1))]t64(t)):con
8904 l@t65a:=th5"l.some"([t:1to(xs)]1to(y),[f:[t:1to(xs)]1to(y)]bijective(1to(xs),1to(y),f),[f:[t:1to(xs)]1to(y)][t:bijective(1to(xs),1to(y),f)]t65(f,t)):not(prop1(xs,y))
8905 p@t66:=[y:nat][t:less(xs,y)]t65a(y,t):prop2(xs)
8906 x@t67:=induction([t:nat]prop2(t),t14,[t:nat][u:prop2(t)]t66(t,u),x):prop2(x)
8907 -8274
8908 [l:less(x,y)]
8909 satz274:=<l><y>t67".8274":not(some"l"([t:1to(x)]1to(y),[f:[t:1to(x)]1to(y)]bijective(1to(x),1to(y),f)))
8910 [f:[t:1to(x)]1to(y)]
8911 satz274a:=th4"l.some"([t:1to(x)]1to(y),[g:[t:1to(x)]1to(y)]bijective(1to(x),1to(y),g),satz274,f):not(bijective(1to(x),1to(y),f))
8912 +*rt
8913 +*rp
8914 +*r
8915 +*c
8916 @[x:nat][u:1to(x)]
8917 inn:=inn"n"(x,u):nat
8918 @[q:[t:cx][u:cx]cx][x:nat][f:[t:1to(x)]cx]
8919 x@[n:1to(x)][o:not(is"e"(1to(x),n,xout(x)))]
8920 +8275
8921 t1:=th3"l.imp"(is"n"(inn(x,n),x),is"e"(1to(x),n,xout(x)),o,[t:is"n"(inn(x,n),x)]tris(1to(x),n,outn(x,inn(x,n),1top(x,n)),xout(x),isoutinn(x,n),isoutni(x,inn(x,n),1top(x,n),x,lessisi3(x),t))):not(is"n"(inn(x,n),x))
8922 t2:=ore1(less"n"(inn(x,n),x),is"n"(inn(x,n),x),1top(x,n),t1):less"n"(inn(x,n),x)
8923 -8275
8924 lemma275:=satz25c(x,inn(x,n),t2".8275"):lessis"n"(<inn(x,n)>suc,x)
8925 f@[g:[t:1to(x)]cx]
8926 recprop:=and(is(<1out(x)>g,<1out(x)>f),[t:1to(x)][u:not(is"e"(1to(x),t,xout(x)))]is(<outn(x,<inn(x,t)>suc,lemma275(t,u))>g,<<outn(x,<inn(x,t)>suc,lemma275(t,u))>f><<t>g>q)):'prop'
8927 +*8275
8928 x@1o:=1out(x):1to(x)
8929 xo:=xout(x):1to(x)
8930 [u:nat][l:lessis"n"(<u>suc,x)]
8931 t11:=satz16b(u,<u>suc,x,satz18c(u),l):less"n"(u,x)
8932 t12:=lessisi1"n"(u,x,t11):lessis"n"(u,x)
8933 ux:=outn(x,u,t12):1to(x)
8934 t13:=ec3e31(is"n"(u,x),more"n"(u,x),less"n"(u,x),satz10b(u,x),t11):nis"n"(u,x)
8935 t14:=th3"l.imp"(is"e"(1to(x),ux,xo),is"n"(u,x),t13,[t:is"e"(1to(x),ux,xo)]isoutne(x,u,t12,x,lessisi3(x),t)):not(is"e"(1to(x),ux,xo))
8936 t15:=isf(nat,nat,suc,u,inn(x,ux),isinoutn(x,u,t12)):is"n"(<u>suc,<inn(x,ux)>suc)
8937 t16:=isoutni(x,<u>suc,l,<inn(x,ux)>suc,lemma275(ux,t14),t15):is"e"(1to(x),outn(x,<u>suc,l),outn(x,<inn(x,ux)>suc,lemma275(ux,t14)))
8938 x@[n:1to(x)][o:not(is"e"(1to(x),n,xo))]
8939 ns:=outn(x,<inn(x,n)>suc,lemma275(n,o)):1to(x)
8940 f@[g:[t:1to(x)]cx]
8941 prop1:=is(<1o>g,<1o>f):'prop'
8942 prop2:=[t:1to(x)][u:not(is"e"(1to(x),t,xo))]is(<ns(t,u)>g,<<ns(t,u)>f><<t>g>q):'prop'
8943 [pg:recprop(g)]
8944 t3:=ande1(prop1,prop2,pg):prop1
8945 [n:1to(x)][o:not(is"e"(1to(x),n,xo))]
8946 t4:=<o><n>ande2(prop1,prop2,pg):is(<ns(n,o)>g,<<ns(n,o)>f><<n>g>q)
8947 pg@[u:nat][l:lessis"n"(<u>suc,x)]
8948 t17:=isf(1to(x),cx,g,outn(x,<u>suc,l),ns(ux(u,l),t14(u,l)),t16(u,l)):is(<outn(x,<u>suc,l)>g,<ns(ux(u,l),t14(u,l))>g)
8949 t18:=tris(cx,<outn(x,<u>suc,l)>g,<ns(ux(u,l),t14(u,l))>g,<<ns(ux(u,l),t14(u,l))>f><<ux(u,l)>g>q,t17,t4(pg,ux(u,l),t14(u,l))):is(<outn(x,<u>suc,l)>g,<<ns(ux(u,l),t14(u,l))>f><<ux(u,l)>g>q)
8950 g@[h:[t:1to(x)]cx][u:nat][l:lessis"n"(u,x)]
8951 prop3:=is(<outn(x,u,l)>g,<outn(x,u,l)>h):'prop'
8952 u@prop4:=and(lessis"n"(u,x),[t:lessis"n"(u,x)]prop3(t)):'prop'
8953 prop5:=or(prop4,more"n"(u,x)):'prop'
8954 h@[pg:recprop(g)][ph:recprop(h)][l:lessis"n"(1,x)]
8955 t5:=isoutni(x,1,l,1,satz24a(x),refis(nat,1)):is"e"(1to(x),outn(x,1,l),1o)
8956 t6:=isf(1to(x),cx,g,outn(x,1,l),1o,t5):is(<outn(x,1,l)>g,<1o>g)
8957 t7:=tris(cx,<outn(x,1,l)>g,<1o>g,<1o>f,t6,t3(pg)):is(<outn(x,1,l)>g,<1o>f)
8958 t8:=tris2(cx,<outn(x,1,l)>g,<outn(x,1,l)>h,<1o>f,t7,t7(h,g,ph,pg,l)):prop3(1,l)
8959 ph@t9:=andi(lessis"n"(1,x),[t:lessis"n"(1,x)]prop3(1,t),satz24a(x),[t:lessis"n"(1,x)]t8(t)):prop4(1)
8960 t10:=ori1(prop4(1),more"n"(1,x),t9):prop5(1)
8961 [u:nat][p:prop5(u)][l:lessis"n"(<u>suc,x)]
8962 t19:=ec3e32(is"n"(u,x),more"n"(u,x),less"n"(u,x),satz10b(u,x),t11(u,l)):not(more"n"(u,x))
8963 t20:=ore1(prop4(u),more"n"(u,x),p,t19):prop4(u)
8964 t21:=<t12(u,l)>ande2(lessis"n"(u,x),[t:lessis"n"(u,x)]prop3(u,t),t20):prop3(u,t12(u,l))
8965 t22:=isf(cx,cx,[t:cx]<<ns(ux(u,l),t14(u,l))>f><t>q,<ux(u,l)>g,<ux(u,l)>h,t21):is(<<ns(ux(u,l),t14(u,l))>f><<ux(u,l)>g>q,<<ns(ux(u,l),t14(u,l))>f><<ux(u,l)>h>q)
8966 t23:=symis(cx,<outn(x,<u>suc,l)>h,<<ns(ux(u,l),t14(u,l))>f><<ux(u,l)>h>q,t18(h,ph,u,l)):is(<<ns(ux(u,l),t14(u,l))>f><<ux(u,l)>h>q,<outn(x,<u>suc,l)>h)
8967 t24:=tr3is(cx,<outn(x,<u>suc,l)>g,<<ns(ux(u,l),t14(u,l))>f><<ux(u,l)>g>q,<<ns(ux(u,l),t14(u,l))>f><<ux(u,l)>h>q,<outn(x,<u>suc,l)>h,t18(g,pg,u,l),t22,t23):prop3(<u>suc,l)
8968 t25:=andi(lessis"n"(<u>suc,x),[t:lessis"n"(<u>suc,x)]prop3(<u>suc,t),l,[t:lessis"n"(<u>suc,x)]t24(t)):prop4(<u>suc)
8969 t26:=ori1(prop4(<u>suc),more"n"(<u>suc,x),t25):prop5(<u>suc)
8970 p@[n:not(lessis"n"(<u>suc,x))]
8971 t27:=satz10k(<u>suc,x,n):more"n"(<u>suc,x)
8972 t28:=ori2(prop4(<u>suc),more"n"(<u>suc,x),t27):prop5(<u>suc)
8973 p@t29:=th1"l.imp"(lessis"n"(<u>suc,x),prop5(<u>suc),[t:lessis"n"(<u>suc,x)]t26(t),[t:not(lessis"n"(<u>suc,x))]t28(t)):prop5(<u>suc)
8974 u@t30:=induction([v:nat]prop5(v),t10,[v:nat][t:prop5(v)]t29(v,t),u):prop5(u)
8975 ph@[n:1to(x)]
8976 t31:=isf(1to(x),cx,g,n,outn(x,inn(x,n),1top(x,n)),isoutinn(x,n)):is(<n>g,<outn(x,inn(x,n),1top(x,n))>g)
8977 t32:=satz10d(inn(x,n),x,1top(x,n)):not(more"n"(inn(x,n),x))
8978 t33:=ore1(prop4(inn(x,n)),more"n"(inn(x,n),x),t30(inn(x,n)),t32):prop4(inn(x,n))
8979 t34:=<1top(x,n)>ande2(lessis"n"(inn(x,n),x),[t:lessis"n"(inn(x,n),x)]prop3(inn(x,n),t),t33):prop3(inn(x,n),1top(x,n))
8980 t35:=symis(cx,<n>h,<outn(x,inn(x,n),1top(x,n))>h,t31(h,g,ph,pg,n)):is(<outn(x,inn(x,n),1top(x,n))>h,<n>h)
8981 t36:=tr3is(cx,<n>g,<outn(x,inn(x,n),1top(x,n))>g,<outn(x,inn(x,n),1top(x,n))>h,<n>h,t31,t34,t35):is(<n>g,<n>h)
8982 ph@t37:=fisi(1to(x),cx,g,h,[t:1to(x)]t36(t)):is"e"([t:1to(x)]cx,g,h)
8983 f@prop6:=some"l"([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(g)):'prop'
8984 x@prop7:=all"l"([t:1to(x)]cx,[f:[t:1to(x)]cx]prop6(f)):'prop'
8985 q@[f:[t:1to(1)]cx]
8986 t38:=refis(cx,<1o(1)>f):prop1(1,f,f)
8987 [n:1to(1)][o:not(is"e"(1to(1),n,xo(1)))]
8988 t39:=<th1"n.singlet"(n)>o:con
8989 t40:=cone(is(<ns(1,n,o)>f,<<ns(1,n,o)>f><<n>f>q),t39):is(<ns(1,n,o)>f,<<ns(1,n,o)>f><<n>f>q)
8990 f@t41:=andi(prop1(1,f,f),prop2(1,f,f),t38,[t:1to(1)][u:not(is"e"(1to(1),t,xo(1)))]t40(t,u)):recprop(1,f,f)
8991 t42:=somei([t:1to(1)]cx,[g:[t:1to(1)]cx]recprop(1,f,g),f,t41):prop6(1,f)
8992 q@t43:=[f:[t:1to(1)]cx]t42(f):prop7(1)
8993 x@[p:prop7(x)]
8994 xs:=<x>suc:nat
8995 [f:[t:1to(xs)]cx]
8996 f1:=left(cx,xs,x,lessisi1"n"(x,xs,satz18c(x)),f):[t:1to(x)]cx
8997 t44:=onei([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(x,f1,g),[g:[t:1to(x)]cx][h:[t:1to(x)]cx][u:recprop(x,f1,g)][v:recprop(x,f1,h)]t37(f1,g,h,u,v),<f1>p):one"e"([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(x,f1,g))
8998 g1:=ind([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(x,f1,g),t44):[t:1to(x)]cx
8999 t45:=oneax([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(x,f1,g),t44):recprop(x,f1,g1)
9000 [n:1to(xs)]
9001 nxs:=is"e"(1to(xs),n,xo(xs)):'prop'
9002 [o:not(nxs)]
9003 t46:=satz26a(x,inn(xs,n),t2(xs,n,o)):lessis"n"(inn(xs,n),x)
9004 n1:=outn(x,inn(xs,n),t46):1to(x)
9005 b:=<n1>g1:cx
9006 [o1:not(nxs)]
9007 t47:=isoutni(x,inn(xs,n),t46(o),inn(xs,n),t46(o1),refis(nat,inn(xs,n))):is"e"(1to(x),n1(o),n1(o1))
9008 t48:=isf(1to(x),cx,g1,n1(o),n1(o1),t47):is(b(o),b(o1))
9009 f@a:=<<xo(xs)>f><<xo(x)>g1>q:cx
9010 n@c:=ite"l.r"(nxs,cx,[t:nxs]a,[t:not(nxs)]b(t),[t:nxs][u:nxs]refis(cx,a),[t:not(nxs)][u:not(nxs)]t48(t,u)):cx
9011 [i:nxs]
9012 t49:=itet"l.r"(nxs,cx,[t:nxs]a,[t:not(nxs)]b(t),[t:nxs][u:nxs]refis(cx,a),[t:not(nxs)][u:not(nxs)]t48(t,u),i):is(c,a)
9013 o@t50:=itef"l.r"(nxs,cx,[t:nxs]a,[t:not(nxs)]b(t),[t:nxs][u:nxs]refis(cx,a),[t:not(nxs)][u:not(nxs)]t48(t,u),o):is(c,b)
9014 f@g2:=[t:1to(xs)]c(t):[t:1to(xs)]cx
9015 t51:=th3"l.imp"(is"e"(1to(xs),1o(xs),xo(xs)),is"n"(1,xs),symnotis(nat,xs,1,<x>ax3),[t:is"e"(1to(xs),1o(xs),xo(xs))]isoutne(xs,1,satz24a(xs),xs,lessisi3(xs),t)):not(is"e"(1to(xs),1o(xs),xo(xs)))
9016 t52:=t50(1o(xs),t51):is(c(1o(xs)),b(1o(xs),t51))
9017 t53:=isinoutn(xs,1,satz24a(xs)):is"n"(1,inn(xs,1o(xs)))
9018 t54:=isoutni(x,1,satz24a(x),inn(xs,1o(xs)),t46(1o(xs),t51),t53):is"e"(1to(x),1o(x),n1(1o(xs),t51))
9019 t55:=isf(1to(x),cx,g1,1o(x),n1(1o(xs),t51),t54):is(<1o(x)>g1,b(1o(xs),t51))
9020 t56:=tris2(cx,c(1o(xs)),<1o(x)>g1,b(1o(xs),t51),t52,t55):is(c(1o(xs)),<1o(x)>g1)
9021 t57:=tris(cx,c(1o(xs)),<1o(x)>g1,<1o(x)>f1,t56,t3(x,f1,g1,t45)):is(c(1o(xs)),<1o(x)>f1)
9022 t58:=isinoutn(x,1,satz24a(x)):is"n"(1,inn(x,1o(x)))
9023 t59:=isoutni(xs,1,satz24a(xs),inn(x,1o(x)),trlessis"n"(inn(x,1o(x)),x,xs,1top(x,1o(x)),lessisi1"n"(x,xs,satz18c(x))),t58):is"e"(1to(xs),1o(xs),left1to(xs,x,lessisi1"n"(x,xs,satz18c(x)),1o(x)))
9024 t60:=isf(1to(xs),cx,f,1o(xs),left1to(xs,x,lessisi1"n"(x,xs,satz18c(x)),1o(x)),t59):is(<1o(xs)>f,<1o(x)>f1)
9025 t61:=tris2(cx,c(1o(xs)),<1o(xs)>f,<1o(x)>f1,t57,t60):prop1(xs,f,g2)
9026 o@[i:is"e"(1to(xs),ns(xs,n,o),xo(xs))]
9027 t62:=isoutne(xs,<inn(xs,n)>suc,lemma275(xs,n,o),xs,lessisi3(xs),i):is"n"(<inn(xs,n)>suc,xs)
9028 t63:=<t62><x><inn(xs,n)>ax4:is"n"(inn(xs,n),x)
9029 t64:=isoutni(x,inn(xs,n),t46,x,lessisi3(x),t63):is"e"(1to(x),n1,xo(x))
9030 t65:=isf(1to(x),cx,g1,xo(x),n1,symis(1to(x),n1,xo(x),t64)):is(<xo(x)>g1,b)
9031 t66:=tris2(cx,<xo(x)>g1,c,b,t65,t50):is(<xo(x)>g1,c)
9032 t67:=isf(cx,cx,[t:cx]<<xo(xs)>f><t>q,<xo(x)>g1,c,t66):is(a,<<xo(xs)>f><c>q)
9033 t68:=isf(1to(xs),cx,f,xo(xs),ns(xs,n,o),symis(1to(xs),ns(xs,n,o),xo(xs),i)):is(<xo(xs)>f,<ns(xs,n,o)>f)
9034 t69:=isf(cx,cx,<c>q,<xo(xs)>f,<ns(xs,n,o)>f,t68):is(<<xo(xs)>f><c>q,<<ns(xs,n,o)>f><c>q)
9035 t70:=tr3is(cx,c(ns(xs,n,o)),a,<<xo(xs)>f><c>q,<<ns(xs,n,o)>f><c>q,t49(ns(xs,n,o),i),t67,t69):is(c(ns(xs,n,o)),<<ns(xs,n,o)>f><c>q)
9036 o@[o1:not(is"e"(1to(xs),ns(xs,n,o),xo(xs)))][i:is"e"(1to(x),n1,xo(x))]
9037 t71:=isoutne(x,inn(xs,n),t46,x,lessisi3(x),i):is"n"(inn(xs,n),x)
9038 t72:=ax2(inn(xs,n),x,t71):is"n"(<inn(xs,n)>suc,xs)
9039 t73:=isoutni(xs,<inn(xs,n)>suc,lemma275(xs,n,o),xs,lessisi3(xs),t72):is"e"(1to(xs),ns(xs,n,o),xo(xs))
9040 o1@t74:=th3"l.imp"(is"e"(1to(x),n1,xo(x)),is"e"(1to(xs),ns(xs,n,o),xo(xs)),o1,[t:is"e"(1to(x),n1,xo(x))]t73(t)):not(is"e"(1to(x),n1,xo(x)))
9041 t75:=isinoutn(x,inn(xs,n),t46):is"n"(inn(xs,n),inn(x,n1))
9042 t76:=ax2(inn(xs,n),inn(x,n1),t75):is"n"(<inn(xs,n)>suc,<inn(x,n1)>suc)
9043 t77:=isinoutn(xs,<inn(xs,n)>suc,lemma275(xs,n,o)):is"n"(<inn(xs,n)>suc,inn(xs,ns(xs,n,o)))
9044 t78:=tris1(nat,inn(xs,ns(xs,n,o)),<inn(x,n1)>suc,<inn(xs,n)>suc,t77,t76):is"n"(inn(xs,ns(xs,n,o)),<inn(x,n1)>suc)
9045 t79:=isoutni(x,inn(xs,ns(xs,n,o)),t46(ns(xs,n,o),o1),<inn(x,n1)>suc,lemma275(x,n1,t74),t78):is"e"(1to(x),n1(ns(xs,n,o),o1),ns(n1,t74))
9046 t80:=isf(1to(x),cx,g1,n1(ns(xs,n,o),o1),ns(n1,t74),t79):is(b(ns(xs,n,o),o1),<ns(n1,t74)>g1)
9047 t81:=isinoutn(x,<inn(x,n1)>suc,lemma275(x,n1,t74)):is"n"(<inn(x,n1)>suc,inn(x,ns(n1,t74)))
9048 t82:=tris(nat,<inn(xs,n)>suc,<inn(x,n1)>suc,inn(x,ns(n1,t74)),t76,t81):is"n"(<inn(xs,n)>suc,inn(x,ns(n1,t74)))
9049 t83:=isoutni(xs,<inn(xs,n)>suc,lemma275(xs,n,o),inn(x,ns(n1,t74)),trlessis"n"(inn(x,ns(n1,t74)),x,xs,1top(x,ns(n1,t74)),lessisi1"n"(x,xs,satz18c(x))),t82):is"e"(1to(xs),ns(xs,n,o),left1to(xs,x,lessisi1"n"(x,xs,satz18c(x)),ns(n1,t74)))
9050 t84:=isf(1to(xs),cx,f,ns(xs,n,o),left1to(xs,x,lessisi1"n"(x,xs,satz18c(x)),ns(n1,t74)),t83):is(<ns(xs,n,o)>f,<ns(n1,t74)>f1)
9051 t85:=isf(cx,cx,<b>q,<ns(xs,n,o)>f,<ns(n1,t74)>f1,t84):is(<<ns(xs,n,o)>f><b>q,<<ns(n1,t74)>f1><b>q)
9052 t86:=isf(cx,cx,[t:cx]<<ns(xs,n,o)>f><t>q,c,b,t50):is(<<ns(xs,n,o)>f><c>q,<<ns(xs,n,o)>f><b>q)
9053 t87:=tr3is(cx,c(ns(xs,n,o)),b(ns(xs,n,o),o1),<ns(n1,t74)>g1,<<ns(n1,t74)>f1><b>q,t50(ns(xs,n,o),o1),t80,t4(f1,g1,t45,n1,t74)):is(c(ns(xs,n,o)),<<ns(n1,t74)>f1><b>q)
9054 t88:=tris(cx,<<ns(xs,n,o)>f><c>q,<<ns(xs,n,o)>f><b>q,<<ns(n1,t74)>f1><b>q,t86,t85):is(<<ns(xs,n,o)>f><c>q,<<ns(n1,t74)>f1><b>q)
9055 t89:=tris2(cx,c(ns(xs,n,o)),<<ns(xs,n,o)>f><c>q,<<ns(n1,t74)>f1><b>q,t87,t88):is(c(ns(xs,n,o)),<<ns(xs,n,o)>f><c>q)
9056 o@t90:=th1"l.imp"(is"e"(1to(xs),ns(xs,n,o),xo(xs)),is(c(ns(xs,n,o)),<<ns(xs,n,o)>f><c>q),[t:is"e"(1to(xs),ns(xs,n,o),xo(xs))]t70(t),[t:not(is"e"(1to(xs),ns(xs,n,o),xo(xs)))]t89(t)):is(c(ns(xs,n,o)),<<ns(xs,n,o)>f><c>q)
9057 f@t91:=[t:1to(xs)][u:not(nxs(t))]t90(t,u):prop2(xs,f,g2)
9058 t92:=andi(prop1(xs,f,g2),prop2(xs,f,g2),t61,t91):recprop(xs,f,g2)
9059 t93:=somei([t:1to(xs)]cx,[g:[t:1to(xs)]cx]recprop(xs,f,g),g2,t92):prop6(xs,f)
9060 p@t94:=[f:[t:1to(xs)]cx]t93(f):prop7(xs)
9061 x@t95:=induction([y:nat]prop7(y),t43,[y:nat][t:prop7(y)]t94(y,t),x):prop7(x)
9062 [f:[t:1to(x)]cx]
9063 t96:=<f>t95:prop6(x,f)
9064 t97:=onei([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(x,f,g),[g:[t:1to(x)]cx][h:[t:1to(x)]cx][u:recprop(x,f,g)][v:recprop(x,f,h)]t37(f,g,h,u,v),t96):one"e"([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(x,f,g))
9065 -8275
9066 f@satz275:=t97".8275"(f):one"e"([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(x,f,g))
9067 recf:=ind([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(x,f,g),satz275):[t:1to(x)]cx
9068 satz275a:=oneax([t:1to(x)]cx,[g:[t:1to(x)]cx]recprop(x,f,g),satz275):recprop(x,f,recf)
9069 [n:1to(x)]
9070 rec:=<n>recf:cx
9071 f@satz275b:=t3".8275"(x,f,recf,satz275a):is(rec(1out(x)),<1out(x)>f)
9072 n@[o:not(is"e"(1to(x),n,xout(x)))]
9073 sucx:=ns".8275"(n,o):1to(x)
9074 satz275c:=t4".8275"(x,f,recf,satz275a,n,o):is(rec(sucx(n,o)),<<sucx(n,o)>f><rec(n)>q)
9075 f@[g:[t:1to(x)]cx][r:recprop(x,f,g)]
9076 satz275d:=t37".8275"(x,f,g,recf,r,satz275a):is"e"([t:1to(x)]cx,g,recf)
9077 [n:1to(x)]
9078 satz275e:=fise(1to(x),cx,g,recf,satz275d,n):is(<n>g,rec(n))
9079 x@[y:nat]
9080 +*8275
9081 y@[l:lessis"n"(y,x)][f:[t:1to(x)]cx]
9082 fl:=left(cx,x,y,l,f):[t:1to(y)]cx
9083 rf:=recf(x,f):[t:1to(x)]cx
9084 rfl:=left(cx,x,y,l,rf):[t:1to(y)]cx
9085 t98:=isinoutn(y,1,satz24a(y)):is"n"(1,inn(y,1out(y)))
9086 t99:=isoutni(x,1,satz24a(x),inn(y,1out(y)),trlessis"n"(inn(y,1out(y)),y,x,1top(y,1out(y)),l),t98):is"e"(1to(x),1out(x),left1to(x,y,l,1out(y)))
9087 t100:=isp(1to(x),[t:1to(x)]is(<t>rf,<t>f),1out(x),left1to(x,y,l,1out(y)),t3(x,f,rf,satz275a(x,f)),t99):prop1(y,fl,rfl)
9088 [n:1to(y)][o:not(is"e"(1to(y),n,xout(y)))]
9089 t100a:=th3"l.imp"(is"n"(inn(y,n),y),is"e"(1to(y),n,xout(y)),o,[t:is"n"(inn(y,n),y)]tris(1to(y),n,outn(y,inn(y,n),1top(y,n)),xout(y),isoutinn(y,n),isoutni(y,inn(y,n),1top(y,n),y,lessisi3(y),t))):not(is"n"(inn(y,n),y))
9090 t100b:=ore1(less"n"(inn(y,n),y),is"n"(inn(y,n),y),1top(y,n),t100a):less"n"(inn(y,n),y)
9091 t101:=satz16b(inn(y,n),y,x,t100b,l):less"n"(inn(y,n),x)
9092 t102:=ec3e31(is"n"(inn(y,n),x),more"n"(inn(y,n),x),less"n"(inn(y,n),x),satz10b(inn(y,n),x),t101):nis"n"(inn(y,n),x)
9093 t103:=th3"l.imp"(is"e"(1to(x),left1to(x,y,l,n),xout(x)),is"n"(inn(y,n),x),t102,[t:is"e"(1to(x),left1to(x,y,l,n),xout(x))]isoutne(x,inn(y,n),trlessis"n"(inn(y,n),y,x,1top(y,n),l),x,lessisi3(x),t)):not(is"e"(1to(x),left1to(x,y,l,n),xout(x)))
9094 t104:=t4(x,f,rf,satz275a(x,f),left1to(x,y,l,n),t103):is(<ns(x,left1to(x,y,l,n),t103)>rf,<<ns(x,left1to(x,y,l,n),t103)>f><<n>rfl>q)
9095 t105:=isinoutn(x,inn(y,n),trlessis"n"(inn(y,n),y,x,1top(y,n),l)):is"n"(inn(y,n),inn(x,left1to(x,y,l,n)))
9096 t106:=ax2(inn(y,n),inn(x,left1to(x,y,l,n)),t105):is"n"(<inn(y,n)>suc,<inn(x,left1to(x,y,l,n))>suc)
9097 t107:=isinoutn(y,<inn(y,n)>suc,lemma275(y,n,o)):is"n"(<inn(y,n)>suc,inn(y,ns(y,n,o)))
9098 t108:=tris1(nat,<inn(x,left1to(x,y,l,n))>suc,inn(y,ns(y,n,o)),<inn(y,n)>suc,t106,t107):is"n"(<inn(x,left1to(x,y,l,n))>suc,inn(y,ns(y,n,o)))
9099 t109:=isoutni(x,<inn(x,left1to(x,y,l,n))>suc,lemma275(x,left1to(x,y,l,n),t103),inn(y,ns(y,n,o)),trlessis"n"(inn(y,ns(y,n,o)),y,x,1top(y,ns(y,n,o)),l),t108):is"e"(1to(x),ns(x,left1to(x,y,l,n),t103),left1to(x,y,l,ns(y,n,o)))
9100 t110:=isp(1to(x),[t:1to(x)]is(<t>rf,<<t>f><<n>rfl>q),ns(x,left1to(x,y,l,n),t103),left1to(x,y,l,ns(y,n,o)),t104,t109):is(<ns(y,n,o)>rfl,<<ns(y,n,o)>fl><<n>rfl>q)
9101 f@t111:=[t:1to(y)][u:not(is"e"(1to(y),t,xout(y)))]t110(t,u):prop2(y,fl,rfl)
9102 t112:=andi(prop1(y,fl,rfl),prop2(y,fl,rfl),t100,t111):recprop(y,fl,rfl)
9103 -8275
9104 y@[l:lessis"n"(y,x)][f:[t:1to(x)]cx]
9105 satz275f:=satz275d(y,fl".8275"(l,f),rfl".8275"(l,f),t112".8275"(l,f)):is"e"([t:1to(y)]cx,left(cx,x,y,l,recf(x,f)),recf(y,left(cx,x,y,l,f)))
9106 x@[f:[t:1to(pl"n"(x,1))]cx]
9107 +8276
9108 xs:=<x>suc:nat
9109 x1:=pl"n"(x,1):nat
9110 t1:=lessisi1"n"(x,x1,satz18a(x,1)):lessis"n"(x,x1)
9111 t2:=lessisi1"n"(x,xs,satz18c(x)):lessis"n"(x,xs)
9112 t3:=lessisi2"n"(xs,x1,satz4e(x)):lessis"n"(xs,x1)
9113 fx:=left(cx,x1,x,t1,f):[t:1to(x)]cx
9114 f1:=left(cx,x1,xs,t3,f):[t:1to(xs)]cx
9115 f1x:=f1"c.8275"(x,t95"c.8275"(x),f1):[t:1to(x)]cx
9116 g1:=g2"c.8275"(x,t95"c.8275"(x),f1):[t:1to(xs)]cx
9117 g1x:=g1"c.8275"(x,t95"c.8275"(x),f1):[t:1to(x)]cx
9118 g:=left(cx,x1,xs,t3,recf(x1,f)):[t:1to(xs)]cx
9119 t4:=t49"c.8275"(x,t95"c.8275"(x),f1,xout(xs),refis(1to(xs),xout(xs))):is(<xout(xs)>g1,<<xout(xs)>f1><<xout(x)>g1x>q)
9120 t5:=satz275d(xs,f1,g1,t92"c.8275"(x,t95"c.8275"(x),f1)):is"e"([t:1to(xs)]cx,g1,recf(xs,f1))
9121 t6:=satz275f(x1,xs,t3,f):is"e"([t:1to(xs)]cx,g,recf(xs,f1))
9122 t7:=tris2([t:1to(xs)]cx,g,g1,recf(xs,f1),t6,t5):is"e"([t:1to(xs)]cx,g,g1)
9123 t8:=fise(1to(xs),cx,g,g1,t7,xout(xs)):is(<xout(xs)>g,<xout(xs)>g1)
9124 t9:=tris(cx,<xout(xs)>g,<xout(xs)>g1,<<xout(xs)>f1><<xout(x)>g1x>q,t8,t4):is(<xout(xs)>g,<<xout(xs)>f1><<xout(x)>g1x>q)
9125 [n:1to(x)]
9126 t10:=isinoutn(xs,inn(x,n),trlessis"n"(inn(x,n),x,xs,1top(x,n),t2)):is"n"(inn(x,n),inn(xs,left1to(xs,x,t2,n)))
9127 t11:=isoutni(x1,inn(x,n),trlessis"n"(inn(x,n),x,x1,1top(x,n),t1),inn(xs,left1to(xs,x,t2,n)),trlessis"n"(inn(xs,left1to(xs,x,t2,n)),xs,x1,1top(xs,left1to(xs,x,t2,n)),t3),t10):is"e"(1to(x1),left1to(x1,x,t1,n),left1to(x1,xs,t3,left1to(xs,x,t2,n)))
9128 t12:=isf(1to(x1),cx,f,left1to(x1,x,t1,n),left1to(x1,xs,t3,left1to(xs,x,t2,n)),t11):is(<n>fx,<n>f1x)
9129 f@t13:=fisi(1to(x),cx,fx,f1x,[t:1to(x)]t12(t)):is"e"([t:1to(x)]cx,fx,f1x)
9130 t14:=isf([t:1to(x)]cx,[t:1to(x)]cx,[u:[t:1to(x)]cx]recf(x,u),fx,f1x,t13):is"e"([t:1to(x)]cx,recf(x,fx),recf(x,f1x))
9131 t15:=satz275d(x,f1x,g1x,t45"c.8275"(x,t95"c.8275"(x),f1)):is"e"([t:1to(x)]cx,g1x,recf(x,f1x))
9132 t16:=tris2([t:1to(x)]cx,g1x,recf(x,fx),recf(x,f1x),t15,t14):is"e"([t:1to(x)]cx,g1x,recf(x,fx))
9133 t17:=fise(1to(x),cx,g1x,recf(x,fx),t16,xout(x)):is(<xout(x)>g1x,<xout(x)>recf(x,fx))
9134 t18:=isinoutn(xs,xs,lessisi3(xs)):is"n"(xs,inn(xs,xout(xs)))
9135 t19:=tris(nat,x1,xs,inn(xs,xout(xs)),satz4a(x),t18):is"n"(x1,inn(xs,xout(xs)))
9136 t20:=isoutni(x1,x1,lessisi3(x1),inn(xs,xout(xs)),trlessis"n"(inn(xs,xout(xs)),xs,x1,1top(xs,xout(xs)),t3),t19):is"e"(1to(x1),xout(x1),left1to(x1,xs,t3,xout(xs)))
9137 t21:=isp1(1to(x1),[t:1to(x1)]is(<t>recf(x1,f),<<t>f><<xout(x)>g1x>q),left1to(x1,xs,t3,xout(xs)),xout(x1),t9,t20):is(<xout(x1)>recf(x1,f),<<xout(x1)>f><<xout(x)>g1x>q)
9138 t22:=isf(cx,cx,[t:cx]<<xout(x1)>f><t>q,<xout(x)>g1x,<xout(x)>recf(x,fx),t17):is(<<xout(x1)>f><<xout(x)>g1x>q,<<xout(x1)>f><<xout(x)>recf(x,fx)>q)
9139 -8276
9140 satz276:=tris(cx,<xout(pl"n"(x,1))>recf(pl"n"(x,1),f),<<xout(pl"n"(x,1))>f><<xout(x)>g1x".8276">q,<<xout(pl"n"(x,1))>f><<xout(x)>recf(x,fx".8276")>q,t21".8276",t22".8276"):is(<xout(pl"n"(x,1))>recf(pl"n"(x,1),f),<<xout(pl"n"(x,1))>f><<xout(x)>recf(x,left(cx,pl"n"(x,1),x,lessisi1"n"(x,pl"n"(x,1),satz18a(x,1)),f))>q)
9141 x@[f:[u:1to(x)]cx]
9142 smpr:=rec(x,f,xout(x)):cx
9143 @[x:nat][f:[u:1to(x)]cx]
9144 sum:=smpr([t:cx][u:cx]pl(t,u),x,f):cx
9145 prod:=smpr([t:cx][u:cx]ts(t,u),x,f):cx
9146 q@[f:[u:1to(1)]cx]
9147 +8277
9148 t1:=isoutni(1,1,satz24a(1),1,lessisi3(1),refis(nat,1)):is"e"(1to(1),1out(1),xout(1))
9149 -8277
9150 satz277:=isp(1to(1),[t:1to(1)]is(rec(1,f,t),<t>f),1out(1),xout(1),satz275b(1,f),t1".8277"):is(smpr(1,f),<xout(1)>f)
9151 q@[x:nat][f:[u:1to(pl"n"(x,1))]cx]
9152 satz278:=satz276(x,f):is(smpr(pl"n"(x,1),f),<<xout(pl"n"(x,1))>f><smpr(x,left(cx,pl"n"(x,1),x,lessisi1"n"(x,pl"n"(x,1),satz18a(x,1)),f))>q)
9153 x@[f:[u:1to(x)]cx][y:nat][i:is"n"(y,x)]
9154 +v8
9155 t1:=lessisi2"n"(y,x,i):lessis"n"(y,x)
9156 f0:=left(cx,x,y,t1,f):[t:1to(y)]cx
9157 t2:=fise(1to(y),cx,left(cx,x,y,t1,recf(x,f)),recf(y,f0),satz275f(x,y,t1,f),xout(y)):is(<left1to(x,y,t1,xout(y))>recf(x,f),smpr(y,f0))
9158 t3:=tris1(nat,inn(y,xout(y)),x,y,isinoutn(y,y,lessisi3(y)),i):is"n"(inn(y,xout(y)),x)
9159 t4:=isoutni(x,inn(y,xout(y)),trlessis"n"(inn(y,xout(y)),y,x,1top(y,xout(y)),t1),x,lessisi3(x),t3):is"e"(1to(x),left1to(x,y,t1,xout(y)),xout(x))
9160 t5:=isf(1to(x),cx,recf(x,f),left1to(x,y,t1,xout(y)),xout(x),t4):is(<left1to(x,y,t1,xout(y))>recf(x,f),smpr(x,f))
9161 -v8
9162 issmpr:=tris1(cx,smpr(y,f0".v8"),smpr(x,f),<left1to(x,y,t1".v8",xout(y))>recf(x,f),t2".v8",t5".v8"):is(smpr(y,left(cx,x,y,lessisi2"n"(y,x,i),f)),smpr(x,f))
9163 @[z:complex][x:nat]
9164 +8279
9165 xr:=rlofnt(x):real
9166 prop1:=is(sum(x,[t:1to(x)]z),ts(z,pli(xr,0))):'prop'
9167 z@t1:=satz277([t:cx][u:cx]pl(t,u),[t:1to(1)]z):is(sum(1,[t:1to(1)]z),z)
9168 t2:=tris(cx,sum(1,[t:1to(1)]z),z,ts(z,1c),t1,satz222a(z)):prop1(1)
9169 x@[p:prop1(x)]
9170 t3:=satz278([t:cx][u:cx]pl(t,u),x,[t:1to(pl"n"(x,1))]z):is(sum(pl"n"(x,1),[t:1to(pl"n"(x,1))]z),pl(sum(x,[t:1to(x)]z),z))
9171 t4:=ispl12(sum(x,[t:1to(x)]z),ts(z,pli(xr,0)),z,ts(z,1c),p,satz222a(z)):is(pl(sum(x,[t:1to(x)]z),z),pl(ts(z,pli(xr,0)),ts(z,1c)))
9172 t5:=distpt2(z,pli(xr,0),1c):is(pl(ts(z,pli(xr,0)),ts(z,1c)),ts(z,pl(pli(xr,0),1c)))
9173 t6:=plis12a(xr,0,1rl,0):is(pl(pli(xr,0),1c),pli(pl"r"(xr,1rl),pl"r"(0,0)))
9174 t7:=isrecx12(pl"r"(xr,1rl),rlofnt(pl"n"(x,1)),pl"r"(0,0),0,satzr155b(x,1),pl01"r"(0,0,refis(real,0))):is(pli(pl"r"(xr,1rl),pl"r"(0,0)),pli(rlofnt(pl"n"(x,1)),0))
9175 t8:=tris(cx,pl(pli(xr,0),1c),pli(pl"r"(xr,1rl),pl"r"(0,0)),pli(rlofnt(pl"n"(x,1)),0),t6,t7):is(pl(pli(xr,0),1c),pli(rlofnt(pl"n"(x,1)),0))
9176 t9:=ists2(pl(pli(xr,0),1c),pli(rlofnt(pl"n"(x,1)),0),z,t8):is(ts(z,pl(pli(xr,0),1c)),ts(z,pli(rlofnt(pl"n"(x,1)),0)))
9177 t10:=tr4is(cx,sum(pl"n"(x,1),[t:1to(pl"n"(x,1))]z),pl(sum(x,[t:1to(x)]z),z),pl(ts(z,pli(xr,0)),ts(z,1c)),ts(z,pl(pli(xr,0),1c)),ts(z,pli(rlofnt(pl"n"(x,1)),0)),t3,t4,t5,t9):prop1(pl"n"(x,1))
9178 t11:=isp(nat,[t:nat]prop1(t),pl"n"(x,1),<x>suc,t10,satz4a(x)):prop1(<x>suc)
9179 -8279
9180 satz279:=induction([t:nat]prop1".8279"(t),t2".8279",[u:nat][t:prop1".8279"(u)]t11".8279"(u,t),x):is(sum(x,[t:1to(x)]z),ts(z,pli(rlofnt(x),0)))
9181 q@[f:[t:1to(2)]cx]
9182 +8280
9183 t1:=lessisi1"n"(1,2,satz18a(1,1)):lessis"n"(1,2)
9184 f1:=left(cx,2,1,t1,f):[t:1to(1)]cx
9185 t2:=satz278(q,1,f):is(smpr(2,f),<<xout(2)>f><smpr(1,f1)>q)
9186 t3:=satz277(q,f1):is(smpr(1,f1),<xout(1)>f1)
9187 t4:=isinoutn(1,1,lessisi3(1)):is"n"(1,inn(1,xout(1)))
9188 t5:=isoutni(2,1,satz24a(2),inn(1,xout(1)),trlessis"n"(inn(1,xout(1)),1,2,1top(1,xout(1)),t1),t4):is"e"(1to(2),1out(2),left1to(2,1,t1,xout(1)))
9189 t6:=isf(1to(2),cx,f,1out(2),left1to(2,1,t1,xout(1)),t5):is(<1out(2)>f,<xout(1)>f1)
9190 t7:=tris2(cx,smpr(1,f1),<1out(2)>f,<xout(1)>f1,t3,t6):is(smpr(1,f1),<1out(2)>f)
9191 t8:=isf(cx,cx,[t:cx]<<xout(2)>f><t>q,smpr(1,f1),<1out(2)>f,t7):is(<<xout(2)>f><smpr(1,f1)>q,<<xout(2)>f><<1out(2)>f>q)
9192 -8280
9193 satz280:=tris(cx,smpr(2,f),<<xout(2)>f><smpr(1,f1".8280")>q,<<xout(2)>f><<1out(2)>f>q,t2".8280",t8".8280"):is(smpr(2,f),<<xout(2)>f><<1out(2)>f>q)
9194 q@assoc:=[x:cx][y:cx][z:cx]is(<z><<y><x>q>q,<<z><y>q><x>q):'prop'
9195 @assocp1:=[x:cx][y:cx][z:cx]asspl1(x,y,z):assoc([x:cx][y:cx]pl(x,y))
9196 assocts:=[x:cx][y:cx][z:cx]assts1(x,y,z):assoc([x:cx][y:cx]ts(x,y))
9197 q@[a:assoc][z:cx][u:cx][v:cx]
9198 assq1:=<v><u><z>a:is(<v><<u><z>q>q,<<v><u>q><z>q)
9199 assq2:=symis(cx,<v><<u><z>q>q,<<v><u>q><z>q,assq1):is(<<v><u>q><z>q,<v><<u><z>q>q)
9200 q@[a:assoc(q)][x:nat][y:nat][f:[t:1to(pl"n"(x,y))]cx]
9201 +8281
9202 y@t1:=lessisi1"n"(x,pl"n"(x,y),satz18a(x,y)):lessis"n"(x,pl"n"(x,y))
9203 f@f1:=left(cx,pl"n"(x,y),x,t1,f):[t:1to(x)]cx
9204 f2:=right(cx,x,y,f):[t:1to(y)]cx
9205 prop1:=is(smpr(pl"n"(x,y),f),<smpr(y,f2)><smpr(x,f1)>q):'prop'
9206 y@prop2:=all"l"([t:1to(pl"n"(x,y))]cx,[u:[t:1to(pl"n"(x,y))]cx]prop1(u)):'prop'
9207 x@[f0:[t:1to(pl"n"(x,1))]cx]
9208 t2:=satz278(q,x,f0):is(smpr(pl"n"(x,1),f0),<<xout(pl"n"(x,1))>f0><smpr(x,f1(1,f0))>q)
9209 t3:=satz277(q,f2(1,f0)):is(smpr(1,f2(1,f0)),<xout(1)>f2(1,f0))
9210 t4:=isinoutn(1,1,lessisi3(1)):is"n"(1,inn(1,xout(1)))
9211 t5:=ispl2"n"(1,inn(1,xout(1)),x,t4):is"n"(pl"n"(x,1),pl"n"(x,inn(1,xout(1))))
9212 t6:=isoutni(pl"n"(x,1),pl"n"(x,1),lessisi3(pl"n"(x,1)),pl"n"(x,inn(1,xout(1))),satz19o(inn(1,xout(1)),1,x,1top(1,xout(1))),t5):is"e"(1to(pl"n"(x,1)),xout(pl"n"(x,1)),right1to(x,1,xout(1)))
9213 t7:=isf(1to(pl"n"(x,1)),cx,f0,xout(pl"n"(x,1)),right1to(x,1,xout(1)),t6):is(<xout(pl"n"(x,1))>f0,<xout(1)>f2(1,f0))
9214 t8:=tris2(cx,<xout(pl"n"(x,1))>f0,smpr(1,f2(1,f0)),<xout(1)>f2(1,f0),t7,t3):is(<xout(pl"n"(x,1))>f0,smpr(1,f2(1,f0)))
9215 t9:=isf(cx,cx,<smpr(x,f1(1,f0))>q,<xout(pl"n"(x,1))>f0,smpr(1,f2(1,f0)),t8):is(<<xout(pl"n"(x,1))>f0><smpr(x,f1(1,f0))>q,<smpr(1,f2(1,f0))><smpr(x,f1(1,f0))>q)
9216 t10:=tris(cx,smpr(pl"n"(x,1),f0),<<xout(pl"n"(x,1))>f0><smpr(x,f1(1,f0))>q,<smpr(1,f2(1,f0))><smpr(x,f1(1,f0))>q,t2,t9):prop1(1,f0)
9217 x@t11:=[u:[t:1to(pl"n"(x,1))]cx]t10(u):prop2(1)
9218 y@yp1:=pl"n"(y,1):nat
9219 xpy:=pl"n"(x,y):nat
9220 xpy1:=pl"n"(x,yp1):nat
9221 xyp1:=pl"n"(xpy,1):nat
9222 t12:=lessisi2"n"(xyp1,xpy1,asspl1"n"(x,y,1)):lessis"n"(xyp1,xpy1)
9223 [p:prop2(y)][f:[t:1to(xpy1)]cx]
9224 t13:=isinoutn(xyp1,xyp1,lessisi3(xyp1)):is"n"(xyp1,inn(xyp1,xout(xyp1)))
9225 t14:=tris(nat,xpy1,xyp1,inn(xyp1,xout(xyp1)),asspl2"n"(x,y,1),t13):is"n"(xpy1,inn(xyp1,xout(xyp1)))
9226 t15:=isoutni(xpy1,xpy1,lessisi3(xpy1),inn(xyp1,xout(xyp1)),trlessis"n"(inn(xyp1,xout(xyp1)),xyp1,xpy1,1top(xyp1,xout(xyp1)),t12),t14):is"e"(1to(xpy1),xout(xpy1),left1to(xpy1,xyp1,t12,xout(xyp1)))
9227 t16:=isf(1to(xpy1),cx,recf(xpy1,f),xout(xpy1),left1to(xpy1,xyp1,t12,xout(xyp1)),t15):is(smpr(xpy1,f),<xout(xyp1)>left(cx,xpy1,xyp1,t12,recf(xpy1,f)))
9228 fr:=left(cx,xpy1,xyp1,t12,f):[t:1to(xyp1)]cx
9229 t17:=satz275f(xpy1,xyp1,t12,f):is"e"([t:1to(xyp1)]cx,left(cx,xpy1,xyp1,t12,recf(xpy1,f)),recf(xyp1,fr))
9230 t18:=fise(1to(xyp1),cx,left(cx,xpy1,xyp1,t12,recf(xpy1,f)),recf(xyp1,fr),t17,xout(xyp1)):is(<xout(xyp1)>left(cx,xpy1,xyp1,t12,recf(xpy1,f)),smpr(xyp1,fr))
9231 t19:=lessisi1"n"(xpy,xyp1,satz18a(xpy,1)):lessis"n"(xpy,xyp1)
9232 frr:=left(cx,xyp1,xpy,t19,fr):[t:1to(xpy)]cx
9233 t20:=satz278(xpy,fr):is(smpr(xyp1,fr),<<xout(xyp1)>fr><smpr(xpy,frr)>q)
9234 t21:=isf(cx,cx,[u:cx]<<xout(xyp1)>fr><u>q,smpr(xpy,frr),<smpr(y,f2(frr))><smpr(x,f1(frr))>q,<frr>p):is(<<xout(xyp1)>fr><smpr(xpy,frr)>q,<<xout(xyp1)>fr><<smpr(y,f2(frr))><smpr(x,f1(frr))>q>q)
9235 t22:=assq1(a,smpr(x,f1(frr)),smpr(y,f2(frr)),<xout(xyp1)>fr):is(<<xout(xyp1)>fr><<smpr(y,f2(frr))><smpr(x,f1(frr))>q>q,<<<xout(xyp1)>fr><smpr(y,f2(frr))>q><smpr(x,f1(frr))>q)
9236 t23:=lessisi1"n"(y,yp1,satz18a(y,1)):lessis"n"(y,yp1)
9237 fy:=left(cx,yp1,y,t23,f2(yp1,f)):[t:1to(y)]cx
9238 t24:=satz278(y,f2(yp1,f)):is(smpr(yp1,f2(yp1,f)),<<xout(yp1)>f2(yp1,f)><smpr(y,fy)>q)
9239 t25:=isinoutn(yp1,yp1,lessisi3(yp1)):is"n"(yp1,inn(yp1,xout(yp1)))
9240 t26:=ispl2"n"(yp1,inn(yp1,xout(yp1)),x,t25):is"n"(xpy1,pl"n"(x,inn(yp1,xout(yp1))))
9241 t27:=tris1(nat,inn(xyp1,xout(xyp1)),pl"n"(x,inn(yp1,xout(yp1))),xpy1,t14,t26):is"n"(inn(xyp1,xout(xyp1)),pl"n"(x,inn(yp1,xout(yp1))))
9242 t28:=isoutni(xpy1,inn(xyp1,xout(xyp1)),trlessis"n"(inn(xyp1,xout(xyp1)),xyp1,xpy1,1top(xyp1,xout(xyp1)),t12),pl"n"(x,inn(yp1,xout(yp1))),satz19o(inn(yp1,xout(yp1)),yp1,x,1top(yp1,xout(yp1))),t27):is"e"(1to(xpy1),left1to(xpy1,xyp1,t12,xout(xyp1)),right1to(x,yp1,xout(yp1)))
9243 t29:=isf(1to(xpy1),cx,f,left1to(xpy1,xyp1,t12,xout(xyp1)),right1to(x,yp1,xout(yp1)),t28):is(<xout(xyp1)>fr,<xout(yp1)>f2(yp1,f))
9244 t30:=isf(cx,cx,<smpr(y,f2(frr))>q,<xout(xyp1)>fr,<xout(yp1)>f2(yp1,f),t29):is(<<xout(xyp1)>fr><smpr(y,f2(frr))>q,<<xout(yp1)>f2(yp1,f)><smpr(y,f2(frr))>q)
9245 [n:1to(y)]
9246 n0:=inn(y,n):nat
9247 nyp1:=left1to(yp1,y,t23,n):1to(yp1)
9248 t31:=isinoutn(yp1,n0,trlessis"n"(n0,y,yp1,1top(y,n),t23)):is"n"(n0,inn(yp1,nyp1))
9249 t32:=ispl2"n"(n0,inn(yp1,nyp1),x,t31):is"n"(pl"n"(x,n0),pl"n"(x,inn(yp1,nyp1)))
9250 nxpy:=right1to(x,y,n):1to(xpy)
9251 nxyp1:=left1to(xyp1,xpy,t19,nxpy):1to(xyp1)
9252 t33:=isinoutn(xyp1,inn(xpy,nxpy),trlessis"n"(inn(xpy,nxpy),xpy,xyp1,1top(xpy,nxpy),t19)):is"n"(inn(xpy,nxpy),inn(xyp1,nxyp1))
9253 t34:=isinoutn(xpy,pl"n"(x,n0),satz19o(n0,y,x,1top(y,n))):is"n"(pl"n"(x,n0),inn(xpy,nxpy))
9254 t35:=tris(nat,pl"n"(x,n0),inn(xpy,nxpy),inn(xyp1,nxyp1),t34,t33):is"n"(pl"n"(x,n0),inn(xyp1,nxyp1))
9255 t36:=tris1(nat,pl"n"(x,inn(yp1,nyp1)),inn(xyp1,nxyp1),pl"n"(x,n0),t32,t35):is"n"(pl"n"(x,inn(yp1,nyp1)),inn(xyp1,nxyp1))
9256 t37:=isoutni(xpy1,pl"n"(x,inn(yp1,nyp1)),satz19o(inn(yp1,nyp1),yp1,x,1top(yp1,nyp1)),inn(xyp1,nxyp1),trlessis"n"(inn(xyp1,nxyp1),xyp1,xpy1,1top(xyp1,nxyp1),t12),t36):is"e"(1to(xpy1),right1to(x,yp1,nyp1),left1to(xpy1,xyp1,t12,nxyp1))
9257 t38:=isf(1to(xpy1),cx,f,right1to(x,yp1,nyp1),left1to(xpy1,xyp1,t12,nxyp1),t37):is(<n>fy,<n>f2(frr))
9258 f@t39:=fisi(1to(y),cx,fy,f2(frr),[u:1to(y)]t38(u)):is"e"([u:1to(y)]cx,fy,f2(frr))
9259 t40:=isf([u:1to(y)]cx,cx,[t:[u:1to(y)]cx]smpr(y,t),fy,f2(frr),t39):is(smpr(y,fy),smpr(y,f2(frr)))
9260 t41:=isf(cx,cx,[t:cx]<<xout(yp1)>f2(yp1,f)><t>q,smpr(y,f2(frr)),smpr(y,fy),symis(cx,smpr(y,fy),smpr(y,f2(frr)),t40)):is(<<xout(yp1)>f2(yp1,f)><smpr(y,f2(frr))>q,<<xout(yp1)>f2(yp1,f)><smpr(y,fy)>q)
9261 t41a:=tris(cx,<<xout(xyp1)>fr><smpr(y,f2(frr))>q,<<xout(yp1)>f2(yp1,f)><smpr(y,f2(frr))>q,<<xout(yp1)>f2(yp1,f)><smpr(y,fy)>q,t30,t41):is(<<xout(xyp1)>fr><smpr(y,f2(frr))>q,<<xout(yp1)>f2(yp1,f)><smpr(y,fy)>q)
9262 t42:=tris2(cx,<<xout(xyp1)>fr><smpr(y,f2(frr))>q,smpr(yp1,f2(yp1,f)),<<xout(yp1)>f2(yp1,f)><smpr(y,fy)>q,t41a,t24):is(<<xout(xyp1)>fr><smpr(y,f2(frr))>q,smpr(yp1,f2(yp1,f)))
9263 t43:=isf(cx,cx,<smpr(x,f1(frr))>q,<<xout(xyp1)>fr><smpr(y,f2(frr))>q,smpr(yp1,f2(yp1,f)),t42):is(<<<xout(xyp1)>fr><smpr(y,f2(frr))>q><smpr(x,f1(frr))>q,<smpr(yp1,f2(yp1,f))><smpr(x,f1(frr))>q)
9264 [m:1to(x)]
9265 m0:=inn(x,m):nat
9266 mxpy:=left1to(xpy,x,t1,m):1to(xpy)
9267 mxyp1:=left1to(xyp1,xpy,t19,mxpy):1to(xyp1)
9268 t44:=isinoutn(xyp1,inn(xpy,mxpy),trlessis"n"(inn(xpy,mxpy),xpy,xyp1,1top(xpy,mxpy),t19)):is"n"(inn(xpy,mxpy),inn(xyp1,mxyp1))
9269 t45:=isinoutn(xpy,m0,trlessis"n"(m0,x,xpy,1top(x,m),t1)):is"n"(m0,inn(xpy,mxpy))
9270 t46:=tris(nat,m0,inn(xpy,mxpy),inn(xyp1,mxyp1),t45,t44):is"n"(m0,inn(xyp1,mxyp1))
9271 t47:=isoutni(xpy1,m0,trlessis"n"(m0,x,xpy1,1top(x,m),t1(yp1)),inn(xyp1,mxyp1),trlessis"n"(inn(xyp1,mxyp1),xyp1,xpy1,1top(xyp1,mxyp1),t12),t46):is"e"(1to(xpy1),left1to(xpy1,x,t1(yp1),m),left1to(xpy1,xyp1,t12,mxyp1))
9272 t48:=isf(1to(xpy1),cx,f,left1to(xpy1,x,t1(yp1),m),left1to(xpy1,xyp1,t12,mxyp1),t47):is(<m>f1(x,yp1,f),<m>f1(x,y,frr))
9273 f@t49:=fisi(1to(x),cx,f1(x,yp1,f),f1(x,y,frr),[u:1to(x)]t48(u)):is"e"([u:1to(x)]cx,f1(x,yp1,f),f1(x,y,frr))
9274 t50:=isf([u:1to(x)]cx,cx,[t:[u:1to(x)]cx]smpr(x,t),f1(yp1,f),f1(frr),t49):is(smpr(x,f1(yp1,f)),smpr(x,f1(frr)))
9275 t51:=isf(cx,cx,[t:cx]<smpr(yp1,f2(yp1,f))><t>q,smpr(x,f1(frr)),smpr(x,f1(yp1,f)),symis(cx,smpr(x,f1(yp1,f)),smpr(x,f1(frr)),t50)):is(<smpr(yp1,f2(yp1,f))><smpr(x,f1(frr))>q,<smpr(yp1,f2(yp1,f))><smpr(x,f1(yp1,f))>q)
9276 t52:=tr4is(cx,smpr(xpy1,f),<xout(xyp1)>left(cx,xpy1,xyp1,t12,recf(xpy1,f)),smpr(xyp1,fr),<<xout(xyp1)>fr><smpr(xpy,frr)>q,<<xout(xyp1)>fr><<smpr(y,f2(frr))><smpr(x,f1(frr))>q>q,t16,t18,t20,t21):is(smpr(xpy1,f),<<xout(xyp1)>fr><<smpr(y,f2(frr))><smpr(x,f1(frr))>q>q)
9277 t53:=tr4is(cx,smpr(xpy1,f),<<xout(xyp1)>fr><<smpr(y,f2(frr))><smpr(x,f1(frr))>q>q,<<<xout(xyp1)>fr><smpr(y,f2(frr))>q><smpr(x,f1(frr))>q,<smpr(yp1,f2(yp1,f))><smpr(x,f1(frr))>q,<smpr(yp1,f2(yp1,f))><smpr(x,f1(yp1,f))>q,t52,t22,t43,t51):prop1(yp1,f)
9278 p@t54:=[u:[t:1to(xpy1)]cx]t53(u):prop2(yp1)
9279 t55:=isp(nat,[t:nat]prop2(t),yp1,<y>suc,t54,satz4a(y)):prop2(<y>suc)
9280 y@t56:=induction([t:nat]prop2(t),t11,[z:nat][t:prop2(z)]t55(z,t),y):prop2(y)
9281 -8281
9282 satz281:=<f>t56".8281":is(smpr(pl"n"(x,y),f),<smpr(y,right(cx,x,y,f))><smpr(x,left(cx,pl"n"(x,y),x,lessisi1"n"(x,pl"n"(x,y),satz18a(x,y)),f))>q)
9283 q@commut:=[x:cx][y:cx]is(<y><x>q,<x><y>q):'prop'
9284 @commutpl:=[x:cx][y:cx]compl(x,y):commut([x:cx][y:cx]pl(x,y))
9285 commutts:=[x:cx][y:cx]comts(x,y):commut([x:cx][y:cx]ts(x,y))
9286 q@[c:commut][z:cx][u:cx]
9287 comq:=<u><z>c:is(<u><z>q,<z><u>q)
9288 a@[c:commut(q)][x:nat][f:[t:1to(x)]cx][g:[t:1to(x)]cx]
9289 +8282
9290 prop1:=is(smpr(x,[t:1to(x)]<<t>g><<t>f>q),<smpr(x,g)><smpr(x,f)>q):'prop'
9291 x@prop2:=all"l"([t:1to(x)]cx,[u:[t:1to(x)]cx]all"l"([t:1to(x)]cx,[v:[t:1to(x)]cx]prop1(u,v))):'prop'
9292 c@[f0:[t:1to(1)]cx][g0:[t:1to(1)]cx]
9293 t1:=satz277([t:1to(1)]<<t>g0><<t>f0>q):is(smpr(1,[t:1to(1)]<<t>g0><<t>f0>q),<<xout(1)>g0><<xout(1)>f0>q)
9294 t2:=isf(cx,cx,<smpr(1,f0)>q,smpr(1,g0),<xout(1)>g0,satz277(g0)):is(<smpr(1,g0)><smpr(1,f0)>q,<<xout(1)>g0><smpr(1,f0)>q)
9295 t3:=isf(cx,cx,[t:cx]<<xout(1)>g0><t>q,smpr(1,f0),<xout(1)>f0,satz277(f0)):is(<<xout(1)>g0><smpr(1,f0)>q,<<xout(1)>g0><<xout(1)>f0>q)
9296 t4:=tris(cx,<smpr(1,g0)><smpr(1,f0)>q,<<xout(1)>g0><smpr(1,f0)>q,<<xout(1)>g0><<xout(1)>f0>q,t2,t3):is(<smpr(1,g0)><smpr(1,f0)>q,<<xout(1)>g0><<xout(1)>f0>q)
9297 t5:=tris2(cx,smpr(1,[t:1to(1)]<<t>g0><<t>f0>q),<smpr(1,g0)><smpr(1,f0)>q,<<xout(1)>g0><<xout(1)>f0>q,t1,t4):prop1(1,f0,g0)
9298 c@t6:=[u:[t:1to(1)]cx][v:[t:1to(1)]cx]t5(u,v):prop2(1)
9299 x@xp1:=pl"n"(x,1):nat
9300 [p:prop2(x)][f:[t:1to(xp1)]cx][g:[t:1to(xp1)]cx]
9301 c@[u:cx][v:cx][w:cx][z:cx]
9302 t7:=assq2(a,<v><u>q,w,z):is(<<z><w>q><<v><u>q>q,<z><<w><<v><u>q>q>q)
9303 t8:=comq(c,<v><u>q,w):is(<w><<v><u>q>q,<<v><u>q><w>q)
9304 t9:=assq2(a,w,u,v):is(<<v><u>q><w>q,<v><<u><w>q>q)
9305 t10:=tris(cx,<w><<v><u>q>q,<<v><u>q><w>q,<v><<u><w>q>q,t8,t9):is(<w><<v><u>q>q,<v><<u><w>q>q)
9306 t11:=isf(cx,cx,[t:cx]<z><t>q,<w><<v><u>q>q,<v><<u><w>q>q,t10):is(<z><<w><<v><u>q>q>q,<z><<v><<u><w>q>q>q)
9307 t12:=assq1(a,<u><w>q,v,z):is(<z><<v><<u><w>q>q>q,<<z><v>q><<u><w>q>q)
9308 t13:=comq(c,w,u):is(<u><w>q,<w><u>q)
9309 t14:=isf(cx,cx,[t:cx]<<z><v>q><t>q,<u><w>q,<w><u>q,t13):is(<<z><v>q><<u><w>q>q,<<z><v>q><<w><u>q>q)
9310 t15:=tr4is(cx,<<z><w>q><<v><u>q>q,<z><<w><<v><u>q>q>q,<z><<v><<u><w>q>q>q,<<z><v>q><<u><w>q>q,<<z><v>q><<w><u>q>q,t7,t11,t12,t14):is(<<z><w>q><<v><u>q>q,<<z><v>q><<w><u>q>q)
9311 g@t16:=lessisi1"n"(x,xp1,satz18a(x,1)):lessis"n"(x,xp1)
9312 sfx:=smpr(x,left(cx,xp1,x,t16,f)):cx
9313 sgx:=smpr(x,left(cx,xp1,x,t16,g)):cx
9314 h:=[t:1to(xp1)]<<t>g><<t>f>q:[t:1to(xp1)]cx
9315 shx:=smpr(x,left(cx,xp1,x,t16,h)):cx
9316 t17:=satz278(x,h):is(smpr(xp1,h),<<<xout(xp1)>g><<xout(xp1)>f>q><shx>q)
9317 t18:=<left(cx,xp1,x,t16,g)><left(cx,xp1,x,t16,f)>p:is(shx,<sgx><sfx>q)
9318 t19:=isf(cx,cx,[t:cx]<<<xout(xp1)>g><<xout(xp1)>f>q><t>q,shx,<sgx><sfx>q,t18):is(<<<xout(xp1)>g><<xout(xp1)>f>q><shx>q,<<<xout(xp1)>g><<xout(xp1)>f>q><<sgx><sfx>q>q)
9319 t20:=t15(sfx,sgx,<xout(xp1)>f,<xout(xp1)>g):is(<<<xout(xp1)>g><<xout(xp1)>f>q><<sgx><sfx>q>q,<<<xout(xp1)>g><sgx>q><<<xout(xp1)>f><sfx>q>q)
9320 t21:=symis(cx,smpr(xp1,f),<<xout(xp1)>f><sfx>q,satz278(x,f)):is(<<xout(xp1)>f><sfx>q,smpr(xp1,f))
9321 t22:=isf(cx,cx,[t:cx]<<<xout(xp1)>g><sgx>q><t>q,<<xout(xp1)>f><sfx>q,smpr(xp1,f),t21):is(<<<xout(xp1)>g><sgx>q><<<xout(xp1)>f><sfx>q>q,<<<xout(xp1)>g><sgx>q><smpr(xp1,f)>q)
9322 t23:=symis(cx,smpr(xp1,g),<<xout(xp1)>g><sgx>q,satz278(x,g)):is(<<xout(xp1)>g><sgx>q,smpr(xp1,g))
9323 t24:=isf(cx,cx,<smpr(xp1,f)>q,<<xout(xp1)>g><sgx>q,smpr(xp1,g),t23):is(<<<xout(xp1)>g><sgx>q><smpr(xp1,f)>q,<smpr(xp1,g)><smpr(xp1,f)>q)
9324 t25:=tr3is(cx,smpr(xp1,h),<<<xout(xp1)>g><<xout(xp1)>f>q><shx>q,<<<xout(xp1)>g><<xout(xp1)>f>q><<sgx><sfx>q>q,<<<xout(xp1)>g><sgx>q><<<xout(xp1)>f><sfx>q>q,t17,t19,t20):is(smpr(xp1,h),<<<xout(xp1)>g><sgx>q><<<xout(xp1)>f><sfx>q>q)
9325 t26:=tr3is(cx,smpr(xp1,h),<<<xout(xp1)>g><sgx>q><<<xout(xp1)>f><sfx>q>q,<<<xout(xp1)>g><sgx>q><smpr(xp1,f)>q,<smpr(xp1,g)><smpr(xp1,f)>q,t25,t22,t24):prop1(xp1,f,g)
9326 p@t27:=[u:[t:1to(xp1)]cx][v:[t:1to(xp1)]cx]t26(u,v):prop2(xp1)
9327 t28:=isp(nat,[t:nat]prop2(t),xp1,<x>suc,t27,satz4a(x)):prop2(<x>suc)
9328 x@t29:=induction([t:nat]prop2(t),t6,[y:nat][t:prop2(y)]t28(y,t),x):prop2(x)
9329 -8282
9330 satz282:=<g><f>t29".8282":is(smpr(x,[t:1to(x)]<<t>g><<t>f>q),<smpr(x,g)><smpr(x,f)>q)
9331 x@[s:[t:1to(x)]1to(x)][b:bijective(1to(x),1to(x),s)][f:[t:1to(x)]cx]
9332 +8283
9333 s@[f:[t:1to(x)]cx]
9334 g:=[t:1to(x)]<<t>s>f:[t:1to(x)]cx
9335 prop1:=is(smpr(x,g),smpr(x,f)):'prop'
9336 x@prop2:=all"l"([t:1to(x)]1to(x),[u:[t:1to(x)]1to(x)]all"l"([t:1to(x)]cx,[v:[t:1to(x)]cx]imp(bijective(1to(x),1to(x),u),prop1(u,v)))):'prop'
9337 c@[s:[t:1to(1)]1to(1)][f:[t:1to(1)]cx]
9338 t1:=tris2(1to(1),<xout(1)>s,xout(1),1o"n",th1"n.singlet"(<xout(1)>s),th1"n.singlet"(xout(1))):is"e"(1to(1),<xout(1)>s,xout(1))
9339 t2:=satz277(g(1,s,f)):is(smpr(1,g(1,s,f)),<xout(1)>g(1,s,f))
9340 t3:=isf(1to(1),cx,f,<xout(1)>s,xout(1),t1):is(<xout(1)>g(1,s,f),<xout(1)>f)
9341 t4:=symis(cx,smpr(1,f),<xout(1)>f,satz277(f)):is(<xout(1)>f,smpr(1,f))
9342 t5:=tr3is(cx,smpr(1,g(1,s,f)),<xout(1)>g(1,s,f),<xout(1)>f,smpr(1,f),t2,t3,t4):prop1(1,s,f)
9343 c@t6:=[u:[t:1to(1)]1to(1)][v:[t:1to(1)]cx][w:bijective(1to(1),1to(1),u)]t5(u,v):prop2(1)
9344 x@xp1:=pl"n"(x,1):nat
9345 t7:=lessisi1"n"(x,xp1,satz18a(x,1)):lessis"n"(x,xp1)
9346 [p:prop2(x)][s:[t:1to(xp1)]1to(xp1)][f:[t:1to(xp1)]cx][b:bijective(1to(xp1),1to(xp1),s)]
9347 t8:=ande1(injective(1to(xp1),1to(xp1),s),surjective(1to(xp1),1to(xp1),s),b):injective(1to(xp1),1to(xp1),s)
9348 [case1:is"e"(1to(xp1),<xout(xp1)>s,xout(xp1))][u:1to(x)]
9349 u1:=left1to(xp1,x,t7,u):1to(xp1)
9350 n1:=inn(xp1,<u1>s):nat
9351 [i:is"n"(n1,xp1)]
9352 t9:=tr3is(1to(xp1),<u1>s,outn(xp1,n1,1top(xp1,<u1>s)),xout(xp1),<xout(xp1)>s,isoutinn(xp1,<u1>s),isoutni(xp1,n1,1top(xp1,<u1>s),xp1,lessisi3(xp1),i),symis(1to(xp1),<xout(xp1)>s,xout(xp1),case1)):is"e"(1to(xp1),<u1>s,<xout(xp1)>s)
9353 t10:=isfe(1to(xp1),1to(xp1),s,t8,u1,xout(xp1),t9):is"e"(1to(xp1),u1,xout(xp1))
9354 t11:=isoutne(xp1,inn(x,u),trlessis"n"(inn(x,u),x,xp1,1top(x,u),t7),xp1,lessisi3(xp1),t10):is"n"(inn(x,u),xp1)
9355 t12:=satz16a(inn(x,u),x,xp1,1top(x,u),satz18a(x,1)):less"n"(inn(x,u),xp1)
9356 t13:=<t11>ec3e31(is"n"(inn(x,u),xp1),more"n"(inn(x,u),xp1),less"n"(inn(x,u),xp1),satz10b(inn(x,u),xp1),t12):con
9357 u@t14:=ore1(less"n"(n1,xp1),is"n"(n1,xp1),1top(xp1,<u1>s),[t:is"n"(n1,xp1)]t13(t)):less"n"(n1,xp1)
9358 t15:=satz26(x,n1,t14):lessis"n"(n1,x)
9359 w1:=outn(x,n1,t15):1to(x)
9360 case1@s01:=[t:1to(x)]w1(t):[t:1to(x)]1to(x)
9361 u@[v:1to(x)][i:is"e"(1to(x),w1(u),w1(v))]
9362 t16:=isoutne(x,n1(u),t15(u),n1(v),t15(v),i):is"n"(n1(u),n1(v))
9363 t17:=<isinne(xp1,<u1(u)>s,<u1(v)>s,t16)><u1(v)><u1(u)>t8:is"e"(1to(xp1),u1(u),u1(v))
9364 t18:=thleft1(xp1,x,t7,u,v,t17):is"e"(1to(x),u,v)
9365 u@u2:=<u1>invf(1to(xp1),1to(xp1),s,b):1to(xp1)
9366 n2:=inn(xp1,u2):nat
9367 [i:is"n"(n2,xp1)]
9368 t19:=tris(1to(xp1),u2,outn(xp1,n2,1top(xp1,u2)),xout(xp1),isoutinn(xp1,u2),isoutni(xp1,n2,1top(xp1,u2),xp1,lessisi3(xp1),i)):is"e"(1to(xp1),u2,xout(xp1))
9369 t20:=tr3is(1to(xp1),u1,<u2>s,<xout(xp1)>s,xout(xp1),thinvf2(1to(xp1),1to(xp1),s,b,u1),isf(1to(xp1),1to(xp1),s,u2,xout(xp1),t19),case1):is"e"(1to(xp1),u1,xout(xp1))
9370 t21:=isoutne(xp1,inn(x,u),trlessis"n"(inn(x,u),x,xp1,1top(x,u),t7),xp1,lessisi3(xp1),t20):is"n"(inn(x,u),xp1)
9371 t22:=satz16a(inn(x,u),x,xp1,1top(x,u),satz18a(x,1)):less"n"(inn(x,u),xp1)
9372 t23:=<t21>ec3e31(is"n"(inn(x,u),xp1),more"n"(inn(x,u),xp1),less"n"(inn(x,u),xp1),satz10b(inn(x,u),xp1),t22):con
9373 u@t24:=ore1(less"n"(n2,xp1),is"n"(n2,xp1),1top(xp1,u2),[t:is"n"(n2,xp1)]t23(t)):less"n"(n2,xp1)
9374 t25:=satz26(x,n2,t24):lessis"n"(n2,x)
9375 w2:=outn(x,n2,t25):1to(x)
9376 t26:=isinoutn(x,n2,t25):is"n"(n2,inn(x,w2))
9377 t27:=tris(1to(xp1),u2,outn(xp1,n2,1top(xp1,u2)),u1(w2),isoutinn(xp1,u2),isoutni(xp1,n2,1top(xp1,u2),inn(x,w2),trlessis"n"(inn(x,w2),x,xp1,1top(x,w2),t7),t26)):is"e"(1to(xp1),u2,u1(w2))
9378 t28:=tris(1to(xp1),u1,<u2>s,<u1(w2)>s,thinvf2(1to(xp1),1to(xp1),s,b,u1),isf(1to(xp1),1to(xp1),s,u2,u1(w2),t27)):is"e"(1to(xp1),u1,<u1(w2)>s)
9379 t29:=tris(nat,inn(x,u),inn(xp1,u1),n1(w2),isinoutn(xp1,inn(x,u),trlessis"n"(inn(x,u),x,xp1,1top(x,u),t7)),isinni(xp1,u1,<u1(w2)>s,t28)):is"n"(inn(x,u),n1(w2))
9380 t30:=tris(1to(x),u,outn(x,inn(x,u),1top(x,u)),w1(w2),isoutinn(x,u),isoutni(x,inn(x,u),1top(x,u),n1(w2),t15(w2),t29)):is"e"(1to(x),u,w1(w2))
9381 t31:=somei(1to(x),[t:1to(x)]is"e"(1to(x),u,w1(t)),w2,t30):image(1to(x),1to(x),s01,u)
9382 case1@t32:=andi(injective(1to(x),1to(x),s01),surjective(1to(x),1to(x),s01),[t:1to(x)][u:1to(x)][v:is"e"(1to(x),<t>s01,<u>s01)]t18(t,u,v),[t:1to(x)]t31(t)):bijective(1to(x),1to(x),s01)
9383 f01:=left(cx,xp1,x,t7,f):[t:1to(x)]cx
9384 t33:=<t32><f01><s01>p:prop1(s01,f01)
9385 g1:=left(cx,xp1,x,t7,g(xp1,s,f)):[t:1to(x)]cx
9386 g2:=g(x,s01,f01):[t:1to(x)]cx
9387 u@t33a:=isoutinn(xp1,<u1>s):is"e"(1to(xp1),<u1>s,outn(xp1,n1,1top(xp1,<u1>s)))
9388 t34:=isinoutn(x,n1,t15):is"n"(n1,inn(x,w1))
9389 t35:=isoutni(xp1,n1,1top(xp1,<u1>s),inn(x,w1),trlessis"n"(inn(x,w1),x,xp1,1top(x,w1),t7),t34):is"e"(1to(xp1),outn(xp1,n1,1top(xp1,<u1>s)),left1to(xp1,x,t7,w1))
9390 t36:=tris(1to(xp1),<u1>s,outn(xp1,n1,1top(xp1,<u1>s)),left1to(xp1,x,t7,w1),t33a,t35):is"e"(1to(xp1),<u1>s,left1to(xp1,x,t7,w1))
9391 t37:=isf(1to(xp1),cx,f,<u1>s,left1to(xp1,x,t7,w1),t36):is(<u>g1,<u>g2)
9392 case1@t38:=fisi(1to(x),cx,g1,g2,[t:1to(x)]t37(t)):is"e"([t:1to(x)]cx,g1,g2)
9393 t39:=isf([t:1to(x)]cx,cx,[u:[t:1to(x)]cx]smpr(x,u),g1,g2,t38):is(smpr(x,g1),smpr(x,g2))
9394 t40:=tris(cx,smpr(x,g1),smpr(x,g2),smpr(x,f01),t39,t33):is(smpr(x,g1),smpr(x,f01))
9395 t41:=isf(1to(xp1),cx,f,<xout(xp1)>s,xout(xp1),case1):is(<xout(xp1)>g(xp1,s,f),<xout(xp1)>f)
9396 t42:=satz278(x,g(xp1,s,f)):is(smpr(xp1,g(xp1,s,f)),<<xout(xp1)>g(xp1,s,f)><smpr(x,g1)>q)
9397 t43:=isf(cx,cx,<smpr(x,g1)>q,<xout(xp1)>g(xp1,s,f),<xout(xp1)>f,t41):is(<<xout(xp1)>g(xp1,s,f)><smpr(x,g1)>q,<<xout(xp1)>f><smpr(x,g1)>q)
9398 t44:=isf(cx,cx,[t:cx]<<xout(xp1)>f><t>q,smpr(x,g1),smpr(x,f01),t40):is(<<xout(xp1)>f><smpr(x,g1)>q,<<xout(xp1)>f><smpr(x,f01)>q)
9399 t45:=symis(cx,smpr(xp1,f),<<xout(xp1)>f><smpr(x,f01)>q,satz278(x,f)):is(<<xout(xp1)>f><smpr(x,f01)>q,smpr(xp1,f))
9400 t46:=tr4is(cx,smpr(xp1,g(xp1,s,f)),<<xout(xp1)>g(xp1,s,f)><smpr(x,g1)>q,<<xout(xp1)>f><smpr(x,g1)>q,<<xout(xp1)>f><smpr(x,f01)>q,smpr(xp1,f),t42,t43,t44,t45):prop1(xp1,s,f)
9401 b@1px:=pl"n"(1,x):nat
9402 [case2:is"e"(1to(xp1),<1out(xp1)>s,1out(xp1))][u:1to(x)]
9403 u3:=right1to(1,x,u):1to(1px)
9404 case2@t47:=lessisi2"n"(1px,xp1,compl"n"(1,x)):lessis"n"(1px,xp1)
9405 s02:=left(1to(xp1),xp1,1px,t47,s):[t:1to(1px)]1to(xp1)
9406 u@n3:=inn(xp1,<u3>s02):nat
9407 [i:is"n"(n3,1)]
9408 t48:=tr3is(1to(xp1),<u3>s02,outn(xp1,n3,1top(xp1,<u3>s02)),1out(xp1),<1out(xp1)>s,isoutinn(xp1,<u3>s02),isoutni(xp1,n3,1top(xp1,<u3>s02),1,satz24a(xp1),i),symis(1to(xp1),<1out(xp1)>s,1out(xp1),case2)):is"e"(1to(xp1),<u3>s02,<1out(xp1)>s)
9409 t49:=isfe(1to(xp1),1to(xp1),s,t8,left1to(xp1,1px,t47,u3),1out(xp1),t48):is"e"(1to(xp1),left1to(xp1,1px,t47,u3),1out(xp1))
9410 t50:=isoutne(xp1,inn(1px,u3),trlessis"n"(inn(1px,u3),1px,xp1,1top(1px,u3),t47),1,satz24a(xp1),t49):is"n"(inn(1px,u3),1)
9411 t51:=isinoutn(1px,pl"n"(1,inn(x,u)),satz19o(inn(x,u),x,1,1top(x,u))):is"n"(pl"n"(1,inn(x,u)),inn(1px,u3))
9412 t52:=ismore1"n"(pl"n"(1,inn(x,u)),inn(1px,u3),1,t51,satz18(1,inn(x,u))):more"n"(inn(1px,u3),1)
9413 t53:=<t50>ec3e21(is"n"(inn(1px,u3),1),more"n"(inn(1px,u3),1),less"n"(inn(1px,u3),1),satz10b(inn(1px,u3),1),t52):con
9414 u@t54:=ore1(more"n"(n3,1),is"n"(n3,1),satz24(n3),[t:is"n"(n3,1)]t53(t)):more"n"(n3,1)
9415 nm1:=mn"n"(n3,1,t54):nat
9416 t54a:=islessis1"n"(n3,pl"n"(nm1,1),xp1,th1c"n.mn"(n3,1,t54),1top(xp1,<u3>s02)):lessis"n"(pl"n"(nm1,1),xp1)
9417 t55:=th9"l.or"(less"n"(pl"n"(nm1,1),xp1),is"n"(pl"n"(nm1,1),xp1),less"n"(nm1,x),is"n"(nm1,x),t54a,[t:less"n"(pl"n"(nm1,1),xp1)]satz20c(nm1,x,1,t),[t:is"n"(pl"n"(nm1,1),xp1)]satz20b(nm1,x,1,t)):lessis"n"(nm1,x)
9418 w3:=outn(x,nm1,t55):1to(x)
9419 case2@s03:=[t:1to(x)]w3(t):[t:1to(x)]1to(x)
9420 u@[v:1to(x)][i:is"e"(1to(x),w3(u),w3(v))]
9421 t56:=isoutne(x,nm1(u),t55(u),nm1(v),t55(v),i):is"n"(nm1(u),nm1(v))
9422 t57:=tr3is(nat,n3(u),pl"n"(nm1(u),1),pl"n"(nm1(v),1),n3(v),th1c"n.mn"(n3(u),1,t54(u)),ispl1"n"(nm1(u),nm1(v),1,t56),th1d"n.mn"(n3(v),1,t54(v))):is"n"(n3(u),n3(v))
9423 t58:=isinne(xp1,<u3(u)>s02,<u3(v)>s02,t57):is"e"(1to(xp1),<u3(u)>s02,<u3(v)>s02)
9424 t59:=isfe(1to(xp1),1to(xp1),s,t8,left1to(xp1,1px,t47,u3(u)),left1to(xp1,1px,t47,u3(v)),t58):is"e"(1to(xp1),left1to(xp1,1px,t47,u3(u)),left1to(xp1,1px,t47,u3(v)))
9425 t60:=thleft1(xp1,1px,t47,u3(u),u3(v),t59):is"e"(1to(1px),u3(u),u3(v))
9426 t61:=thright1(1,x,u,v,t60):is"e"(1to(x),u,v)
9427 case2@s04:=left(1to(xp1),xp1,1px,t47,invf(1to(xp1),1to(xp1),s,b)):[t:1to(1px)]1to(xp1)
9428 u@u4:=<u3>s04:1to(xp1)
9429 n4:=inn(xp1,u4):nat
9430 [i:is"n"(n4,1)]
9431 t62:=tris(1to(xp1),u4,outn(xp1,n4,1top(xp1,u4)),1out(xp1),isoutinn(xp1,u4),isoutni(xp1,n4,1top(xp1,u4),1,satz24a(xp1),i)):is"e"(1to(xp1),u4,1out(xp1))
9432 t63:=tr3is(1to(xp1),left1to(xp1,1px,t47,u3),<u4>s,<1out(xp1)>s,1out(xp1),thinvf2(1to(xp1),1to(xp1),s,b,left1to(xp1,1px,t47,u3)),isf(1to(xp1),1to(xp1),s,u4,1out(xp1),t62),case2):is"e"(1to(xp1),left1to(xp1,1px,t47,u3),1out(xp1))
9433 t64:=isoutne(xp1,inn(1px,u3),trlessis"n"(inn(1px,u3),1px,xp1,1top(1px,u3),t47),1,satz24a(xp1),t63):is"n"(inn(1px,u3),1)
9434 t65:=tris(nat,pl"n"(1,inn(x,u)),inn(1px,u3),1,isinoutn(1px,pl"n"(1,inn(x,u)),satz19o(inn(x,u),x,1,1top(x,u))),t64):is"n"(pl"n"(1,inn(x,u)),1)
9435 t66:=<t65>ec3e21(is"n"(pl"n"(1,inn(x,u)),1),more"n"(pl"n"(1,inn(x,u)),1),less"n"(pl"n"(1,inn(x,u)),1),satz10b(pl"n"(1,inn(x,u)),1),satz18(1,inn(x,u))):con
9436 u@t67:=ore1(more"n"(n4,1),is"n"(n4,1),satz24(n4),[t:is"n"(n4,1)]t66(t)):more"n"(n4,1)
9437 nm2:=mn"n"(n4,1,t67):nat
9438 t68:=islessis1"n"(n4,pl"n"(nm2,1),xp1,th1c"n.mn"(n4,1,t67),1top(xp1,u4)):lessis"n"(pl"n"(nm2,1),xp1)
9439 t69:=th9"l.or"(less"n"(pl"n"(nm2,1),xp1),is"n"(pl"n"(nm2,1),xp1),less"n"(nm2,x),is"n"(nm2,x),t68,[t:less"n"(pl"n"(nm2,1),xp1)]satz20c(nm2,x,1,t),[t:is"n"(pl"n"(nm2,1),xp1)]satz20b(nm2,x,1,t)):lessis"n"(nm2,x)
9440 w4:=outn(x,nm2,t69):1to(x)
9441 t70:=isinoutn(x,nm2,t69):is"n"(nm2,inn(x,w4))
9442 t71:=tr3is(nat,n4,pl"n"(1,nm2),pl"n"(1,inn(x,w4)),inn(1px,u3(w4)),th1a"n.mn"(n4,1,t67),ispl2"n"(nm2,inn(x,w4),1,t70),isinoutn(1px,pl"n"(1,inn(x,w4)),satz19o(inn(x,w4),x,1,1top(x,w4)))):is"n"(n4,inn(1px,u3(w4)))
9443 t72:=tris(1to(xp1),u4,outn(xp1,n4,1top(xp1,u4)),left1to(xp1,1px,t47,u3(w4)),isoutinn(xp1,u4),isoutni(xp1,n4,1top(xp1,u4),inn(1px,u3(w4)),trlessis"n"(inn(1px,u3(w4)),1px,xp1,1top(1px,u3(w4)),t47),t71)):is"e"(1to(xp1),u4,left1to(xp1,1px,t47,u3(w4)))
9444 t73:=tris(1to(xp1),left1to(xp1,1px,t47,u3),<u4>s,<u3(w4)>s02,thinvf2(1to(xp1),1to(xp1),s,b,left1to(xp1,1px,t47,u3)),isf(1to(xp1),1to(xp1),s,u4,left1to(xp1,1px,t47,u3(w4)),t72)):is"e"(1to(xp1),left1to(xp1,1px,t47,u3),<u3(w4)>s02)
9445 t74:=tr3is(nat,pl"n"(1,inn(x,u)),inn(1px,u3),inn(xp1,left1to(xp1,1px,t47,u3)),n3(w4),isinoutn(1px,pl"n"(1,inn(x,u)),satz19o(inn(x,u),x,1,1top(x,u))),isinoutn(xp1,inn(1px,u3),trlessis"n"(inn(1px,u3),1px,xp1,1top(1px,u3),t47)),isinni(xp1,left1to(xp1,1px,t47,u3),<u3(w4)>s02,t73)):is"n"(pl"n"(1,inn(x,u)),n3(w4))
9446 t75:=th1e"n.mn"(n3(w4),1,inn(x,u),t54(w4),t74):is"n"(inn(x,u),nm1(w4))
9447 t76:=tris(1to(x),u,outn(x,inn(x,u),1top(x,u)),w3(w4),isoutinn(x,u),isoutni(x,inn(x,u),1top(x,u),nm1(w4),t55(w4),t75)):is"e"(1to(x),u,w3(w4))
9448 t77:=somei(1to(x),[t:1to(x)]is"e"(1to(x),u,w3(t)),w4,t76):image(1to(x),1to(x),s03,u)
9449 case2@t78:=andi(injective(1to(x),1to(x),s03),surjective(1to(x),1to(x),s03),[t:1to(x)][u:1to(x)][v:is"e"(1to(x),<t>s03,<u>s03)]t61(t,u,v),[t:1to(x)]t77(t)):bijective(1to(x),1to(x),s03)
9450 f02:=left(cx,xp1,1px,t47,f):[t:1to(1px)]cx
9451 f03:=right(cx,1,x,f02):[t:1to(x)]cx
9452 t79:=<t78><f03><s03>p:prop1(s03,f03)
9453 g3:=left(cx,xp1,1px,t47,g(xp1,s,f)):[t:1to(1px)]cx
9454 g4:=right(cx,1,x,g3):[t:1to(x)]cx
9455 g5:=g(x,s03,f03):[t:1to(x)]cx
9456 u@t80:=tr3is(nat,n3,pl"n"(1,nm1),pl"n"(1,inn(x,w3)),inn(1px,right1to(1,x,w3)),th1a"n.mn"(n3,1,t54),ispl2"n"(nm1,inn(x,w3),1,isinoutn(x,nm1,t55)),isinoutn(1px,pl"n"(1,inn(x,w3)),satz19o(inn(x,w3),x,1,1top(x,w3)))):is"n"(n3,inn(1px,right1to(1,x,w3)))
9457 t81:=tris(1to(xp1),<u3>s02,outn(xp1,n3,1top(xp1,<u3>s02)),left1to(xp1,1px,t47,right1to(1,x,w3)),isoutinn(xp1,<u3>s02),isoutni(xp1,n3,1top(xp1,<u3>s02),inn(1px,right1to(1,x,w3)),trlessis"n"(inn(1px,right1to(1,x,w3)),1px,xp1,1top(1px,right1to(1,x,w3)),t47),t80)):is"e"(1to(xp1),<u3>s02,left1to(xp1,1px,t47,right1to(1,x,w3)))
9458 t82:=isf(1to(xp1),cx,f,<u3>s02,left1to(xp1,1px,t47,right1to(1,x,w3)),t81):is(<u>g4,<u>g5)
9459 case2@t83:=fisi(1to(x),cx,g4,g5,[t:1to(x)]t82(t)):is"e"([t:1to(x)]cx,g4,g5)
9460 t85:=isf([t:1to(x)]cx,cx,[u:[t:1to(x)]cx]smpr(x,u),g4,g5,t83):is(smpr(x,g4),smpr(x,g5))
9461 t86:=tris(cx,smpr(x,g4),smpr(x,g5),smpr(x,f03),t85,t79):is(smpr(x,g4),smpr(x,f03))
9462 t87:=lessisi1"n"(1,1px,satz18a(1,x)):lessis"n"(1,1px)
9463 g6:=left(cx,1px,1,t87,g3):[t:1to(1)]cx
9464 f04:=left(cx,1px,1,t87,f02):[t:1to(1)]cx
9465 t87a:=isinoutn(1,1,lessisi3(1)):is"n"(1,inn(1,xout(1)))
9466 1d:=left1to(1px,1,t87,xout(1)):1to(1px)
9467 t87b:=isinoutn(1px,inn(1,xout(1)),trlessis"n"(inn(1,xout(1)),1,1px,1top(1,xout(1)),t87)):is"n"(inn(1,xout(1)),inn(1px,1d))
9468 t88:=tris(nat,1,inn(1,xout(1)),inn(1px,1d),t87a,t87b):is"n"(1,inn(1px,1d))
9469 1e:=left1to(xp1,1px,t47,1d):1to(xp1)
9470 t88a:=isoutni(xp1,1,satz24a(xp1),inn(1px,1d),trlessis"n"(inn(1px,1d),1px,xp1,1top(1px,1d),t47),t88):is"e"(1to(xp1),1out(xp1),1e)
9471 t88b:=isf(1to(xp1),1to(xp1),s,1e,1out(xp1),symis(1to(xp1),1out(xp1),1e,t88a)):is"e"(1to(xp1),<1e>s,<1out(xp1)>s)
9472 t89:=tr3is(1to(xp1),<1e>s,<1out(xp1)>s,1out(xp1),1e,t88b,case2,t88a):is"e"(1to(xp1),<1e>s,1e)
9473 t89a:=tr3is(cx,smpr(1,g6),<xout(1)>g6,<xout(1)>f04,smpr(1,f04),satz277(g6),isf(1to(xp1),cx,f,<1e>s,1e,t89),symis(cx,smpr(1,f04),<xout(1)>f04,satz277(f04))):is(smpr(1,g6),smpr(1,f04))
9474 t90:=satz281(1,x,g3):is(smpr(1px,g3),<smpr(x,g4)><smpr(1,g6)>q)
9475 t91:=isf(cx,cx,<smpr(1,g6)>q,smpr(x,g4),smpr(x,f03),t86):is(<smpr(x,g4)><smpr(1,g6)>q,<smpr(x,f03)><smpr(1,g6)>q)
9476 t92:=isf(cx,cx,[t:cx]<smpr(x,f03)><t>q,smpr(1,g6),smpr(1,f04),t89a):is(<smpr(x,f03)><smpr(1,g6)>q,<smpr(x,f03)><smpr(1,f04)>q)
9477 t93:=symis(cx,smpr(1px,f02),<smpr(x,f03)><smpr(1,f04)>q,satz281(1,x,f02)):is(<smpr(x,f03)><smpr(1,f04)>q,smpr(1px,f02))
9478 t94:=tr4is(cx,smpr(1px,g3),<smpr(x,g4)><smpr(1,g6)>q,<smpr(x,f03)><smpr(1,g6)>q,<smpr(x,f03)><smpr(1,f04)>q,smpr(1px,f02),t90,t91,t92,t93):is(smpr(1px,g3),smpr(1px,f02))
9479 t95:=issmpr(xp1,f,1px,compl"n"(1,x)):is(smpr(1px,f02),smpr(xp1,f))
9480 t96:=symis(cx,smpr(1px,g3),smpr(xp1,g(xp1,s,f)),issmpr(xp1,g(xp1,s,f),1px,compl"n"(1,x))):is(smpr(xp1,g(xp1,s,f)),smpr(1px,g3))
9481 t97:=tr3is(cx,smpr(xp1,g(xp1,s,f)),smpr(1px,g3),smpr(1px,f02),smpr(xp1,f),t96,t94,t95):prop1(xp1,s,f)
9482 b@[not1:not(is"e"(1to(xp1),<xout(xp1)>s,xout(xp1)))][not2:not(is"e"(1to(xp1),<1out(xp1)>s,1out(xp1)))]
9483 a0:=<1out(xp1)>s:1to(xp1)
9484 b0:=<1out(xp1)>invf(1to(xp1),1to(xp1),s,b):1to(xp1)
9485 t98:=thinvf2(1to(xp1),1to(xp1),s,b,1out(xp1)):is"e"(1to(xp1),1out(xp1),<b0>s)
9486 t99:=not2:not(is"e"(1to(xp1),a0,1out(xp1)))
9487 t100:=symnotis(1to(xp1),<1out(xp1)>s,<b0>s,th3"e.notis"(1to(xp1),<1out(xp1)>s,1out(xp1),<b0>s,not2,t98)):not(is"e"(1to(xp1),<b0>s,<1out(xp1)>s))
9488 t101:=th3"l.imp"(is"e"(1to(xp1),b0,1out(xp1)),is"e"(1to(xp1),<b0>s,<1out(xp1)>s),t100,[t:is"e"(1to(xp1),b0,1out(xp1))]isf(1to(xp1),1to(xp1),s,b0,1out(xp1),t)):not(is"e"(1to(xp1),b0,1out(xp1)))
9489 s1:=changef(1to(xp1),1to(xp1),s,1out(xp1),b0):[t:1to(xp1)]1to(xp1)
9490 [u:1to(xp1)][i:is"e"(1to(xp1),u,1out(xp1))]
9491 t102:=changef1(1to(xp1),1to(xp1),s,1out(xp1),b0,u,i):is"e"(1to(xp1),<u>s1,<b0>s)
9492 t103:=tris2(1to(xp1),<u>s1,1out(xp1),<b0>s,t102,t98):is"e"(1to(xp1),<u>s1,1out(xp1))
9493 u@[i:is"e"(1to(xp1),u,b0)]
9494 t104:=changef2(1to(xp1),1to(xp1),s,1out(xp1),b0,u,i):is"e"(1to(xp1),<u>s1,a0)
9495 u@[n:not(is"e"(1to(xp1),u,1out(xp1)))][o:not(is"e"(1to(xp1),u,b0))]
9496 t105:=changef3(1to(xp1),1to(xp1),s,1out(xp1),b0,u,n,o):is"e"(1to(xp1),<u>s1,<u>s)
9497 not2@t106:=th6"e.wissel"(1to(xp1),1to(xp1),s,1out(xp1),b0,b):bijective(1to(xp1),1to(xp1),s1)
9498 [alpha:not(is"e"(1to(xp1),a0,xout(xp1)))]
9499 s2:=wissel(1to(xp1),1out(xp1),a0):[t:1to(xp1)]1to(xp1)
9500 t107:=th3"e.wissel"(1to(xp1),1out(xp1),a0):bijective(1to(xp1),1to(xp1),s2)
9501 [u:1to(xp1)][i:is"e"(1to(xp1),u,1out(xp1))]
9502 t108:=iswissel1(1to(xp1),1out(xp1),a0,<u>s1,t103(u,i)):is"e"(1to(xp1),<<u>s1>s2,a0)
9503 t109:=tris2(1to(xp1),<u>s,<<u>s1>s2,a0,isf(1to(xp1),1to(xp1),s,u,1out(xp1),i),t108):is"e"(1to(xp1),<u>s,<<u>s1>s2)
9504 u@[i:is"e"(1to(xp1),u,b0)]
9505 t110:=iswissel2(1to(xp1),1out(xp1),a0,<u>s1,t104(u,i)):is"e"(1to(xp1),<<u>s1>s2,1out(xp1))
9506 t111:=tris2(1to(xp1),<u>s,<<u>s1>s2,1out(xp1),tris2(1to(xp1),<u>s,1out(xp1),<b0>s,isf(1to(xp1),1to(xp1),s,u,b0,i),t98),t110):is"e"(1to(xp1),<u>s,<<u>s1>s2)
9507 u@[n:not(is"e"(1to(xp1),u,1out(xp1)))][o:not(is"e"(1to(xp1),u,b0))][i:is"e"(1to(xp1),<u>s1,1out(xp1))]
9508 t112:=isfe(1to(xp1),1to(xp1),s1,t8(s1,f,t106),u,1out(xp1),tris2(1to(xp1),<u>s1,<1out(xp1)>s1,1out(xp1),i,t103(1out(xp1),refis(1to(xp1),1out(xp1))))):is"e"(1to(xp1),u,1out(xp1))
9509 o@t113:=th3"l.imp"(is"e"(1to(xp1),<u>s1,1out(xp1)),is"e"(1to(xp1),u,1out(xp1)),n,[t:is"e"(1to(xp1),<u>s1,1out(xp1))]t112(t)):not(is"e"(1to(xp1),<u>s1,1out(xp1)))
9510 [i:is"e"(1to(xp1),<u>s1,a0)]
9511 t114:=isfe(1to(xp1),1to(xp1),s1,t8(s1,f,t106),u,b0,tris2(1to(xp1),<u>s1,<b0>s1,a0,i,t104(b0,refis(1to(xp1),b0)))):is"e"(1to(xp1),u,b0)
9512 o@t115:=th3"l.imp"(is"e"(1to(xp1),<u>s1,a0),is"e"(1to(xp1),u,b0),o,[t:is"e"(1to(xp1),<u>s1,a0)]t114(t)):not(is"e"(1to(xp1),<u>s1,a0))
9513 t116:=iswissel3(1to(xp1),1out(xp1),a0,<u>s1,t113,t115):is"e"(1to(xp1),<<u>s1>s2,<u>s1)
9514 t117:=symis(1to(xp1),<<u>s1>s2,<u>s,tris(1to(xp1),<<u>s1>s2,<u>s1,<u>s,t116,t105(u,n,o))):is"e"(1to(xp1),<u>s,<<u>s1>s2)
9515 n@t118:=th1"l.imp"(is"e"(1to(xp1),u,b0),is"e"(1to(xp1),<u>s,<<u>s1>s2),[t:is"e"(1to(xp1),u,b0)]t111(t),[t:not(is"e"(1to(xp1),u,b0))]t117(t)):is"e"(1to(xp1),<u>s,<<u>s1>s2)
9516 u@t119:=th1"l.imp"(is"e"(1to(xp1),u,1out(xp1)),is"e"(1to(xp1),<u>s,<<u>s1>s2),[t:is"e"(1to(xp1),u,1out(xp1))]t109(t),[t:not(is"e"(1to(xp1),u,1out(xp1)))]t118(t)):is"e"(1to(xp1),<u>s,<<u>s1>s2)
9517 alpha@t120:=fisi(1to(xp1),1to(xp1),s,[t:1to(xp1)]<<t>s1>s2,[t:1to(xp1)]t119(t)):is"e"([t:1to(xp1)]1to(xp1),s,[t:1to(xp1)]<<t>s1>s2)
9518 not2@t121:=t103(1out(xp1),refis(1to(xp1),1out(xp1))):is"e"(1to(xp1),<1out(xp1)>s1,1out(xp1))
9519 t121a:=ec3e21(is"n"(xp1,1),more"n"(xp1,1),less"n"(xp1,1),satz10b(xp1,1),ismore1"n"(1px,xp1,1,compl"n"(1,x),satz18(1,x))):nis"n"(xp1,1)
9520 t122:=th3"l.imp"(is"e"(1to(xp1),xout(xp1),1out(xp1)),is"n"(xp1,1),t121a,[t:is"e"(1to(xp1),xout(xp1),1out(xp1))]isoutne(xp1,xp1,lessisi3(xp1),1,satz24a(xp1),t)):not(is"e"(1to(xp1),xout(xp1),1out(xp1)))
9521 alpha@t123:=symnotis(1to(xp1),a0,xout(xp1),alpha):not(is"e"(1to(xp1),xout(xp1),a0))
9522 t124:=iswissel3(1to(xp1),1out(xp1),a0,xout(xp1),t122,t123):is"e"(1to(xp1),<xout(xp1)>s2,xout(xp1))
9523 t125:=isf([t:1to(xp1)]1to(xp1),cx,[u:[t:1to(xp1)]1to(xp1)]smpr(xp1,g(xp1,u,f)),s,[t:1to(xp1)]<<t>s1>s2,t120):is(smpr(xp1,g(xp1,s,f)),smpr(xp1,g(xp1,s1,g(xp1,s2,f))))
9524 t126:=t97(s1,g(xp1,s2,f),t106,t121):is(smpr(xp1,g(xp1,s1,g(xp1,s2,f))),smpr(xp1,g(xp1,s2,f)))
9525 t127:=t46(s2,f,t107,t124):is(smpr(xp1,g(xp1,s2,f)),smpr(xp1,f))
9526 t128:=tr3is(cx,smpr(xp1,g(xp1,s,f)),smpr(xp1,g(xp1,s1,g(xp1,s2,f))),smpr(xp1,g(xp1,s2,f)),smpr(xp1,f),t125,t126,t127):prop1(xp1,s,f)
9527 not2@[i3:is"e"(1to(xp1),a0,xout(xp1))][beta:not(is"e"(1to(xp1),b0,xout(xp1)))]
9528 s3:=wissel(1to(xp1),1out(xp1),b0):[t:1to(xp1)]1to(xp1)
9529 t129:=th3"e.wissel"(1to(xp1),1out(xp1),b0):bijective(1to(xp1),1to(xp1),s3)
9530 [u:1to(xp1)][i:is"e"(1to(xp1),u,1out(xp1))]
9531 t130:=t104(<u>s3,iswissel1(1to(xp1),1out(xp1),b0,u,i)):is"e"(1to(xp1),<<u>s3>s1,a0)
9532 t131:=tris2(1to(xp1),<u>s,<<u>s3>s1,a0,isf(1to(xp1),1to(xp1),s,u,1out(xp1),i),t130):is"e"(1to(xp1),<u>s,<<u>s3>s1)
9533 u@[i:is"e"(1to(xp1),u,b0)]
9534 t132:=t103(<u>s3,iswissel2(1to(xp1),1out(xp1),b0,u,i)):is"e"(1to(xp1),<<u>s3>s1,1out(xp1))
9535 t133:=tris2(1to(xp1),<u>s,<<u>s3>s1,<b0>s,isf(1to(xp1),1to(xp1),s,u,b0,i),tris(1to(xp1),<<u>s3>s1,1out(xp1),<b0>s,t132,t98)):is"e"(1to(xp1),<u>s,<<u>s3>s1)
9536 u@[n:not(is"e"(1to(xp1),u,1out(xp1)))][o:not(is"e"(1to(xp1),u,b0))]
9537 t134:=iswissel3(1to(xp1),1out(xp1),b0,u,n,o):is"e"(1to(xp1),<u>s3,u)
9538 t135:=isf(1to(xp1),1to(xp1),s1,<u>s3,u,t134):is"e"(1to(xp1),<<u>s3>s1,<u>s1)
9539 t136:=t105(u,n,o):is"e"(1to(xp1),<u>s1,<u>s)
9540 t139:=symis(1to(xp1),<<u>s3>s1,<u>s,tris(1to(xp1),<<u>s3>s1,<u>s1,<u>s,t135,t136)):is"e"(1to(xp1),<u>s,<<u>s3>s1)
9541 n@t140:=th1"l.imp"(is"e"(1to(xp1),u,b0),is"e"(1to(xp1),<u>s,<<u>s3>s1),[t:is"e"(1to(xp1),u,b0)]t133(t),[t:not(is"e"(1to(xp1),u,b0))]t139(t)):is"e"(1to(xp1),<u>s,<<u>s3>s1)
9542 u@t141:=th1"l.imp"(is"e"(1to(xp1),u,1out(xp1)),is"e"(1to(xp1),<u>s,<<u>s3>s1),[t:is"e"(1to(xp1),u,1out(xp1))]t131(t),[t:not(is"e"(1to(xp1),u,1out(xp1)))]t140(t)):is"e"(1to(xp1),<u>s,<<u>s3>s1)
9543 beta@t142:=fisi(1to(xp1),1to(xp1),s,[t:1to(xp1)]<<t>s3>s1,[t:1to(xp1)]t141(t)):is"e"([t:1to(xp1)]1to(xp1),s,[t:1to(xp1)]<<t>s3>s1)
9544 t143:=symnotis(1to(xp1),b0,xout(xp1),beta):not(is"e"(1to(xp1),xout(xp1),b0))
9545 t144:=iswissel3(1to(xp1),1out(xp1),b0,xout(xp1),t122,t143):is"e"(1to(xp1),<xout(xp1)>s3,xout(xp1))
9546 t145:=isf([t:1to(xp1)]1to(xp1),cx,[u:[t:1to(xp1)]1to(xp1)]smpr(xp1,g(xp1,u,f)),s,[t:1to(xp1)]<<t>s3>s1,t142):is(smpr(xp1,g(xp1,s,f)),smpr(xp1,g(xp1,s3,g(xp1,s1,f))))
9547 t146:=t46(s3,g(xp1,s1,f),t129,t144):is(smpr(xp1,g(xp1,s3,g(xp1,s1,f))),smpr(xp1,g(xp1,s1,f)))
9548 t147:=t97(s1,f,t106,t121):is(smpr(xp1,g(xp1,s1,f)),smpr(xp1,f))
9549 t148:=tr3is(cx,smpr(xp1,g(xp1,s,f)),smpr(xp1,g(xp1,s3,g(xp1,s1,f))),smpr(xp1,g(xp1,s1,f)),smpr(xp1,f),t145,t146,t147):prop1(xp1,s,f)
9550 i3@[gamma:is"e"(1to(xp1),b0,xout(xp1))][i:is"n"(x,1)]
9551 t149:=ispl1"n"(1,x,1,symis(nat,x,1,i)):is"n"(2,xp1)
9552 t150:=lessisi2"n"(2,xp1,t149):lessis"n"(2,xp1)
9553 f05:=left(cx,xp1,2,t150,f):[t:1to(2)]cx
9554 s05:=left(1to(xp1),xp1,2,t150,s):[t:1to(2)]1to(xp1)
9555 g7:=left(cx,xp1,2,t150,g(xp1,s,f)):[t:1to(2)]cx
9556 t151:=issmpr(xp1,f,2,t149):is(smpr(2,f05),smpr(xp1,f))
9557 t152:=issmpr(xp1,g(xp1,s,f),2,t149):is(smpr(2,g7),smpr(xp1,g(xp1,s,f)))
9558 t153:=satz280(f05):is(smpr(2,f05),<<xout(2)>f05><<1out(2)>f05>q)
9559 t154:=satz280(g7):is(smpr(2,g7),<<xout(2)>g7><<1out(2)>g7>q)
9560 t155:=tris1(nat,inn(2,xout(2)),xp1,2,isinoutn(2,2,lessisi3(2)),t149):is"n"(inn(2,xout(2)),xp1)
9561 t156:=isoutni(xp1,inn(2,xout(2)),trlessis"n"(inn(2,xout(2)),2,xp1,1top(2,xout(2)),t150),xp1,lessisi3(xp1),t155):is"e"(1to(xp1),left1to(xp1,2,t150,xout(2)),xout(xp1))
9562 t157:=isf(1to(xp1),1to(xp1),s,left1to(xp1,2,t150,xout(2)),xout(xp1),t156):is"e"(1to(xp1),<xout(2)>s05,<xout(xp1)>s)
9563 t158:=symis(nat,1,inn(2,1out(2)),isinoutn(2,1,satz24a(2))):is"n"(inn(2,1out(2)),1)
9564 t159:=isoutni(xp1,inn(2,1out(2)),trlessis"n"(inn(2,1out(2)),2,xp1,1top(2,1out(2)),t150),1,satz24a(xp1),t158):is"e"(1to(xp1),left1to(xp1,2,t150,1out(2)),1out(xp1))
9565 t160:=tr3is(1to(xp1),left1to(xp1,2,t150,1out(2)),1out(xp1),<b0>s,<xout(xp1)>s,t159,t98,isf(1to(xp1),1to(xp1),s,b0,xout(xp1),gamma)):is"e"(1to(xp1),left1to(xp1,2,t150,1out(2)),<xout(xp1)>s)
9566 t161:=tris2(1to(xp1),<xout(2)>s05,left1to(xp1,2,t150,1out(2)),<xout(xp1)>s,t157,t160):is"e"(1to(xp1),<xout(2)>s05,left1to(xp1,2,t150,1out(2)))
9567 t163:=isf(1to(xp1),1to(xp1),s,left1to(xp1,2,t150,1out(2)),1out(xp1),t159):is"e"(1to(xp1),<1out(2)>s05,<1out(xp1)>s)
9568 t164:=tris(1to(xp1),<1out(2)>s05,<1out(xp1)>s,xout(xp1),t163,i3):is"e"(1to(xp1),<1out(2)>s05,xout(xp1))
9569 t165:=tris2(1to(xp1),<1out(2)>s05,left1to(xp1,2,t150,xout(2)),xout(xp1),t164,t156):is"e"(1to(xp1),<1out(2)>s05,left1to(xp1,2,t150,xout(2)))
9570 t166:=isf(1to(xp1),cx,f,<xout(2)>s05,left1to(xp1,2,t150,1out(2)),t161):is(<xout(2)>g7,<1out(2)>f05)
9571 t167:=isf(1to(xp1),cx,f,<1out(2)>s05,left1to(xp1,2,t150,xout(2)),t165):is(<1out(2)>g7,<xout(2)>f05)
9572 t168:=isf(cx,cx,<<1out(2)>g7>q,<xout(2)>g7,<1out(2)>f05,t166):is(<<xout(2)>g7><<1out(2)>g7>q,<<1out(2)>f05><<1out(2)>g7>q)
9573 t169:=comq(c,<1out(2)>g7,<1out(2)>f05):is(<<1out(2)>f05><<1out(2)>g7>q,<<1out(2)>g7><<1out(2)>f05>q)
9574 t170:=isf(cx,cx,<<1out(2)>f05>q,<1out(2)>g7,<xout(2)>f05,t167):is(<<1out(2)>g7><<1out(2)>f05>q,<<xout(2)>f05><<1out(2)>f05>q)
9575 t171:=tr4is(cx,smpr(xp1,g(xp1,s,f)),smpr(2,g7),<<xout(2)>g7><<1out(2)>g7>q,<<1out(2)>f05><<1out(2)>g7>q,<<1out(2)>g7><<1out(2)>f05>q,symis(cx,smpr(2,g7),smpr(xp1,g(xp1,s,f)),t152),t154,t168,t169):is(smpr(xp1,g(xp1,s,f)),<<1out(2)>g7><<1out(2)>f05>q)
9576 t172:=tr4is(cx,smpr(xp1,g(xp1,s,f)),<<1out(2)>g7><<1out(2)>f05>q,<<xout(2)>f05><<1out(2)>f05>q,smpr(2,f05),smpr(xp1,f),t171,t170,symis(cx,smpr(2,f05),<<xout(2)>f05><<1out(2)>f05>q,t153),t151):prop1(xp1,s,f)
9577 trivial:=t172:prop1(xp1,s,f)
9578 gamma@[n:nis"n"(x,1)]
9579 t173:=ore1(more"n"(x,1),is"n"(x,1),satz24(x),n):more"n"(x,1)
9580 xm1:=mn"n"(x,1,t173):nat
9581 s4:=changef(1to(xp1),1to(xp1),s,1out(xp1),xout(xp1)):[t:1to(xp1)]1to(xp1)
9582 t174:=th6"e.wissel"(1to(xp1),1to(xp1),s,1out(xp1),xout(xp1),b):bijective(1to(xp1),1to(xp1),s4)
9583 t175:=changef2(1to(xp1),1to(xp1),s,1out(xp1),xout(xp1),xout(xp1),refis(1to(xp1),xout(xp1))):is"e"(1to(xp1),<xout(xp1)>s4,<1out(xp1)>s)
9584 t176:=symis(1to(xp1),a0,xout(xp1),i3):is"e"(1to(xp1),xout(xp1),<1out(xp1)>s)
9585 t177:=tris2(1to(xp1),<xout(xp1)>s4,xout(xp1),<1out(xp1)>s,t175,t176):is"e"(1to(xp1),<xout(xp1)>s4,xout(xp1))
9586 t178:=lessisi1"n"(x,xp1,satz18a(x,1)):lessis"n"(x,xp1)
9587 g8:=left(cx,xp1,x,t178,g(xp1,s,f)):[t:1to(x)]cx
9588 t179:=satz278(x,g(xp1,s,f)):is(smpr(xp1,g(xp1,s,f)),<<xout(xp1)>g(xp1,s,f)><smpr(x,g8)>q)
9589 t180:=th1b"n.mn"(x,1,t173):is"n"(pl"n"(1,xm1),x)
9590 t181:=lessisi2"n"(pl"n"(1,xm1),x,t180):lessis"n"(pl"n"(1,xm1),x)
9591 g9:=left(cx,x,pl"n"(1,xm1),t181,g8):[t:1to(pl"n"(1,xm1))]cx
9592 t182:=symis(cx,smpr(pl"n"(1,xm1),g9),smpr(x,g8),issmpr(x,g8,pl"n"(1,xm1),t180)):is(smpr(x,g8),smpr(pl"n"(1,xm1),g9))
9593 g10:=right(cx,1,xm1,g9):[t:1to(xm1)]cx
9594 t183:=lessisi1"n"(1,pl"n"(1,xm1),satz18a(1,xm1)):lessis"n"(1,pl"n"(1,xm1))
9595 g11:=left(cx,pl"n"(1,xm1),1,t183,g9):[t:1to(1)]cx
9596 t184:=satz281(a,1,xm1,g9):is(smpr(pl"n"(1,xm1),g9),<smpr(xm1,g10)><smpr(1,g11)>q)
9597 t185:=satz277(g11):is(smpr(1,g11),<xout(1)>g11)
9598 t186:=isinoutn(1,1,lessisi3(1)):is"n"(1,inn(1,xout(1)))
9599 1a:=left1to(pl"n"(1,xm1),1,t183,xout(1)):1to(pl"n"(1,xm1))
9600 t187:=isinoutn(pl"n"(1,xm1),inn(1,xout(1)),trlessis"n"(inn(1,xout(1)),1,pl"n"(1,xm1),1top(1,xout(1)),t183)):is"n"(inn(1,xout(1)),inn(pl"n"(1,xm1),1a))
9601 1b:=left1to(x,pl"n"(1,xm1),t181,1a):1to(x)
9602 t188:=isinoutn(x,inn(pl"n"(1,xm1),1a),trlessis"n"(inn(pl"n"(1,xm1),1a),pl"n"(1,xm1),x,1top(pl"n"(1,xm1),1a),t181)):is"n"(inn(pl"n"(1,xm1),1a),inn(x,1b))
9603 t189:=tr3is(nat,1,inn(1,xout(1)),inn(pl"n"(1,xm1),1a),inn(x,1b),t186,t187,t188):is"n"(1,inn(x,1b))
9604 1c:=left1to(xp1,x,t178,1b):1to(xp1)
9605 t190:=isoutni(xp1,1,satz24a(xp1),inn(x,1b),trlessis"n"(inn(x,1b),x,xp1,1top(x,1b),t178),t189):is"e"(1to(xp1),1out(xp1),1c)
9606 t191:=isf(1to(xp1),cx,g(xp1,s,f),1out(xp1),1c,t190):is(<1out(xp1)>g(xp1,s,f),<xout(1)>g11)
9607 t192:=tris2(cx,smpr(1,g11),<1out(xp1)>g(xp1,s,f),<xout(1)>g11,t185,t191):is(smpr(1,g11),<1out(xp1)>g(xp1,s,f))
9608 t193:=symis(1to(xp1),<xout(xp1)>s4,<1out(xp1)>s,t175):is"e"(1to(xp1),<1out(xp1)>s,<xout(xp1)>s4)
9609 t194:=isf(1to(xp1),cx,f,<1out(xp1)>s,<xout(xp1)>s4,t193):is(<1out(xp1)>g(xp1,s,f),<xout(xp1)>g(xp1,s4,f))
9610 t195:=tris(cx,smpr(1,g11),<1out(xp1)>g(xp1,s,f),<xout(xp1)>g(xp1,s4,f),t192,t194):is(smpr(1,g11),<xout(xp1)>g(xp1,s4,f))
9611 t196:=isf(cx,cx,[t:cx]<smpr(xm1,g10)><t>q,smpr(1,g11),<xout(xp1)>g(xp1,s4,f),t195):is(<smpr(xm1,g10)><smpr(1,g11)>q,<smpr(xm1,g10)><<xout(xp1)>g(xp1,s4,f)>q)
9612 t197:=tr3is(cx,smpr(x,g8),smpr(pl"n"(1,xm1),g9),<smpr(xm1,g10)><smpr(1,g11)>q,<smpr(xm1,g10)><<xout(xp1)>g(xp1,s4,f)>q,t182,t184,t196):is(smpr(x,g8),<smpr(xm1,g10)><<xout(xp1)>g(xp1,s4,f)>q)
9613 t198:=isf(cx,cx,[t:cx]<<xout(xp1)>g(xp1,s,f)><t>q,smpr(x,g8),<smpr(xm1,g10)><<xout(xp1)>g(xp1,s4,f)>q,t197):is(<<xout(xp1)>g(xp1,s,f)><smpr(x,g8)>q,<<xout(xp1)>g(xp1,s,f)><<smpr(xm1,g10)><<xout(xp1)>g(xp1,s4,f)>q>q)
9614 t199:=assq1(a,<xout(xp1)>g(xp1,s4,f),smpr(xm1,g10),<xout(xp1)>g(xp1,s,f)):is(<<xout(xp1)>g(xp1,s,f)><<smpr(xm1,g10)><<xout(xp1)>g(xp1,s4,f)>q>q,<<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q><<xout(xp1)>g(xp1,s4,f)>q)
9615 t200:=comq(c,<xout(xp1)>g(xp1,s4,f),<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q):is(<<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q><<xout(xp1)>g(xp1,s4,f)>q,<<xout(xp1)>g(xp1,s4,f)><<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q>q)
9616 g12:=left(cx,xp1,x,t178,g(xp1,s4,f)):[t:1to(x)]cx
9617 g13:=left(cx,x,pl"n"(1,xm1),t181,g12):[t:1to(pl"n"(1,xm1))]cx
9618 g14:=right(cx,1,xm1,g13):[t:1to(xm1)]cx
9619 g15:=left(cx,pl"n"(1,xm1),1,lessisi1"n"(1,pl"n"(1,xm1),satz18a(1,xm1)),g13):[t:1to(1)]cx
9620 t201:=satz278(x,g(xp1,s4,f)):is(smpr(xp1,g(xp1,s4,f)),<<xout(xp1)>g(xp1,s4,f)><smpr(x,g12)>q)
9621 t202:=symis(cx,smpr(pl"n"(1,xm1),g13),smpr(x,g12),issmpr(x,g12,pl"n"(1,xm1),t180)):is(smpr(x,g12),smpr(pl"n"(1,xm1),g13))
9622 t203:=satz281(a,1,xm1,g13):is(smpr(pl"n"(1,xm1),g13),<smpr(xm1,g14)><smpr(1,g15)>q)
9623 t204:=satz277(g15):is(smpr(1,g15),<xout(1)>g15)
9624 t205:=isf(1to(xp1),cx,g(xp1,s4,f),1out(xp1),1c,t190):is(<1out(xp1)>g(xp1,s4,f),<xout(1)>g15)
9625 t206:=tris2(cx,smpr(1,g15),<1out(xp1)>g(xp1,s4,f),<xout(1)>g15,t204,t205):is(smpr(1,g15),<1out(xp1)>g(xp1,s4,f))
9626 t207:=changef1(1to(xp1),1to(xp1),s,1out(xp1),xout(xp1),1out(xp1),refis(1to(xp1),1out(xp1))):is"e"(1to(xp1),<1out(xp1)>s4,<xout(xp1)>s)
9627 t208:=isf(1to(xp1),cx,f,<1out(xp1)>s4,<xout(xp1)>s,t207):is(<1out(xp1)>g(xp1,s4,f),<xout(xp1)>g(xp1,s,f))
9628 t209:=tris(cx,smpr(1,g15),<1out(xp1)>g(xp1,s4,f),<xout(xp1)>g(xp1,s,f),t206,t208):is(smpr(1,g15),<xout(xp1)>g(xp1,s,f))
9629 t210:=isf(cx,cx,[t:cx]<smpr(xm1,g14)><t>q,smpr(1,g15),<xout(xp1)>g(xp1,s,f),t209):is(<smpr(xm1,g14)><smpr(1,g15)>q,<smpr(xm1,g14)><<xout(xp1)>g(xp1,s,f)>q)
9630 t211:=tr3is(cx,smpr(x,g12),smpr(pl"n"(1,xm1),g13),<smpr(xm1,g14)><smpr(1,g15)>q,<smpr(xm1,g14)><<xout(xp1)>g(xp1,s,f)>q,t202,t203,t210):is(smpr(x,g12),<smpr(xm1,g14)><<xout(xp1)>g(xp1,s,f)>q)
9631 [u:1to(xm1)]
9632 ua:=right1to(1,xm1,u):1to(pl"n"(1,xm1))
9633 ub:=left1to(x,pl"n"(1,xm1),t181,ua):1to(x)
9634 uc:=left1to(xp1,x,t178,ub):1to(xp1)
9635 [i:is"e"(1to(xp1),uc,xout(xp1))]
9636 t212:=isoutne(xp1,inn(x,ub),trlessis"n"(inn(x,ub),x,xp1,1top(x,ub),t178),xp1,lessisi3(xp1),i):is"n"(inn(x,ub),xp1)
9637 t213:=satz16a(inn(x,ub),x,xp1,1top(x,ub),satz18a(x,1)):less"n"(inn(x,ub),xp1)
9638 t214:=<t212>ec3e31(is"n"(inn(x,ub),xp1),more"n"(inn(x,ub),xp1),less"n"(inn(x,ub),xp1),satz10b(inn(x,ub),xp1),t213):con
9639 u@t215:=[t:is"e"(1to(xp1),uc,xout(xp1))]t214(t):not(is"e"(1to(xp1),uc,xout(xp1)))
9640 [i:is"e"(1to(xp1),uc,1out(xp1))]
9641 t216:=isoutne(xp1,inn(x,ub),trlessis"n"(inn(x,ub),x,xp1,1top(x,ub),t178),1,satz24a(xp1),i):is"n"(inn(x,ub),1)
9642 t217:=isinoutn(x,inn(pl"n"(1,xm1),ua),trlessis"n"(inn(pl"n"(1,xm1),ua),pl"n"(1,xm1),x,1top(pl"n"(1,xm1),ua),t181)):is"n"(inn(pl"n"(1,xm1),ua),inn(x,ub))
9643 t218:=isinoutn(pl"n"(1,xm1),pl"n"(1,inn(xm1,u)),satz19o(inn(xm1,u),xm1,1,1top(xm1,u))):is"n"(pl"n"(1,inn(xm1,u)),inn(pl"n"(1,xm1),ua))
9644 t219:=tr3is(nat,pl"n"(1,inn(xm1,u)),inn(pl"n"(1,xm1),ua),inn(x,ub),1,t218,t217,t216):is"n"(pl"n"(1,inn(xm1,u)),1)
9645 t220:=satz18(1,inn(xm1,u)):more"n"(pl"n"(1,inn(xm1,u)),1)
9646 t221:=<t219>ec3e21(is"n"(pl"n"(1,inn(xm1,u)),1),more"n"(pl"n"(1,inn(xm1,u)),1),less"n"(pl"n"(1,inn(xm1,u)),1),satz10b(pl"n"(1,inn(xm1,u)),1),t220):con
9647 u@t222:=[t:is"e"(1to(xp1),uc,1out(xp1))]t221(t):not(is"e"(1to(xp1),uc,1out(xp1)))
9648 t223:=changef3(1to(xp1),1to(xp1),s,1out(xp1),xout(xp1),uc,t222,t215):is"e"(1to(xp1),<uc>s4,<uc>s)
9649 t224:=isf(1to(xp1),cx,f,<uc>s4,<uc>s,t223):is(<u>g14,<u>g10)
9650 n@t225:=fisi(1to(xm1),cx,g10,g14,[t:1to(xm1)]symis(cx,<t>g14,<t>g10,t224(t))):is"e"([t:1to(xm1)]cx,g10,g14)
9651 t226:=isf([t:1to(xm1)]cx,cx,[u:[t:1to(xm1)]cx]smpr(xm1,u),g10,g14,t225):is(smpr(xm1,g10),smpr(xm1,g14))
9652 t227:=comq(c,smpr(xm1,g10),<xout(xp1)>g(xp1,s,f)):is(<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q,<smpr(xm1,g10)><<xout(xp1)>g(xp1,s,f)>q)
9653 t228:=isf(cx,cx,<<xout(xp1)>g(xp1,s,f)>q,smpr(xm1,g10),smpr(xm1,g14),t226):is(<smpr(xm1,g10)><<xout(xp1)>g(xp1,s,f)>q,<smpr(xm1,g14)><<xout(xp1)>g(xp1,s,f)>q)
9654 t229:=tris(cx,<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q,<smpr(xm1,g10)><<xout(xp1)>g(xp1,s,f)>q,<smpr(xm1,g14)><<xout(xp1)>g(xp1,s,f)>q,t227,t228):is(<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q,<smpr(xm1,g14)><<xout(xp1)>g(xp1,s,f)>q)
9655 t230:=tris2(cx,<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q,smpr(x,g12),<smpr(xm1,g14)><<xout(xp1)>g(xp1,s,f)>q,t229,t211):is(<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q,smpr(x,g12))
9656 t231:=isf(cx,cx,[t:cx]<<xout(xp1)>g(xp1,s4,f)><t>q,<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q,smpr(x,g12),t230):is(<<xout(xp1)>g(xp1,s4,f)><<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q>q,<<xout(xp1)>g(xp1,s4,f)><smpr(x,g12)>q)
9657 t232:=symis(cx,smpr(xp1,g(xp1,s4,f)),<<xout(xp1)>g(xp1,s4,f)><smpr(x,g12)>q,t201):is(<<xout(xp1)>g(xp1,s4,f)><smpr(x,g12)>q,smpr(xp1,g(xp1,s4,f)))
9658 t233:=t46(s4,f,t174,t177):is(smpr(xp1,g(xp1,s4,f)),smpr(xp1,f))
9659 t234:=tr4is(cx,smpr(xp1,g(xp1,s,f)),<<xout(xp1)>g(xp1,s,f)><smpr(x,g8)>q,<<xout(xp1)>g(xp1,s,f)><<smpr(xm1,g10)><<xout(xp1)>g(xp1,s4,f)>q>q,<<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q><<xout(xp1)>g(xp1,s4,f)>q,<<xout(xp1)>g(xp1,s4,f)><<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q>q,t179,t198,t199,t200):is(smpr(xp1,g(xp1,s,f)),<<xout(xp1)>g(xp1,s4,f)><<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q>q)
9660 t235:=tr4is(cx,smpr(xp1,g(xp1,s,f)),<<xout(xp1)>g(xp1,s4,f)><<<xout(xp1)>g(xp1,s,f)><smpr(xm1,g10)>q>q,<<xout(xp1)>g(xp1,s4,f)><smpr(x,g12)>q,smpr(xp1,g(xp1,s4,f)),smpr(xp1,f),t234,t231,t232,t233):prop1(xp1,s,f)
9661 gamma@t236:=th1"l.imp"(is"n"(x,1),prop1(xp1,s,f),[t:is"n"(x,1)]t172(t),[t:not(is"n"(x,1))]t235(t)):prop1(xp1,s,f)
9662 i3@t237:=th1"l.imp"(is"e"(1to(xp1),b0,xout(xp1)),prop1(xp1,s,f),[t:is"e"(1to(xp1),b0,xout(xp1))]t236(t),[t:not(is"e"(1to(xp1),b0,xout(xp1)))]t148(t)):prop1(xp1,s,f)
9663 not2@t238:=th1"l.imp"(is"e"(1to(xp1),a0,xout(xp1)),prop1(xp1,s,f),[t:is"e"(1to(xp1),a0,xout(xp1))]t237(t),[t:not(is"e"(1to(xp1),a0,xout(xp1)))]t128(t)):prop1(xp1,s,f)
9664 not1@t239:=th1"l.imp"(is"e"(1to(xp1),<1out(xp1)>s,1out(xp1)),prop1(xp1,s,f),[t:is"e"(1to(xp1),<1out(xp1)>s,1out(xp1))]t97(t),[t:not(is"e"(1to(xp1),<1out(xp1)>s,1out(xp1)))]t238(t)):prop1(xp1,s,f)
9665 b@t240:=th1"l.imp"(is"e"(1to(xp1),<xout(xp1)>s,xout(xp1)),prop1(xp1,s,f),[t:is"e"(1to(xp1),<xout(xp1)>s,xout(xp1))]t46(t),[t:not(is"e"(1to(xp1),<xout(xp1)>s,xout(xp1)))]t239(t)):prop1(xp1,s,f)
9666 p@t241:=[u:[t:1to(xp1)]1to(xp1)][v:[t:1to(xp1)]cx][w:bijective(1to(xp1),1to(xp1),u)]t240(u,v,w):prop2(xp1)
9667 t242:=isp(nat,[t:nat]prop2(t),xp1,<x>suc,t241,satz4a(x)):prop2(<x>suc)
9668 x@t243:=induction([t:nat]prop2(t),t6,[t:nat][u:prop2(t)]t242(t,u),x):prop2(x)
9669 -8283
9670 satz283:=<b><f><s>t243".8283":is(smpr(x,[t:1to(x)]<<t>s>f),smpr(x,f))
9671 @[x:real][ix:intrl(x)][y:real][iy:intrl(y)][ly:lessis(y,x)]
9672 shiftl:=shiftl"r"(x,ix,y,iy,ly):nat
9673 [n:1to(shiftl)]
9674 shiftr:=shiftr"r"(x,ix,y,iy,ly,n):real
9675 intshiftr:=intshiftr"r"(x,ix,y,iy,ly,n):intrl(shiftr)
9676 shiftrls:=shiftrls"r"(x,ix,y,iy,ly,n):lessis(shiftr,x)
9677 lsshiftr:=lsshiftr"r"(x,ix,y,iy,ly,n):lessis(y,shiftr)
9678 [m:1to(shiftl)][i:is"r"(shiftr(n),shiftr(m))]
9679 iseshiftr:=iseshiftr"r"(x,ix,y,iy,ly,n,m,i):is"e"(1to(shiftl),n,m)
9680 ly@[u:real][a:and3(intrl(u),lessis(y,u),lessis(u,x))]
9681 shiftl1:=shiftl1"r"(x,ix,y,iy,ly,u,a):1to(shiftl)
9682 shiftinv1:=shiftinv1"r"(x,ix,y,iy,ly,u,a):is"r"(u,shiftr(shiftl1))
9683 shiftinv2:=shiftinv2"r"(x,ix,y,iy,ly,u,a):is"r"(shiftr(shiftl1),u)
9684 ly@[f:seq(x,ix,y,iy,ly,cx)]
9685 shiftf:=shiftf(x,ix,y,iy,ly,cx,f):[t:1to(shiftl)]cx
9686 [q:[t:cx][u:cx]cx]
9687 smpri:=smpr(q,shiftl,shiftf):cx
9688 f@[pi:proofsirrelevant(x,ix,y,iy,ly,cx,f)][q:[t:cx][u:cx]cx][a:assoc(q)][u:real][iu:intrl(u)][l:lessis(y,u)][k:less(u,x)][v:real][iv:intrl(v)][lv:lessis(y,v)][kv:lessis(v,u)]
9689 +8284
9690 t1:=lessisi1(v,x,satz172a(v,u,x,kv,k)):lessis(v,x)
9691 -8284
9692 k@lft:=[t:real][v:intrl(t)][lt:lessis(y,t)][kt:lessis(t,u)]<t1".8284"(t,v,lt,kt)><lt><v><t>f:[t:real][v:intrl(t)][lt:lessis(y,t)][kt:lessis(t,u)]cx
9693 iv@[lv:lessis(pl"r"(u,1rl),v)][kv:lessis(v,x)]
9694 +*8284
9695 x@[y:real]
9696 pl:=pl"r"(x,y):real
9697 mn:=mn"r"(x,y):real
9698 x@p1:=pl"r"(x,1rl):real
9699 kv@t2:=satz190c(u,u,0,1rl,lessisi2(u,u,refis(real,u)),lemma1"r"(1rl,0,satz169a(1rl,natpos(1rl,natrl1)))):less(pl(u,0),p1(u))
9700 t3:=isless1(pl(u,0),u,p1(u),pl02"r"(u,0,refis(real,0)),t2):less(u,p1(u))
9701 t4:=lessisi1(y,v,satz172b(y,p1(u),v,satz172a(y,u,p1(u),l,t3),lv)):lessis(y,v)
9702 -8284
9703 k@rgt:=[t:real][v:intrl(t)][lt:lessis(pl"r"(u,1rl),t)][kt:lessis(t,x)]<kt><t4".8284"(t,v,lt,kt)><v><t>f:[t:real][v:intrl(t)][lt:lessis(pl"r"(u,1rl),t)][kt:lessis(t,x)]cx
9704 +*8284
9705 k@t5:=intpl(u,iu,1rl,intrl1):intrl(p1(u))
9706 t6:=satzr25(u,iu,x,ix,k):lessis(p1(u),x)
9707 t7:=tr3is(real,pl(mn(p1(u),y),mn(p1(x),p1(u))),pl(mn(p1(x),p1(u)),mn(p1(u),y)),mn(pl(mn(p1(x),p1(u)),p1(u)),y),mn(p1(x),y),compl"r"(mn(p1(u),y),mn(p1(x),p1(u))),asspl2"r"(mn(p1(x),p1(u)),p1(u),m0"r"(y)),ismn1"r"(pl(mn(p1(x),p1(u)),p1(u)),p1(x),y,plmn(p1(x),p1(u)))):is"r"(pl(mn(p1(u),y),mn(p1(x),p1(u))),mn(p1(x),y))
9708 sxy:=shiftl(x,ix,y,iy,ly):nat
9709 suy:=shiftl(u,iu,y,iy,l):nat
9710 sxu:=shiftl(x,ix,p1(u),t5,t6):nat
9711 t8:=tr4is(real,rlofnt(pl"n"(suy,sxu)),pl(rlofnt(suy),rlofnt(sxu)),pl(mn(p1(u),y),mn(p1(x),p1(u))),mn(p1(x),y),rlofnt(sxy),satzr155a(suy,sxu),ispl12"r"(rlofnt(suy),mn(p1(u),y),rlofnt(sxu),mn(p1(x),p1(u)),isrlnt2(mn(p1(u),y),t6"r.shift"(u,iu,y,iy,l)),isrlnt2(mn(p1(x),p1(u)),t6"r.shift"(x,ix,p1(u),t5,t6))),t7,isrlnt1(mn(p1(x),y),t6"r.shift"(x,ix,y,iy,ly))):is"r"(rlofnt(pl"n"(suy,sxu)),rlofnt(sxy))
9712 t9:=isntirl(pl"n"(suy,sxu),sxy,t8):is"n"(pl"n"(suy,sxu),sxy)
9713 t10:=lessisi2"n"(pl"n"(suy,sxu),sxy,t9):lessis"n"(pl"n"(suy,sxu),sxy)
9714 f1:=left(cx,sxy,pl"n"(suy,sxu),t10,shiftf(x,ix,y,iy,ly,f)):[t:1to(pl"n"(suy,sxu))]cx
9715 t11:=issmpr(q,sxy,shiftf(x,ix,y,iy,ly,f),pl"n"(suy,sxu),t9):is(smpr(q,pl"n"(suy,sxu),f1),smpri(x,ix,y,iy,ly,f,q))
9716 fr:=right(cx,suy,sxu,f1):[t:1to(sxu)]cx
9717 t12:=lessisi1"n"(suy,pl"n"(suy,sxu),satz18a(suy,sxu)):lessis"n"(suy,pl"n"(suy,sxu))
9718 fl:=left(cx,pl"n"(suy,sxu),suy,t12,f1):[t:1to(suy)]cx
9719 t12a:=satz281(q,a,suy,sxu,f1):is(smpr(q,pl"n"(suy,sxu),f1),<smpr(q,sxu,fr)><smpr(q,suy,fl)>q)
9720 [n:1to(sxu)]
9721 t13:=isinoutn(pl"n"(suy,sxu),pl"n"(suy,inn(sxu,n)),satz19o(inn(sxu,n),sxu,suy,1top(sxu,n))):is"n"(pl"n"(suy,inn(sxu,n)),inn(pl"n"(suy,sxu),right1to(suy,sxu,n)))
9722 t14:=isinoutn(sxy,inn(pl"n"(suy,sxu),right1to(suy,sxu,n)),trlessis"n"(inn(pl"n"(suy,sxu),right1to(suy,sxu,n)),pl"n"(suy,sxu),sxy,1top(pl"n"(suy,sxu),right1to(suy,sxu,n)),t10)):is"n"(inn(pl"n"(suy,sxu),right1to(suy,sxu,n)),inn(sxy,left1to(sxy,pl"n"(suy,sxu),t10,right1to(suy,sxu,n))))
9723 n1:=left1to(sxy,pl"n"(suy,sxu),t10,right1to(suy,sxu,n)):1to(sxy)
9724 t15:=tris(nat,pl"n"(suy,inn(sxu,n)),inn(pl"n"(suy,sxu),right1to(suy,sxu,n)),inn(sxy,n1),t13,t14):is"n"(pl"n"(suy,inn(sxu,n)),inn(sxy,n1))
9725 t16:=isnterl(pl"n"(suy,inn(sxu,n)),inn(sxy,n1),t15):is"r"(rlofnt(pl"n"(suy,inn(sxu,n))),rlofnt(inn(sxy,n1)))
9726 t17:=satzr155b(suy,inn(sxu,n)):is"r"(pl(rlofnt(suy),rlofnt(inn(sxu,n))),rlofnt(pl"n"(suy,inn(sxu,n))))
9727 t18:=ispl1"r"(mn(p1(u),y),rlofnt(suy),rlofnt(inn(sxu,n)),isrlnt1(mn(p1(u),y),t6"r.shift"(u,iu,y,iy,l))):is"r"(pl(mn(p1(u),y),rlofnt(inn(sxu,n))),pl(rlofnt(suy),rlofnt(inn(sxu,n))))
9728 t19:=tr3is(real,pl(mn(p1(u),y),rlofnt(inn(sxu,n))),pl(rlofnt(suy),rlofnt(inn(sxu,n))),rlofnt(pl"n"(suy,inn(sxu,n))),rlofnt(inn(sxy,n1)),t18,t17,t16):is"r"(pl(mn(p1(u),y),rlofnt(inn(sxu,n))),rlofnt(inn(sxy,n1)))
9729 t20:=ispl1"r"(pl(mn(p1(u),y),rlofnt(inn(sxu,n))),rlofnt(inn(sxy,n1)),y,t19):is"r"(pl(pl(mn(p1(u),y),rlofnt(inn(sxu,n))),y),pl(rlofnt(inn(sxy,n1)),y))
9730 t21:=tr3is(real,pl(pl(mn(p1(u),y),rlofnt(inn(sxu,n))),y),pl(pl(rlofnt(inn(sxu,n)),mn(p1(u),y)),y),pl(rlofnt(inn(sxu,n)),pl(mn(p1(u),y),y)),pl(rlofnt(inn(sxu,n)),p1(u)),ispl1"r"(pl(mn(p1(u),y),rlofnt(inn(sxu,n))),pl(rlofnt(inn(sxu,n)),mn(p1(u),y)),y,compl"r"(mn(p1(u),y),rlofnt(inn(sxu,n)))),asspl1"r"(rlofnt(inn(sxu,n)),mn(p1(u),y),y),ispl2"r"(pl(mn(p1(u),y),y),p1(u),rlofnt(inn(sxu,n)),plmn(p1(u),y))):is"r"(pl(pl(mn(p1(u),y),rlofnt(inn(sxu,n))),y),pl(rlofnt(inn(sxu,n)),p1(u)))
9731 t22:=tris1(real,pl(rlofnt(inn(sxy,n1)),y),pl(rlofnt(inn(sxu,n)),p1(u)),pl(pl(mn(p1(u),y),rlofnt(inn(sxu,n))),y),t20,t21):is"r"(pl(rlofnt(inn(sxy,n1)),y),pl(rlofnt(inn(sxu,n)),p1(u)))
9732 t23:=ismn1"r"(pl(rlofnt(inn(sxy,n1)),y),pl(rlofnt(inn(sxu,n)),p1(u)),1rl,t22):is"r"(shiftr(x,ix,y,iy,ly,n1),shiftr(x,ix,p1(u),t5,t6,n))
9733 t24:=intshiftr(x,ix,y,iy,ly,n1):intrl(shiftr(x,ix,y,iy,ly,n1))
9734 t25:=lsshiftr(x,ix,y,iy,ly,n1):lessis(y,shiftr(x,ix,y,iy,ly,n1))
9735 t26:=shiftrls(x,ix,y,iy,ly,n1):lessis(shiftr(x,ix,y,iy,ly,n1),x)
9736 t27:=intshiftr(x,ix,p1(u),t5,t6,n):intrl(shiftr(x,ix,p1(u),t5,t6,n))
9737 t28:=lsshiftr(x,ix,p1(u),t5,t6,n):lessis(p1(u),shiftr(x,ix,p1(u),t5,t6,n))
9738 t29:=shiftrls(x,ix,p1(u),t5,t6,n):lessis(shiftr(x,ix,p1(u),t5,t6,n),x)
9739 t30:=t4(shiftr(x,ix,p1(u),t5,t6,n),t27,t28,t29):lessis(y,shiftr(x,ix,p1(u),t5,t6,n))
9740 t31:=<t23><t29><t30><t27><shiftr(x,ix,p1(u),t5,t6,n)><t26><t25><t24><shiftr(x,ix,y,iy,ly,n1)>pi:is(<n>fr,<n>shiftf(x,ix,p1(u),t5,t6,rgt))
9741 k@t32:=fisi(1to(sxu),cx,fr,shiftf(x,ix,p1(u),t5,t6,rgt),[t:1to(sxu)]t31(t)):is"e"([t:1to(sxu)]cx,fr,shiftf(x,ix,p1(u),t5,t6,rgt))
9742 t33:=isf([t:1to(sxu)]cx,cx,[v:[t:1to(sxu)]cx]smpr(q,sxu,v),fr,shiftf(x,ix,p1(u),t5,t6,rgt),t32):is(smpr(q,sxu,fr),smpri(x,ix,p1(u),t5,t6,rgt,q))
9743 t34:=isf(cx,cx,<smpr(q,suy,fl)>q,smpr(q,sxu,fr),smpri(x,ix,p1(u),t5,t6,rgt,q),t33):is(<smpr(q,sxu,fr)><smpr(q,suy,fl)>q,<smpri(x,ix,p1(u),t5,t6,rgt,q)><smpr(q,suy,fl)>q)
9744 [n:1to(suy)]
9745 t35:=isinoutn(pl"n"(suy,sxu),inn(suy,n),trlessis"n"(inn(suy,n),suy,pl"n"(suy,sxu),1top(suy,n),t12)):is"n"(inn(suy,n),inn(pl"n"(suy,sxu),left1to(pl"n"(suy,sxu),suy,t12,n)))
9746 t36:=isinoutn(sxy,inn(pl"n"(suy,sxu),left1to(pl"n"(suy,sxu),suy,t12,n)),trlessis"n"(inn(pl"n"(suy,sxu),left1to(pl"n"(suy,sxu),suy,t12,n)),pl"n"(suy,sxu),sxy,1top(pl"n"(suy,sxu),left1to(pl"n"(suy,sxu),suy,t12,n)),t10)):is"n"(inn(pl"n"(suy,sxu),left1to(pl"n"(suy,sxu),suy,t12,n)),inn(sxy,left1to(sxy,pl"n"(suy,sxu),t10,left1to(pl"n"(suy,sxu),suy,t12,n))))
9747 n2:=left1to(sxy,pl"n"(suy,sxu),t10,left1to(pl"n"(suy,sxu),suy,t12,n)):1to(sxy)
9748 t37:=tris(nat,inn(suy,n),inn(pl"n"(suy,sxu),left1to(pl"n"(suy,sxu),suy,t12,n)),inn(sxy,n2),t35,t36):is"n"(inn(suy,n),inn(sxy,n2))
9749 t38:=isnterl(inn(suy,n),inn(sxy,n2),t37):is"r"(rlofnt(inn(suy,n)),rlofnt(inn(sxy,n2)))
9750 t39:=ispl1"r"(rlofnt(inn(suy,n)),rlofnt(inn(sxy,n2)),y,t38):is"r"(pl(rlofnt(inn(suy,n)),y),pl(rlofnt(inn(sxy,n2)),y))
9751 t40:=ismn1"r"(pl(rlofnt(inn(suy,n)),y),pl(rlofnt(inn(sxy,n2)),y),1rl,t39):is"r"(shiftr(u,iu,y,iy,l,n),shiftr(x,ix,y,iy,ly,n2))
9752 t41:=intshiftr(u,iu,y,iy,l,n):intrl(shiftr(u,iu,y,iy,l,n))
9753 t42:=lsshiftr(u,iu,y,iy,l,n):lessis(y,shiftr(u,iu,y,iy,l,n))
9754 t43:=shiftrls(u,iu,y,iy,l,n):lessis(shiftr(u,iu,y,iy,l,n),u)
9755 t44:=t1(shiftr(u,iu,y,iy,l,n),t41,t42,t43):lessis(shiftr(u,iu,y,iy,l,n),x)
9756 t45:=intshiftr(x,ix,y,iy,ly,n2):intrl(shiftr(x,ix,y,iy,ly,n2))
9757 t46:=lsshiftr(x,ix,y,iy,ly,n2):lessis(y,shiftr(x,ix,y,iy,ly,n2))
9758 t47:=shiftrls(x,ix,y,iy,ly,n2):lessis(shiftr(x,ix,y,iy,ly,n2),x)
9759 t48:=<t40><t47><t46><t45><shiftr(x,ix,y,iy,ly,n2)><t44><t42><t41><shiftr(u,iu,y,iy,l,n)>pi:is(<n>shiftf(u,iu,y,iy,l,lft),<n>fl)
9760 t49:=symis(cx,<n>shiftf(u,iu,y,iy,l,lft),<n>fl,t48):is(<n>fl,<n>shiftf(u,iu,y,iy,l,lft))
9761 k@t50:=fisi(1to(suy),cx,fl,shiftf(u,iu,y,iy,l,lft),[t:1to(suy)]t49(t)):is"e"([t:1to(suy)]cx,fl,shiftf(u,iu,y,iy,l,lft))
9762 t51:=isf([t:1to(suy)]cx,cx,[v:[t:1to(suy)]cx]smpr(q,suy,v),fl,shiftf(u,iu,y,iy,l,lft),t50):is(smpr(q,suy,fl),smpri(u,iu,y,iy,l,lft,q))
9763 t52:=isf(cx,cx,[t:cx]<smpri(x,ix,p1(u),t5,t6,rgt,q)><t>q,smpr(q,suy,fl),smpri(u,iu,y,iy,l,lft,q),t51):is(<smpri(x,ix,p1(u),t5,t6,rgt,q)><smpr(q,suy,fl)>q,<smpri(x,ix,p1(u),t5,t6,rgt,q)><smpri(u,iu,y,iy,l,lft,q)>q)
9764 t53:=tr3is(cx,smpr(q,pl"n"(suy,sxu),f1),<smpr(q,sxu,fr)><smpr(q,suy,fl)>q,<smpri(x,ix,p1(u),t5,t6,rgt,q)><smpr(q,suy,fl)>q,<smpri(x,ix,p1(u),t5,t6,rgt,q)><smpri(u,iu,y,iy,l,lft,q)>q,t12a,t34,t52):is(smpr(q,pl"n"(suy,sxu),f1),<smpri(x,ix,p1(u),t5,t6,rgt,q)><smpri(u,iu,y,iy,l,lft,q)>q)
9765 -8284
9766 k@satz284:=tris1(cx,smpri(x,ix,y,iy,ly,f,q),<smpri(x,ix,pl"r"(u,1rl),intpl(u,iu,1rl,intrl1),satzr25(u,iu,x,ix,k),rgt,q)><smpri(u,iu,y,iy,l,lft,q)>q,smpr(q,pl"n"(suy".8284",sxu".8284"),f1".8284"),t11".8284",t53".8284"):is(smpri(x,ix,y,iy,ly,f,q),<smpri(x,ix,pl"r"(u,1rl),intpl(u,iu,1rl,intrl1),satzr25(u,iu,x,ix,k),rgt,q)><smpri(u,iu,y,iy,l,lft,q)>q)
9767 q@[v:real][iv:intrl(v)][w:real][iw:intrl(w)][lw:lessis(pl"r"(y,v),w)][kw:lessis(w,pl"r"(x,v))]
9768 +8285
9769 x@[y:real]
9770 pl:=pl"r"(x,y):real
9771 mn:=mn"r"(x,y):real
9772 x@p1:=pl"r"(x,1rl):real
9773 kw@t1:=th9"l.or"(less(pl(y,v),w),is"r"(pl(y,v),w),less(mn(pl(y,v),v),mn(w,v)),is"r"(mn(pl(y,v),v),mn(w,v)),lw,[t:less(pl(y,v),w)]satz188f(pl(y,v),w,m0"r"(v),t),[t:is"r"(pl(y,v),w)]ismn1"r"(pl(y,v),w,v,t)):lessis(mn(pl(y,v),v),mn(w,v))
9774 t2:=islessis1(mn(pl(y,v),v),y,mn(w,v),mnpl(y,v),t1):lessis(y,mn(w,v))
9775 t3:=th9"l.or"(less(w,pl(x,v)),is"r"(w,pl(x,v)),less(mn(w,v),mn(pl(x,v),v)),is"r"(mn(w,v),mn(pl(x,v),v)),kw,[t:less(w,pl(x,v))]satz188f(w,pl(x,v),m0"r"(v),t),[t:is"r"(w,pl(x,v))]ismn1"r"(w,pl(x,v),v,t)):lessis(mn(w,v),mn(pl(x,v),v))
9776 t4:=islessis2(mn(pl(x,v),v),x,mn(w,v),mnpl(x,v),t3):lessis(mn(w,v),x)
9777 -8285
9778 iv@sft:=[t:real][w:intrl(t)][lt:lessis(pl"r"(y,v),t)][kt:lessis(t,pl"r"(x,v))]<t4".8285"(t,w,lt,kt)><t2".8285"(t,w,lt,kt)><intmn(t,w,v,iv)><mn"r"(t,v)>f:[t:real][w:intrl(t)][lt:lessis(pl"r"(y,v),t)][kt:lessis(t,pl"r"(x,v))]cx
9779 +*8285
9780 iv@t5:=tris(real,m0"r"(pl(y,v)),m0"r"(pl(v,y)),pl(m0"r"(v),m0"r"(y)),ism0"r"(pl(y,v),pl(v,y),compl"r"(y,v)),satz180(v,y)):is"r"(m0"r"(pl(y,v)),pl(m0"r"(v),m0"r"(y)))
9781 t6:=tr3is(real,mn(pl(1rl,pl(x,v)),v),pl(1rl,mn(pl(x,v),v)),pl(1rl,x),p1(x),asspl1"r"(1rl,pl(x,v),m0"r"(v)),ispl2"r"(mn(pl(x,v),v),x,1rl,mnpl(x,v)),compl"r"(1rl,x)):is"r"(mn(pl(1rl,pl(x,v)),v),p1(x))
9782 t7:=tr3is(real,mn(p1(pl(x,v)),pl(y,v)),pl(pl(1rl,pl(x,v)),pl(m0"r"(v),m0"r"(y))),mn(mn(pl(1rl,pl(x,v)),v),y),mn(p1(x),y),ispl12"r"(p1(pl(x,v)),pl(1rl,pl(x,v)),m0"r"(pl(y,v)),pl(m0"r"(v),m0"r"(y)),compl"r"(pl(x,v),1rl),t5),asspl2"r"(pl(1rl,pl(x,v)),m0"r"(v),m0"r"(y)),ismn1"r"(mn(pl(1rl,pl(x,v)),v),p1(x),y,t6)):is"r"(mn(p1(pl(x,v)),pl(y,v)),mn(p1(x),y))
9783 t8:=th9"l.or"(less(y,x),is"r"(y,x),less(pl(y,v),pl(x,v)),is"r"(pl(y,v),pl(x,v)),ly,[t:less(y,x)]satz188f(y,x,v,t),[t:is"r"(y,x)]ispl1"r"(y,x,v,t)):lessis(pl(y,v),pl(x,v))
9784 s0:=shiftl(x,ix,y,iy,ly):nat
9785 sv:=shiftl(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8):nat
9786 t9:=isrlent(mn(p1(pl(x,v)),pl(y,v)),t6"r.shift"(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8),mn(p1(x),y),t6"r.shift"(x,ix,y,iy,ly),t7):is"n"(sv,s0)
9787 t10:=lessisi2"n"(sv,s0,t9):lessis"n"(sv,s0)
9788 f1:=left(cx,s0,sv,t10,shiftf(x,ix,y,iy,ly,f)):[t:1to(sv)]cx
9789 t11:=issmpr(q,s0,shiftf(x,ix,y,iy,ly,f),sv,t9):is(smpr(q,sv,f1),smpri(x,ix,y,iy,ly,f,q))
9790 [n:1to(sv)]
9791 t12:=isinoutn(s0,inn(sv,n),trlessis"n"(inn(sv,n),sv,s0,1top(sv,n),t10)):is"n"(inn(sv,n),inn(s0,left1to(s0,sv,t10,n)))
9792 n1:=left1to(s0,sv,t10,n):1to(s0)
9793 t13:=isnterl(inn(sv,n),inn(s0,n1),t12):is"r"(rlofnt(inn(sv,n)),rlofnt(inn(s0,n1)))
9794 t14:=tris(real,mn(pl(rlofnt(inn(sv,n)),pl(y,v)),v),pl(rlofnt(inn(sv,n)),mn(pl(y,v),v)),pl(rlofnt(inn(s0,n1)),y),asspl1"r"(rlofnt(inn(sv,n)),pl(y,v),m0"r"(v)),ispl12"r"(rlofnt(inn(sv,n)),rlofnt(inn(s0,n1)),mn(pl(y,v),v),y,t13,mnpl(y,v))):is"r"(mn(pl(rlofnt(inn(sv,n)),pl(y,v)),v),pl(rlofnt(inn(s0,n1)),y))
9795 st0:=shiftr(x,ix,y,iy,ly,n1):real
9796 stv:=shiftr(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8,n):real
9797 t15:=tr4is(real,mn(stv,v),pl(pl(rlofnt(inn(sv,n)),pl(y,v)),pl(m0"r"(1rl),m0"r"(v))),pl(pl(rlofnt(inn(sv,n)),pl(y,v)),pl(m0"r"(v),m0"r"(1rl))),mn(mn(pl(rlofnt(inn(sv,n)),pl(y,v)),v),1rl),st0,asspl1"r"(pl(rlofnt(inn(sv,n)),pl(y,v)),m0"r"(1rl),m0"r"(v)),ispl2"r"(pl(m0"r"(1rl),m0"r"(v)),pl(m0"r"(v),m0"r"(1rl)),pl(rlofnt(inn(sv,n)),pl(y,v)),compl"r"(m0"r"(1rl),m0"r"(v))),asspl2"r"(pl(rlofnt(inn(sv,n)),pl(y,v)),m0"r"(v),m0"r"(1rl)),ismn1"r"(mn(pl(rlofnt(inn(sv,n)),pl(y,v)),v),pl(rlofnt(inn(s0,n1)),y),1rl,t14)):is"r"(mn(stv,v),st0)
9798 t16:=intshiftr(x,ix,y,iy,ly,n1):intrl(st0)
9799 t17:=shiftrls(x,ix,y,iy,ly,n1):lessis(st0,x)
9800 t18:=lsshiftr(x,ix,y,iy,ly,n1):lessis(y,st0)
9801 t19:=intshiftr(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8,n):intrl(stv)
9802 t20:=shiftrls(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8,n):lessis(stv,pl(x,v))
9803 t21:=lsshiftr(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8,n):lessis(pl(y,v),stv)
9804 t22:=intmn(stv,t19,v,iv):intrl(mn(stv,v))
9805 t23:=t2(stv,t19,t21,t20):lessis(y,mn(stv,v))
9806 t24:=t4(stv,t19,t21,t20):lessis(mn(stv,v),x)
9807 t25:=<t15><t17><t18><t16><st0><t24><t23><t22><mn(stv,v)>pi:is(<n>shiftf(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8,sft),<n>f1)
9808 iv@t26:=fisi(1to(sv),cx,shiftf(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8,sft),f1,[t:1to(sv)]t25(t)):is"e"([t:1to(sv)]cx,shiftf(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8,sft),f1)
9809 t27:=isf([t:1to(sv)]cx,cx,[w:[t:1to(sv)]cx]smpr(q,sv,w),shiftf(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8,sft),f1,t26):is(smpri(pl(x,v),intpl(x,ix,v,iv),pl(y,v),intpl(y,iy,v,iv),t8,sft,q),smpr(q,sv,f1))
9810 -8285
9811 iv@lemma285:=t8".8285":lessis(pl"r"(y,v),pl"r"(x,v))
9812 satz285:=tris(cx,smpri(pl"r"(x,v),intpl(x,ix,v,iv),pl"r"(y,v),intpl(y,iy,v,iv),lemma285,sft,q),smpr(q,sv".8285",f1".8285"),smpri(x,ix,y,iy,ly,f,q),t27".8285",t11".8285"):is(smpri(pl"r"(x,v),intpl(x,ix,v,iv),pl"r"(y,v),intpl(y,iy,v,iv),lemma285,sft,q),smpri(x,ix,y,iy,ly,f,q))
9813 ly@[s:seq(x,ix,y,iy,ly,real)][ins:inseq(x,ix,y,iy,ly,s)][f:seq(x,ix,y,iy,ly,cx)][u:real][iu:intrl(u)][lu:lessis(y,u)][ul:lessis(u,x)]
9814 us:=<ul><lu><iu><u>s:real
9815 +8286
9816 t1:=<ul><lu><iu><u>ins:and3(intrl(us),lessis(y,us),lessis(us,x))
9817 -8286
9818 inseqe1:=t22"r.shift"(x,ix,y,iy,ly,us,t1".8286"):intrl(us)
9819 inseqe2:=t23"r.shift"(x,ix,y,iy,ly,us,t1".8286"):lessis(y,us)
9820 inseqe3:=t24"r.shift"(x,ix,y,iy,ly,us,t1".8286"):lessis(us,x)
9821 usf:=<inseqe3><inseqe2><inseqe1><us>f:cx
9822 f@permseq:=[t:real][u:intrl(t)][v:lessis(y,t)][w:lessis(t,x)]usf(t,u,v,w):[t:real][u:intrl(t)][v:lessis(y,t)][w:lessis(t,x)]cx
9823 q@[a:assoc(q)][c:commut(q)][s:seq(x,ix,y,iy,ly,real)][ins:inseq(x,ix,y,iy,ly,s)][pri:proofsirrelevant(x,ix,y,iy,ly,real,s)][ps:perm(x,ix,y,iy,ly,s)]
9824 +*8286
9825 ps@ss:=shiftseq(x,ix,y,iy,ly,s,ins):[t:1to(shiftl)]1to(shiftl)
9826 t2:=satz283(q,a,c,shiftl,ss,bijshiftseq(x,ix,y,iy,ly,s,ins,pri,ps),shiftf(f)):is(smpr(q,shiftl,[t:1to(shiftl)]<<t>ss>shiftf(f)),smpri(f,q))
9827 [n:1to(shiftl)]
9828 ns:=us(s,ins,f,shiftr(n),intshiftr(n),lsshiftr(n),shiftrls(n)):real
9829 t3:=shiftinv1(ns,t34"r.shift"(x,ix,y,iy,ly,s,ins,n)):is"r"(ns,shiftr(<n>ss))
9830 t4:=inseqe1(s,ins,f,shiftr(n),intshiftr(n),lsshiftr(n),shiftrls(n)):intrl(ns)
9831 t5:=inseqe2(s,ins,f,shiftr(n),intshiftr(n),lsshiftr(n),shiftrls(n)):lessis(y,ns)
9832 t6:=inseqe3(s,ins,f,shiftr(n),intshiftr(n),lsshiftr(n),shiftrls(n)):lessis(ns,x)
9833 t7:=intshiftr(<n>ss):intrl(shiftr(<n>ss))
9834 t8:=lsshiftr(<n>ss):lessis(y,shiftr(<n>ss))
9835 t9:=shiftrls(<n>ss):lessis(shiftr(<n>ss),x)
9836 t10:=<t3><t9><t8><t7><shiftr(<n>ss)><t6><t5><t4><ns>pi:is(<n>shiftf(permseq(s,ins,f)),<<n>ss>shiftf(f))
9837 ps@t11:=fisi(1to(shiftl),cx,shiftf(permseq(s,ins,f)),[t:1to(shiftl)]<<t>ss>shiftf(f),[t:1to(shiftl)]t10(t)):is"e"([t:1to(shiftl)]cx,shiftf(permseq(s,ins,f)),[t:1to(shiftl)]<<t>ss>shiftf(f))
9838 t12:=isf([t:1to(shiftl)]cx,cx,[u:[t:1to(shiftl)]cx]smpr(q,shiftl,u),shiftf(permseq(s,ins,f)),[t:1to(shiftl)]<<t>ss>shiftf(f),t11):is(smpri(permseq(s,ins,f),q),smpr(q,shiftl,[t:1to(shiftl)]<<t>ss>shiftf(f)))
9839 -8286
9840 ps@satz286:=tris(cx,smpri(permseq(s,ins,f),q),smpr(q,shiftl,[t:1to(shiftl)]<<t>ss".8286">shiftf(f)),smpri(f,q),t12".8286",t2".8286"):is(smpri(permseq(s,ins,f),q),smpri(f,q))
9841 @[x:nat][f:[t:1to(x)]cx]
9842 modf:=[t:1to(x)]pli(mod(<t>f),0):[t:1to(x)]cx
9843 +8287
9844 [r:real]
9845 prop1:=lessis(mod(sum(x,f)),r):'prop'
9846 prop2:=is(sum(x,modf(f)),pli(r,0)):'prop'
9847 prop3:=and(prop1,prop2):'prop'
9848 f@prop4:=some"r"([t:real]prop3(t)):'prop'
9849 x@prop5:=[u:[t:1to(x)]cx]prop4(u):'prop'
9850 @[f:[t:1to(1)]cx]
9851 t1:=satz277([t:cx][u:cx]pl(t,u),f):is(sum(1,f),<xout(1)>f)
9852 t2:=ismod(sum(1,f),<xout(1)>f,t1):is"r"(mod(sum(1,f)),mod(<xout(1)>f))
9853 t3:=lessisi2(mod(sum(1,f)),mod(<xout(1)>f),t2):prop1(1,f,mod(<xout(1)>f))
9854 t4:=satz277([t:cx][u:cx]pl(t,u),modf(1,f)):prop2(1,f,mod(<xout(1)>f))
9855 t5:=andi(prop1(1,f,mod(<xout(1)>f)),prop2(1,f,mod(<xout(1)>f)),t3,t4):prop3(1,f,mod(<xout(1)>f))
9856 t6:=somei(real,[t:real]prop3(1,f,t),mod(<xout(1)>f),t5):prop4(1,f)
9857 @t7:=[u:[t:1to(1)]cx]t6(u):prop5(1)
9858 x@[p:prop5(x)][f:[t:1to(pl"n"(x,1))]cx]
9859 t8:=lessisi1"n"(x,pl"n"(x,1),satz18a(x,1)):lessis"n"(x,pl"n"(x,1))
9860 lf:=left(cx,pl"n"(x,1),x,t8,f):[t:1to(x)]cx
9861 [r:real][pr:prop3(lf,r)]
9862 t9:=ande1(prop1(lf,r),prop2(lf,r),pr):prop1(lf,r)
9863 t10:=ande2(prop1(lf,r),prop2(lf,r),pr):prop2(lf,r)
9864 t11:=satz278([t:cx][u:cx]pl(t,u),x,f):is(sum(pl"n"(x,1),f),pl(sum(x,lf),<xout(pl"n"(x,1))>f))
9865 t12:=ismod(pl(sum(x,lf),<xout(pl"n"(x,1))>f),sum(pl"n"(x,1),f),symis(cx,sum(pl"n"(x,1),f),pl(sum(x,lf),<xout(pl"n"(x,1))>f),t11)):is"r"(mod(pl(sum(x,lf),<xout(pl"n"(x,1))>f)),mod(sum(pl"n"(x,1),f)))
9866 m:=mod(<xout(pl"n"(x,1))>f):real
9867 t13:=islessis1(mod(pl(sum(x,lf),<xout(pl"n"(x,1))>f)),mod(sum(pl"n"(x,1),f)),pl"r"(mod(sum(x,lf)),m),t12,satz271(sum(x,lf),<xout(pl"n"(x,1))>f)):lessis(mod(sum(pl"n"(x,1),f)),pl"r"(mod(sum(x,lf)),m))
9868 t14:=th9"l.or"(less(mod(sum(x,lf)),r),is"r"(mod(sum(x,lf)),r),less(pl"r"(mod(sum(x,lf)),m),pl"r"(r,m)),is"r"(pl"r"(mod(sum(x,lf)),m),pl"r"(r,m)),t9,[t:less(mod(sum(x,lf)),r)]satz188f(mod(sum(x,lf)),r,m,t),[t:is"r"(mod(sum(x,lf)),r)]ispl1"r"(mod(sum(x,lf)),r,m,t)):lessis(pl"r"(mod(sum(x,lf)),m),pl"r"(r,m))
9869 t15:=trlessis(mod(sum(pl"n"(x,1),f)),pl"r"(mod(sum(x,lf)),m),pl"r"(r,m),t13,t14):prop1(pl"n"(x,1),f,pl"r"(r,m))
9870 lmf:=left(cx,pl"n"(x,1),x,t8,modf(pl"n"(x,1),f)):[t:1to(x)]cx
9871 t16:=satz278([t:cx][u:cx]pl(t,u),x,modf(pl"n"(x,1),f)):is(sum(pl"n"(x,1),modf(pl"n"(x,1),f)),pl(sum(x,lmf),pli(m,0)))
9872 t17:=ispl1(sum(x,lmf),pli(r,0),pli(m,0),t10):is(pl(sum(x,lmf),pli(m,0)),pl(pli(r,0),pli(m,0)))
9873 t18:=plis12a(r,0,m,0):is(pl(pli(r,0),pli(m,0)),pli(pl"r"(r,m),pl"r"(0,0)))
9874 t19:=isrecx2(pl"r"(0,0),0,pl"r"(r,m),pl01"r"(0,0,refis(real,0))):is(pli(pl"r"(r,m),pl"r"(0,0)),pli(pl"r"(r,m),0))
9875 t20:=tr4is(cx,sum(pl"n"(x,1),modf(pl"n"(x,1),f)),pl(sum(x,lmf),pli(m,0)),pl(pli(r,0),pli(m,0)),pli(pl"r"(r,m),pl"r"(0,0)),pli(pl"r"(r,m),0),t16,t17,t18,t19):prop2(pl"n"(x,1),f,pl"r"(r,m))
9876 t21:=andi(prop1(pl"n"(x,1),f,pl"r"(r,m)),prop2(pl"n"(x,1),f,pl"r"(r,m)),t15,t20):prop3(pl"n"(x,1),f,pl"r"(r,m))
9877 t22:=somei(real,[t:real]prop3(pl"n"(x,1),f,t),pl"r"(r,m),t21):prop4(pl"n"(x,1),f)
9878 f@t23:=someapp(real,[t:real]prop3(lf,t),<lf>p,prop4(pl"n"(x,1),f),[t:real][u:prop3(lf,t)]t22(t,u)):prop4(pl"n"(x,1),f)
9879 p@t25:=[u:[t:1to(pl"n"(x,1))]cx]t23(u):prop5(pl"n"(x,1))
9880 t26:=isp(nat,[t:nat]prop5(t),pl"n"(x,1),<x>suc,t25,satz4a(x)):prop5(<x>suc)
9881 -8287
9882 satz287:=<f>induction([t:nat]prop5".8287"(t),t7".8287",[t:nat][u:prop5".8287"(t)]t26".8287"(t,u),x):some"r"([t:real]and(lessis(mod(sum(x,f)),t),is(sum(x,modf(f)),pli(t,0))))
9883 +8288
9884 prop1:=is(pli(mod(prod(x,f)),0),prod(x,modf(f))):'prop'
9885 x@prop2:=[u:[t:1to(x)]cx]prop1(u):'prop'
9886 @[f:[t:1to(1)]cx]
9887 t1:=satz277([t:cx][u:cx]ts(t,u),f):is(prod(1,f),<xout(1)>f)
9888 t2:=ismod(prod(1,f),<xout(1)>f,t1):is"r"(mod(prod(1,f)),mod(<xout(1)>f))
9889 t3:=isrecx1(mod(prod(1,f)),mod(<xout(1)>f),0,t2):is(pli(mod(prod(1,f)),0),pli(mod(<xout(1)>f),0))
9890 t4:=satz277([t:cx][u:cx]ts(t,u),modf(1,f)):is(prod(1,modf(1,f)),pli(mod(<xout(1)>f),0))
9891 t5:=tris2(cx,pli(mod(prod(1,f)),0),prod(1,modf(1,f)),pli(mod(<xout(1)>f),0),t3,t4):prop1(1,f)
9892 @t6:=[u:[t:1to(1)]cx]t5(u):prop2(1)
9893 x@[p:prop2(x)][f:[t:1to(pl"n"(x,1))]cx]
9894 t7:=lessisi1"n"(x,pl"n"(x,1),satz18a(x,1)):lessis"n"(x,pl"n"(x,1))
9895 lf:=left(cx,pl"n"(x,1),x,t7,f):[t:1to(x)]cx
9896 t8:=satz278([t:cx][u:cx]ts(t,u),x,f):is(prod(pl"n"(x,1),f),ts(prod(x,lf),<xout(pl"n"(x,1))>f))
9897 m:=mod(<xout(pl"n"(x,1))>f):real
9898 t9:=ismod(prod(pl"n"(x,1),f),ts(prod(x,lf),<xout(pl"n"(x,1))>f),t8):is"r"(mod(prod(pl"n"(x,1),f)),mod(ts(prod(x,lf),<xout(pl"n"(x,1))>f)))
9899 t10:=satz268(prod(x,lf),<xout(pl"n"(x,1))>f):is"r"(mod(ts(prod(x,lf),<xout(pl"n"(x,1))>f)),ts"r"(mod(prod(x,lf)),m))
9900 t11:=tris(real,mod(prod(pl"n"(x,1),f)),mod(ts(prod(x,lf),<xout(pl"n"(x,1))>f)),ts"r"(mod(prod(x,lf)),m),t9,t10):is"r"(mod(prod(pl"n"(x,1),f)),ts"r"(mod(prod(x,lf)),m))
9901 t12:=isrecx1(mod(prod(pl"n"(x,1),f)),ts"r"(mod(prod(x,lf)),m),0,t11):is(pli(mod(prod(pl"n"(x,1),f)),0),pli(ts"r"(mod(prod(x,lf)),m),0))
9902 lmf:=left(cx,pl"n"(x,1),x,t7,modf(pl"n"(x,1),f)):[t:1to(x)]cx
9903 t13:=satz278([t:cx][u:cx]ts(t,u),x,modf(pl"n"(x,1),f)):is(prod(pl"n"(x,1),modf(pl"n"(x,1),f)),ts(prod(x,lmf),pli(m,0)))
9904 t14:=symis(cx,pli(mod(prod(x,lf)),0),prod(x,lmf),<lf>p):is(prod(x,lmf),pli(mod(prod(x,lf)),0))
9905 t15:=ists1(prod(x,lmf),pli(mod(prod(x,lf)),0),pli(m,0),t14):is(ts(prod(x,lmf),pli(m,0)),ts(pli(mod(prod(x,lf)),0),pli(m,0)))
9906 t16:=tsis12a(mod(prod(x,lf)),0,m,0):is(ts(pli(mod(prod(x,lf)),0),pli(m,0)),pli(mn"r"(ts"r"(mod(prod(x,lf)),m),ts"r"(0,0)),pl"r"(ts"r"(mod(prod(x,lf)),0),ts"r"(0,m))))
9907 t17:=tris(real,mn"r"(ts"r"(mod(prod(x,lf)),m),ts"r"(0,0)),mn"r"(ts"r"(mod(prod(x,lf)),m),0),ts"r"(mod(prod(x,lf)),m),ismn2"r"(ts"r"(0,0),0,ts"r"(mod(prod(x,lf)),m),ts01"r"(0,0,refis(real,0))),pl02"r"(ts"r"(mod(prod(x,lf)),m),m0"r"(0),satz176b(0,refis(real,0)))):is"r"(mn"r"(ts"r"(mod(prod(x,lf)),m),ts"r"(0,0)),ts"r"(mod(prod(x,lf)),m))
9908 t18:=tris(real,pl"r"(ts"r"(mod(prod(x,lf)),0),ts"r"(0,m)),pl"r"(0,0),0,ispl12"r"(ts"r"(mod(prod(x,lf)),0),0,ts"r"(0,m),0,ts02"r"(mod(prod(x,lf)),0,refis(real,0)),ts01"r"(0,m,refis(real,0))),pl01"r"(0,0,refis(real,0))):is"r"(pl"r"(ts"r"(mod(prod(x,lf)),0),ts"r"(0,m)),0)
9909 t19:=isrecx12(mn"r"(ts"r"(mod(prod(x,lf)),m),ts"r"(0,0)),ts"r"(mod(prod(x,lf)),m),pl"r"(ts"r"(mod(prod(x,lf)),0),ts"r"(0,m)),0,t17,t18):is(pli(mn"r"(ts"r"(mod(prod(x,lf)),m),ts"r"(0,0)),pl"r"(ts"r"(mod(prod(x,lf)),0),ts"r"(0,m))),pli(ts"r"(mod(prod(x,lf)),m),0))
9910 t20:=tr4is(cx,prod(pl"n"(x,1),modf(pl"n"(x,1),f)),ts(prod(x,lmf),pli(m,0)),ts(pli(mod(prod(x,lf)),0),pli(m,0)),pli(mn"r"(ts"r"(mod(prod(x,lf)),m),ts"r"(0,0)),pl"r"(ts"r"(mod(prod(x,lf)),0),ts"r"(0,m))),pli(ts"r"(mod(prod(x,lf)),m),0),t13,t15,t16,t19):is(prod(pl"n"(x,1),modf(pl"n"(x,1),f)),pli(ts"r"(mod(prod(x,lf)),m),0))
9911 t21:=tris2(cx,pli(mod(prod(pl"n"(x,1),f)),0),prod(pl"n"(x,1),modf(pl"n"(x,1),f)),pli(ts"r"(mod(prod(x,lf)),m),0),t12,t20):prop1(pl"n"(x,1),f)
9912 p@t21a:=[u:[t:1to(pl"n"(x,1))]cx]t21(u):prop2(pl"n"(x,1))
9913 t22:=isp(nat,[t:nat]prop2(t),pl"n"(x,1),<x>suc,t21a,satz4a(x)):prop2(<x>suc)
9914 -8288
9915 satz288:=<f>induction([t:nat]prop2".8288"(t),t6".8288",[t:nat][u:prop2".8288"(t)]t22".8288"(t,u),x):is(pli(mod(prod(x,f)),0),prod(x,modf(f)))
9916 +8289
9917 prop1:=is(prod(x,f),0c):'prop'
9918 prop2:=some"l"(1to(x),[t:1to(x)]is(<t>f,0c)):'prop'
9919 prop3:=iff(prop1,prop2):'prop'
9920 x@prop4:=[u:[t:1to(x)]cx]prop3(u):'prop'
9921 @[f:[t:1to(1)]cx]
9922 t1:=satz277([t:cx][u:cx]ts(t,u),f):is(prod(1,f),<xout(1)>f)
9923 [p:prop1(1,f)]
9924 t2:=tris1(cx,<xout(1)>f,0c,prod(1,f),t1,p):is(<xout(1)>f,0c)
9925 t3:=somei(1to(1),[t:1to(1)]is(<t>f,0c),xout(1),t2):prop2(1,f)
9926 f@[p:prop2(1,f)][u:1to(1)][i:is(<u>f,0c)]
9927 t4:=th1"n.singlet"(u):is"e"(1to(1),u,xout(1))
9928 t5:=tr3is(cx,prod(1,f),<xout(1)>f,<u>f,0c,t1,isf(1to(1),cx,f,xout(1),u,symis(1to(1),u,xout(1),t4)),i):prop1(1,f)
9929 p@t6:=someapp(1to(1),[t:1to(1)]is(<t>f,0c),p,prop1(1,f),[t:1to(1)][u:is(<t>f,0c)]t5(t,u)):prop1(1,f)
9930 f@t7:=iffi(prop1(1,f),prop2(1,f),[t:prop1(1,f)]t3(t),[t:prop2(1,f)]t6(t)):prop3(1,f)
9931 @t8:=[u:[t:1to(1)]cx]t7(u):prop4(1)
9932 x@[p:prop4(x)][f:[t:1to(pl"n"(x,1))]cx]
9933 t9:=lessisi1"n"(x,pl"n"(x,1),satz18a(x,1)):lessis"n"(x,pl"n"(x,1))
9934 lf:=left(cx,pl"n"(x,1),x,t9,f):[t:1to(x)]cx
9935 t10:=satz278([t:cx][u:cx]ts(t,u),x,f):is(prod(pl"n"(x,1),f),ts(prod(x,lf),<xout(pl"n"(x,1))>f))
9936 [q:prop1(pl"n"(x,1),f)]
9937 t11:=tris1(cx,ts(prod(x,lf),<xout(pl"n"(x,1))>f),0c,prod(pl"n"(x,1),f),t10,q):is(ts(prod(x,lf),<xout(pl"n"(x,1))>f),0c)
9938 t12:=satz221c(prod(x,lf),<xout(pl"n"(x,1))>f,t11):or(is(prod(x,lf),0c),is(<xout(pl"n"(x,1))>f,0c))
9939 [i:is(prod(x,lf),0c)]
9940 t13:=th3"l.iff"(prop1(x,lf),prop2(x,lf),<lf>p,i):prop2(x,lf)
9941 [n:1to(x)][j:is(<n>lf,0c)]
9942 t14:=somei(1to(pl"n"(x,1)),[t:1to(pl"n"(x,1))]is(<t>f,0c),left1to(pl"n"(x,1),x,t9,n),j):prop2(pl"n"(x,1),f)
9943 i@t15:=someapp(1to(x),[t:1to(x)]is(<t>lf,0c),t13,prop2(pl"n"(x,1),f),[t:1to(x)][u:is(<t>lf,0c)]t14(t,u)):prop2(pl"n"(x,1),f)
9944 q@[i:is(<xout(pl"n"(x,1))>f,0c)]
9945 t16:=somei(1to(pl"n"(x,1)),[t:1to(pl"n"(x,1))]is(<t>f,0c),xout(pl"n"(x,1)),i):prop2(pl"n"(x,1),f)
9946 q@t17:=orapp(is(prod(x,lf),0c),is(<xout(pl"n"(x,1))>f,0c),prop2(pl"n"(x,1),f),t12,[t:is(prod(x,lf),0c)]t15(t),[t:is(<xout(pl"n"(x,1))>f,0c)]t16(t)):prop2(pl"n"(x,1),f)
9947 f@[q:prop2(pl"n"(x,1),f)][n:1to(pl"n"(x,1))][i:is(<n>f,0c)][j:is"e"(1to(pl"n"(x,1)),n,xout(pl"n"(x,1)))]
9948 t18:=tris1(cx,<xout(pl"n"(x,1))>f,0c,<n>f,isf(1to(pl"n"(x,1)),cx,f,n,xout(pl"n"(x,1)),j),i):is(<xout(pl"n"(x,1))>f,0c)
9949 t20:=satz221b(prod(x,lf),<xout(pl"n"(x,1))>f,t18):is(ts(prod(x,lf),<xout(pl"n"(x,1))>f),0c)
9950 i@[m:not(is"e"(1to(pl"n"(x,1)),n,xout(pl"n"(x,1))))]
9951 n1:=inn(pl"n"(x,1),n):nat
9952 [j:is"n"(n1,pl"n"(x,1))]
9953 t21:=isoutni(pl"n"(x,1),n1,1top(pl"n"(x,1),n),pl"n"(x,1),lessisi3(pl"n"(x,1)),j):is"e"(1to(pl"n"(x,1)),outn(pl"n"(x,1),n1,1top(pl"n"(x,1),n)),xout(pl"n"(x,1)))
9954 t22:=tris(1to(pl"n"(x,1)),n,outn(pl"n"(x,1),n1,1top(pl"n"(x,1),n)),xout(pl"n"(x,1)),isoutinn(pl"n"(x,1),n),t21):is"e"(1to(pl"n"(x,1)),n,xout(pl"n"(x,1)))
9955 m@t23:=th3"l.imp"(is"n"(n1,pl"n"(x,1)),is"e"(1to(pl"n"(x,1)),n,xout(pl"n"(x,1))),m,[t:is"n"(n1,pl"n"(x,1))]t22(t)):nis"n"(n1,pl"n"(x,1))
9956 t24:=ore1(less"n"(n1,pl"n"(x,1)),is"n"(n1,pl"n"(x,1)),1top(pl"n"(x,1),n),t23):less"n"(n1,pl"n"(x,1))
9957 t25:=satz26(x,n1,t24):lessis"n"(n1,x)
9958 n2:=outn(x,n1,t25):1to(x)
9959 t26:=isinoutn(x,n1,t25):is"n"(n1,inn(x,n2))
9960 t27:=isoutni(pl"n"(x,1),n1,1top(pl"n"(x,1),n),inn(x,n2),trlessis"n"(inn(x,n2),x,pl"n"(x,1),1top(x,n2),t9),t26):is"e"(1to(pl"n"(x,1)),outn(pl"n"(x,1),n1,1top(pl"n"(x,1),n)),left1to(pl"n"(x,1),x,t9,n2))
9961 t28:=tris(1to(pl"n"(x,1)),n,outn(pl"n"(x,1),n1,1top(pl"n"(x,1),n)),left1to(pl"n"(x,1),x,t9,n2),isoutinn(pl"n"(x,1),n),t27):is"e"(1to(pl"n"(x,1)),n,left1to(pl"n"(x,1),x,t9,n2))
9962 t29:=isf(1to(pl"n"(x,1)),cx,f,n,left1to(pl"n"(x,1),x,t9,n2),t28):is(<n>f,<n2>lf)
9963 t30:=tris1(cx,<n2>lf,0c,<n>f,t29,i):is(<n2>lf,0c)
9964 t31:=somei(1to(x),[t:1to(x)]is(<t>lf,0c),n2,t30):prop2(x,lf)
9965 t32:=th4"l.iff"(prop1(x,lf),prop2(x,lf),<lf>p,t31):prop1(x,lf)
9966 t34:=satz221a(prod(x,lf),<xout(pl"n"(x,1))>f,t32):is(ts(prod(x,lf),<xout(pl"n"(x,1))>f),0c)
9967 i@t35:=th1"l.imp"(is"e"(1to(pl"n"(x,1)),n,xout(pl"n"(x,1))),is(ts(prod(x,lf),<xout(pl"n"(x,1))>f),0c),[t:is"e"(1to(pl"n"(x,1)),n,xout(pl"n"(x,1)))]t20(t),[t:not(is"e"(1to(pl"n"(x,1)),n,xout(pl"n"(x,1))))]t34(t)):is(ts(prod(x,lf),<xout(pl"n"(x,1))>f),0c)
9968 t36:=tris(cx,prod(pl"n"(x,1),f),ts(prod(x,lf),<xout(pl"n"(x,1))>f),0c,t10,t35):prop1(pl"n"(x,1),f)
9969 q@t37:=someapp(1to(pl"n"(x,1)),[t:1to(pl"n"(x,1))]is(<t>f,0c),q,prop1(pl"n"(x,1),f),[t:1to(pl"n"(x,1))][u:is(<t>f,0c)]t36(t,u)):prop1(pl"n"(x,1),f)
9970 f@t38:=iffi(prop1(pl"n"(x,1),f),prop2(pl"n"(x,1),f),[t:prop1(pl"n"(x,1),f)]t17(t),[t:prop2(pl"n"(x,1),f)]t37(t)):prop3(pl"n"(x,1),f)
9971 p@t39:=[u:[t:1to(pl"n"(x,1))]cx]t38(u):prop4(pl"n"(x,1))
9972 t40:=isp(nat,[t:nat]prop4(t),pl"n"(x,1),<x>suc,t39,satz4a(x)):prop4(<x>suc)
9973 -8289
9974 satz289:=<f>induction([t:nat]prop4".8289"(t),t8".8289",[t:nat][u:prop4".8289"(t)]t40".8289"(t,u),x):iff(is(prod(x,f),0c),some"l"(1to(x),[t:1to(x)]is(<t>f,0c)))
9975 [i:is(prod(x,f),0c)]
9976 satz289a:=th3"l.iff"(prop1".8289",prop2".8289",satz289,i):some"l"(1to(x),[t:1to(x)]is(<t>f,0c))
9977 f@[n:1to(x)][i:is(<n>f,0c)]
9978 +*8289
9979 i"c"@t41:=somei(1to(x),[t:1to(x)]is(<t>f,0c),n,i):prop2
9980 -8289
9981 i@satz289b:=th4"l.iff"(prop1".8289",prop2".8289",satz289,t41".8289"):is(prod(x,f),0c)
9982 @[x:complex][m:real][mi:intrl(m)][o:or(nis(x,0c),pos(m))]
9983 +v9
9984 [p:pos(m)]
9985 t1:=posintnatrl(m,p,mi):natrl(m)
9986 m1:=ntofrl(m,t1):nat
9987 pw1:=prod(m1,[t:1to(m1)]x):cx
9988 -v9
9989 x@[y:complex][m:real][n:real][i:is(x,y)][j:is"r"(m,n)][mi1:intrl(m)][ni1:intrl(n)][ox:or(nis(x,0c),pos(m))][oy:or(nis(y,0c),pos(n))]
9990 +*v9
9991 oy@[mp:pos(m)][np:pos(n)]
9992 m0:=m1(x,m,mi1,ox,mp):nat
9993 n0:=m1(y,n,ni1,oy,np):nat
9994 t2:=isrlent(m,t1(x,m,mi1,ox,mp),n,t1(y,n,ni1,oy,np),j):is"n"(m0,n0)
9995 t3:=lessisi2"n"(m0,n0,t2):lessis"n"(m0,n0)
9996 t4:=issmpr([t:cx][u:cx]ts(t,u),n0,[t:1to(n0)]y,m0,t2):is(prod(m0,left(cx,n0,m0,t3,[t:1to(n0)]y)),pw1(y,n,ni1,oy,np))
9997 t5:=fisi(1to(m0),cx,[t:1to(m0)]x,left(cx,n0,m0,t3,[t:1to(n0)]y),[t:1to(m0)]i):is"e"([t:1to(m0)]cx,[t:1to(m0)]x,left(cx,n0,m0,t3,[t:1to(n0)]y))
9998 t6:=isf([t:1to(m0)]cx,cx,[u:[t:1to(m0)]cx]prod(m0,u),[t:1to(m0)]x,left(cx,n0,m0,t3,[t:1to(n0)]y),t5):is(pw1(x,m,mi1,ox,mp),prod(m0,left(cx,n0,m0,t3,[t:1to(n0)]y)))
9999 t7:=tris(cx,pw1(x,m,mi1,ox,mp),prod(m0,left(cx,n0,m0,t3,[t:1to(n0)]y)),pw1(y,n,ni1,oy,np),t6,t4):is(pw1(x,m,mi1,ox,mp),pw1(y,n,ni1,oy,np))
10000 p@[p1:pos(m)]
10001 t8:=t7(x,x,m,m,refis(cx,x),refis(real,m),mi,mi,o,o,p,p1):is(pw1(p),pw1(p1))
10002 p@[n:nis(x,0c)]
10003 t9:=th5"l.some"(1to(m1),[t:1to(m1)]is(<t>[u:1to(m1)]x,0c),[t:1to(m1)]n):not(some"l"(1to(m1),[t:1to(m1)]is(<t>[u:1to(m1)]x,0c)))
10004 t10:=th3"l.imp"(is(pw1,0c),some"l"(1to(m1),[t:1to(m1)]is(<t>[u:1to(m1)]x,0c)),t9,[t:is(pw1,0c)]satz289a(m1,[u:1to(m1)]x,t)):nis(pw1,0c)
10005 o@[n:neg(m)]
10006 mi@t11:=intabs(m,mi):intrl(abs(m))
10007 n@t12:=satz166b(m,n):pos(abs(m))
10008 t13:=ori2(nis(x,0c),pos(abs(m)),t12):or(nis(x,0c),pos(abs(m)))
10009 t14:=ore1(nis(x,0c),pos(m),o,nnotp(m,n)):nis(x,0c)
10010 t15:=t10(abs(m),t11,t13,t12,t14):nis(pw1(abs(m),t11,t13,t12),0c)
10011 pw2:=ov(1c,pw1(abs(m),t11,t13,t12),t15):cx
10012 oy@[nm:neg(m)][nn:neg(n)]
10013 pwm:=pw1(x,abs(m),t11(x,m,mi1),t13(x,m,mi1,ox,nm),t12(x,m,mi1,ox,nm)):cx
10014 pwn:=pw1(y,abs(n),t11(y,n,ni1),t13(y,n,ni1,oy,nn),t12(y,n,ni1,oy,nn)):cx
10015 t16:=t7(abs(m),abs(n),i,isabs(m,n,j),t11(x,m,mi1),t11(y,n,ni1),t13(x,m,mi1,ox,nm),t13(y,n,ni1,oy,nn),t12(x,m,mi1,ox,nm),t12(y,n,ni1,oy,nn)):is(pwm,pwn)
10016 t17:=isov2(pwm,pwn,1c,t16,t15(x,m,mi1,ox,nm),t15(y,n,ni1,oy,nn)):is(pw2(x,m,mi1,ox,nm),pw2(y,n,ni1,oy,nn))
10017 n@[n1:neg(m)]
10018 t18:=t17(x,x,m,m,refis(cx,x),refis(real,m),mi,mi,o,o,n,n1):is(pw2(n),pw2(n1))
10019 o@pw3:=ite"l.r"(neg(m),cx,[t:neg(m)]pw2(t),[t:not(neg(m))]1c,[t:neg(m)][u:neg(m)]t18(t,u),[t:not(neg(m))][u:not(neg(m))]refis(cx,1c)):cx
10020 n@t19:=itet"l.r"(neg(m),cx,[t:neg(m)]pw2(t),[t:not(neg(m))]1c,[t:neg(m)][u:neg(m)]t18(t,u),[t:not(neg(m))][u:not(neg(m))]refis(cx,1c),n):is(pw3,pw2(n))
10021 o@[nn:not(neg(m))]
10022 t20:=itef"l.r"(neg(m),cx,[t:neg(m)]pw2(t),[t:not(neg(m))]1c,[t:neg(m)][u:neg(m)]t18(t,u),[t:not(neg(m))][u:not(neg(m))]refis(cx,1c),nn):is(pw3,1c)
10023 nm@t21:=isp(real,[t:real]neg(t),m,n,nm,j):neg(n)
10024 t22:=t19(x,m,mi1,ox,nm):is(pw3(x,m,mi1,ox),pw2(x,m,mi1,ox,nm))
10025 t23:=symis(cx,pw3(y,n,ni1,oy),pw2(y,n,ni1,oy,t21),t19(y,n,ni1,oy,t21)):is(pw2(y,n,ni1,oy,t21),pw3(y,n,ni1,oy))
10026 t24:=tr3is(cx,pw3(x,m,mi1,ox),pw2(x,m,mi1,ox,nm),pw2(y,n,ni1,oy,t21),pw3(y,n,ni1,oy),t22,t17(t21),t23):is(pw3(x,m,mi1,ox),pw3(y,n,ni1,oy))
10027 oy@[nn:not(neg(m))]
10028 t25:=isp(real,[t:real]not(neg(t)),m,n,nn,j):not(neg(n))
10029 t26:=t20(x,m,mi1,ox,nn):is(pw3(x,m,mi1,ox),1c)
10030 t27:=t20(y,n,ni1,oy,t25):is(pw3(y,n,ni1,oy),1c)
10031 t28:=tris2(cx,pw3(x,m,mi1,ox),pw3(y,n,ni1,oy),1c,t26,t27):is(pw3(x,m,mi1,ox),pw3(y,n,ni1,oy))
10032 oy@t29:=th1"l.imp"(neg(m),is(pw3(x,m,mi1,ox),pw3(y,n,ni1,oy)),[t:neg(m)]t24(t),[t:not(neg(m))]t28(t)):is(pw3(x,m,mi1,ox),pw3(y,n,ni1,oy))
10033 -v9
10034 o@pw:=ite"l.r"(pos(m),cx,[t:pos(m)]pw1".v9"(t),[t:not(pos(m))]pw3".v9",[t:pos(m)][u:pos(m)]t8".v9"(t,u),[t:not(pos(m))][u:not(pos(m))]refis(cx,pw3".v9")):cx
10035 +*v9
10036 p@t30:=itet"l.r"(pos(m),cx,[t:pos(m)]pw1(t),[t:not(pos(m))]pw3,[t:pos(m)][u:pos(m)]t8(t,u),[t:not(pos(m))][u:not(pos(m))]refis(cx,pw3),p):is(pw,pw1(p))
10037 o@[n:not(pos(m))]
10038 t31:=itef"l.r"(pos(m),cx,[t:pos(m)]pw1(t),[t:not(pos(m))]pw3,[t:pos(m)][u:pos(m)]t8(t,u),[t:not(pos(m))][u:not(pos(m))]refis(cx,pw3),n):is(pw,pw3)
10039 o@[i:is"r"(m,0)]
10040 t32:=tris(cx,pw,pw3,1c,t31(0notp(m,i)),t20(0notn(m,i))):is(pw,1c)
10041 o@[n:neg(m)]
10042 t33:=tris(cx,pw,pw3,pw2(n),t31(nnotp(m,n)),t19(n)):is(pw,pw2(n))
10043 -v9
10044 o@[p:pos(m)]
10045 posexp:=t30".v9"(p):is(pw(x,m,mi,o),prod(ntofrl(m,posintnatrl(m,p,mi)),[t:1to(ntofrl(m,posintnatrl(m,p,mi)))]x))
10046 [n:nis(x,0c)]
10047 lemmapw1:=th2"e.notis"(cx,pw1".v9"(p),0c,pw,t10".v9"(p,n),posexp):nis(pw(x,m,mi,o),0c)
10048 o@[i:is"r"(m,0)]
10049 0exp:=t32".v9"(i):is(pw(x,m,mi,o),1c)
10050 o@[n:neg(m)]
10051 lemmapw2:=t14".v9"(n):nis(x,0c)
10052 lemmapw3:=t13".v9"(n):or(nis(x,0c),pos(abs(m)))
10053 +*v9
10054 n@t34:=t30(abs(m),t11,t13(n),t12(n)):is(pw(x,abs(m),t11,t13(n)),pw1(x,abs(m),t11,t13(n),t12(n)))
10055 t35:=isov2(pw(x,abs(m),t11,t13(n)),pw1(x,abs(m),t11,t13(n),t12(n)),1c,t34,lemmapw1(abs(m),t11,t13(n),t12(n),t14(n)),t15(n)):is(ov(1c,pw(x,abs(m),t11,t13(n)),lemmapw1(abs(m),t11,t13(n),t12(n),t14(n))),pw2(n))
10056 -v9
10057 n@negexp:=tris2(cx,pw(x,m,mi,o),ov(1c,pw(x,abs(m),intabs(m,mi),lemmapw3),lemmapw1(x,abs(m),intabs(m,mi),lemmapw3,satz166b(m,n),lemmapw2)),pw2".v9"(n),t33".v9"(n),t35".v9"(n)):is(pw(x,m,mi,o),ov(1c,pw(x,abs(m),intabs(m,mi),lemmapw3),lemmapw1(x,abs(m),intabs(m,mi),lemmapw3,satz166b(m,n),lemmapw2)))
10058 +*v9
10059 mp@t36:=isp(real,[t:real]pos(t),m,n,mp,j):pos(n)
10060 t37:=t30(x,m,mi1,ox,mp):is(pw(x,m,mi1,ox),pw1(x,m,mi1,ox,mp))
10061 t38:=symis(cx,pw(y,n,ni1,oy),pw1(y,n,ni1,oy,t36),t30(y,n,ni1,oy,t36)):is(pw1(y,n,ni1,oy,t36),pw(y,n,ni1,oy))
10062 t39:=tr3is(cx,pw(x,m,mi1,ox),pw1(x,m,mi1,ox,mp),pw1(y,n,ni1,oy,t36),pw(y,n,ni1,oy),t37,t7(t36),t38):is(pw(x,m,mi1,ox),pw(y,n,ni1,oy))
10063 oy@[np:not(pos(m))]
10064 t40:=isp(real,[t:real]not(pos(t)),m,n,np,j):not(pos(n))
10065 t41:=t31(x,m,mi1,ox,np):is(pw(x,m,mi1,ox),pw3(x,m,mi1,ox))
10066 t42:=symis(cx,pw(y,n,ni1,oy),pw3(y,n,ni1,oy),t31(y,n,ni1,oy,t40)):is(pw3(y,n,ni1,oy),pw(y,n,ni1,oy))
10067 t43:=tr3is(cx,pw(x,m,mi1,ox),pw3(x,m,mi1,ox),pw3(y,n,ni1,oy),pw(y,n,ni1,oy),t41,t29,t42):is(pw(x,m,mi1,ox),pw(y,n,ni1,oy))
10068 -v9
10069 oy@ispw12:=th1"l.imp"(pos(m),is(pw(x,m,mi1,ox),pw(y,n,ni1,oy)),[t:pos(m)]t39".v9"(t),[t:not(pos(m))]t43".v9"(t)):is(pw(x,m,mi1,ox),pw(y,n,ni1,oy))
10070 m@[i:is(x,y)][mi:intrl(m)][ox:or(nis(x,0c),pos(m))][oy:or(nis(y,0c),pos(m))]
10071 ispw1:=ispw12(x,y,m,m,i,refis(real,m),mi,mi,ox,oy):is(pw(x,m,mi,ox),pw(y,m,mi,oy))
10072 x@[m:real][n:real][i:is"r"(m,n)][mi:intrl(m)][ni:intrl(n)][om:or(nis(x,0c),pos(m))][on:or(nis(x,0c),pos(n))]
10073 ispw2:=ispw12(x,x,m,n,refis(cx,x),i,mi,ni,om,on):is(pw(x,m,mi,om),pw(x,n,ni,on))
10074 o@[n:nis(x,0c)]
10075 +9290
10076 [p:pos(m)]
10077 t1:=lemmapw1(p,n):nis(pw(x,m,mi,o),0c)
10078 @[i:is(1c,0c)]
10079 t2:=tr3is(real,1rl,re(1c),re(0c),0,isre(1rl,0),iscere(1c,0c,i),reis(0,0)):is"r"(1rl,0)
10080 -9290
10081 @1not0:=th3"l.imp"(is(1c,0c),is"r"(1rl,0),pnot0(1rl,pos1),[t:is(1c,0c)]t2".9290"(t)):nis(1c,0c)
10082 +*9290
10083 n@[i:is"r"(m,0)]
10084 t4:=th2"e.notis"(cx,1c,0c,pw(x,m,mi,o),1not0,0exp(i)):nis(pw(x,m,mi,o),0c)
10085 n@[nm:neg(m)]
10086 p0:=pw(x,abs(m),intabs(m,mi),lemmapw3(nm)):cx
10087 t5:=lemmapw1(x,abs(m),intabs(m,mi),lemmapw3(nm),satz166b(m,nm),lemmapw2(nm)):nis(p0,0c)
10088 t6:=tris(cx,ts(pw(x,m,mi,o),p0),ts(ov(1c,p0,t5),p0),1c,ists1(pw(x,m,mi,o),ov(1c,p0,t5),p0,negexp(nm)),satz229e(1c,p0,t5)):is(ts(pw(x,m,mi,o),p0),1c)
10089 t7:=th2"e.notis"(cx,1c,0c,ts(pw(x,m,mi,o),p0),1not0,t6):nis(ts(pw(x,m,mi,o),p0),0c)
10090 t8:=th3"l.imp"(is(pw(x,m,mi,o),0c),is(ts(pw(x,m,mi,o),p0),0c),t7,[t:is(pw(x,m,mi,o),0c)]satz221a(pw(x,m,mi,o),p0,t)):nis(pw(x,m,mi,o),0c)
10091 -9290
10092 n@satz290:=rapp(m,nis(pw(x,m,mi,o),0c),[t:pos(m)]t1".9290"(t),[t:is"r"(m,0)]t4".9290"(t),[t:neg(m)]t8".9290"(t)):nis(pw(x,m,mi,o),0c)
10093 x@lemma291:=ori2(nis(x,0c),pos(1rl),pos1):or(nis(x,0c),pos(1rl))
10094 +9291
10095 1a:=ntofrl(1rl,posintnatrl(1rl,pos1,intrl1)):nat
10096 t1:=posexp(x,1rl,intrl1,lemma291,pos1):is(pw(x,1rl,intrl1,lemma291),prod(1a,[t:1to(1a)]x))
10097 t2:=tris(nat,1,ntofrl(1rl,natrl1),ntofrl(1rl,posintnatrl(1rl,pos1,intrl1)),isntrl1(1),isrlent(1rl,natrl1,1rl,posintnatrl(1rl,pos1,intrl1),refis(real,1rl))):is"n"(1,1a)
10098 t3:=lessisi2"n"(1,1a,t2):lessis"n"(1,1a)
10099 t4:=issmpr([t:cx][u:cx]ts(t,u),1a,[t:1to(1a)]x,1,t2):is(prod(1,left(cx,1a,1,t3,[t:1to(1a)]x)),prod(1a,[t:1to(1a)]x))
10100 t5:=satz277([t:cx][u:cx]ts(t,u),left(cx,1a,1,t3,[t:1to(1a)]x)):is(prod(1,left(cx,1a,1,t3,[t:1to(1a)]x)),x)
10101 t6:=tris1(cx,prod(1a,[t:1to(1a)]x),x,prod(1,left(cx,1a,1,t3,[t:1to(1a)]x)),t4,t5):is(prod(1a,[t:1to(1a)]x),x)
10102 -9291
10103 satz291:=tris(cx,pw(x,1rl,intrl1,lemma291),prod(1a".9291",[t:1to(1a".9291")]x),x,t1".9291",t6".9291"):is(pw(x,1rl,intrl1,lemma291),x)
10104 [y:cx][m:real][mi:intrl(m)][o:or(and(nis(x,0c),nis(y,0c)),pos(m))]
10105 +9292
10106 [a:and(nis(x,0c),nis(y,0c))]
10107 t1:=ande1(nis(x,0c),nis(y,0c),a):nis(x,0c)
10108 t2:=ande2(nis(x,0c),nis(y,0c),a):nis(y,0c)
10109 t3:=satz221d(x,y,t1,t2):nis(ts(x,y),0c)
10110 -9292
10111 lemma292a:=th7"l.or"(and(nis(x,0c),nis(y,0c)),pos(m),nis(x,0c),o,[t:and(nis(x,0c),nis(y,0c))]t1".9292"(t)):or(nis(x,0c),pos(m))
10112 lemma292b:=th7"l.or"(and(nis(x,0c),nis(y,0c)),pos(m),nis(y,0c),o,[t:and(nis(x,0c),nis(y,0c))]t2".9292"(t)):or(nis(y,0c),pos(m))
10113 lemma292c:=th7"l.or"(and(nis(x,0c),nis(y,0c)),pos(m),nis(ts(x,y),0c),o,[t:and(nis(x,0c),nis(y,0c))]t3".9292"(t)):or(nis(ts(x,y),0c),pos(m))
10114 +*9292
10115 x@[n:nat]
10116 nr:=rlofnt(n):real
10117 t4:=natintrl(nr,natrli(n)):intrl(nr)
10118 t5:=ori2(nis(x,0c),pos(nr),natpos(nr,natrli(n))):or(nis(x,0c),pos(nr))
10119 p0:=pw(x,nr,t4,t5):cx
10120 x@t6:=tris(cx,p0(1),pw(x,1rl,intrl1,lemma291(x)),x,ispw1(x,x,1rl,refis(cx,x),intrl1,t5(1),lemma291(x)),satz291(x)):is(p0(1),x)
10121 n@n0:=ntofrl(nr,posintnatrl(nr,natpos(nr,natrli(n)),t4)):nat
10122 t7:=tris(nat,n,ntofrl(nr,natrli(n)),n0,isntrl1(n),isrlent(nr,natrli(n),nr,posintnatrl(nr,natpos(nr,natrli(n)),t4),refis(real,nr))):is"n"(n,n0)
10123 t8:=lessisi2"n"(n,n0,t7):lessis"n"(n,n0)
10124 t9:=posexp(x,nr,t4,t5,natpos(nr,natrli(n))):is(p0,prod(n0,[t:1to(n0)]x))
10125 t10:=issmpr([t:cx][u:cx]ts(t,u),n0,[t:1to(n0)]x,n,t7):is(prod(n,left(cx,n0,n,t8,[t:1to(n0)]x)),prod(n0,[t:1to(n0)]x))
10126 t11:=tris2(cx,p0,prod(n,[t:1to(n)]x),prod(n0,[t:1to(n0)]x),t9,t10):is(p0,prod(n,[t:1to(n)]x))
10127 n1:=pl"n"(n,1):nat
10128 t12:=lessisi1"n"(n,n1,satz18a(n,1)):lessis"n"(n,n1)
10129 t13:=satz278([t:cx][u:cx]ts(t,u),n,[t:1to(n1)]x):is(prod(n1,[t:1to(n1)]x),ts(prod(n,left(cx,n1,n,t12,[t:1to(n1)]x)),x))
10130 t14:=ists1(p0,prod(n,[t:1to(n)]x),x,t11):is(ts(p0,x),ts(prod(n,[t:1to(n)]x),x))
10131 t15:=tris2(cx,prod(n1,[t:1to(n1)]x),ts(p0,x),ts(prod(n,[t:1to(n)]x),x),t13,t14):is(prod(n1,[t:1to(n1)]x),ts(p0,x))
10132 t16:=tris(cx,p0(n1),prod(n1,[t:1to(n1)]x),ts(p0,x),t11(n1),t15):is(p0(n1),ts(p0(n),x))
10133 y@[n:nat]
10134 prop1:=is(p0(ts(x,y),n),ts(p0(x,n),p0(y,n))):'prop'
10135 y@t17:=ists12(p0(x,1),x,p0(y,1),y,t6(x),t6(y)):is(ts(p0(x,1),p0(y,1)),ts(x,y))
10136 t18:=tris2(cx,p0(ts(x,y),1),ts(p0(x,1),p0(y,1)),ts(x,y),t6(ts(x,y)),t17):prop1(1)
10137 n@[p:prop1(n)]
10138 t19:=ists1(p0(ts(x,y),n),ts(p0(x,n),p0(y,n)),ts(x,y),p):is(ts(p0(ts(x,y),n),ts(x,y)),ts(ts(p0(x,n),p0(y,n)),ts(x,y)))
10139 t20:=tr3is(cx,ts(ts(p0(x,n),p0(y,n)),x),ts(p0(x,n),ts(p0(y,n),x)),ts(p0(x,n),ts(x,p0(y,n))),ts(ts(p0(x,n),x),p0(y,n)),assts1(p0(x,n),p0(y,n),x),ists2(ts(p0(y,n),x),ts(x,p0(y,n)),p0(x,n),comts(p0(y,n),x)),assts2(p0(x,n),x,p0(y,n))):is(ts(ts(p0(x,n),p0(y,n)),x),ts(ts(p0(x,n),x),p0(y,n)))
10140 t21:=tr3is(cx,ts(ts(p0(x,n),p0(y,n)),ts(x,y)),ts(ts(ts(p0(x,n),p0(y,n)),x),y),ts(ts(ts(p0(x,n),x),p0(y,n)),y),ts(ts(p0(x,n),x),ts(p0(y,n),y)),assts2(ts(p0(x,n),p0(y,n)),x,y),ists1(ts(ts(p0(x,n),p0(y,n)),x),ts(ts(p0(x,n),x),p0(y,n)),y,t20),assts1(ts(p0(x,n),x),p0(y,n),y)):is(ts(ts(p0(x,n),p0(y,n)),ts(x,y)),ts(ts(p0(x,n),x),ts(p0(y,n),y)))
10141 t22:=tr3is(cx,p0(ts(x,y),n1(n)),ts(p0(ts(x,y),n),ts(x,y)),ts(ts(p0(x,n),p0(y,n)),ts(x,y)),ts(ts(p0(x,n),x),ts(p0(y,n),y)),t16(ts(x,y),n),t19,t21):is(p0(ts(x,y),n1(n)),ts(ts(p0(x,n),x),ts(p0(y,n),y)))
10142 t23:=ists12(p0(x,n1(n)),ts(p0(x,n),x),p0(y,n1(n)),ts(p0(y,n),y),t16(x,n),t16(y,n)):is(ts(p0(x,n1(n)),p0(y,n1(n))),ts(ts(p0(x,n),x),ts(p0(y,n),y)))
10143 t24:=tris2(cx,p0(ts(x,y),n1(n)),ts(p0(x,n1(n)),p0(y,n1(n))),ts(ts(p0(x,n),x),ts(p0(y,n),y)),t22,t23):prop1(n1(n))
10144 t25:=isp(nat,[t:nat]prop1(t),n1(n),<n>suc,t24,satz4a(n)):prop1(<n>suc)
10145 n@t26:=induction([t:nat]prop1(t),t18,[t:nat][u:prop1(t)]t25(t,u),n):prop1
10146 o@prop2:=is(pw(ts(x,y),m,mi,lemma292c),ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b))):'prop'
10147 [p:pos(m)]
10148 t28:=posintnatrl(m,p,mi):natrl(m)
10149 m0:=ntofrl(m,t28):nat
10150 t29:=isrlnt1(m,t28):is"r"(m,nr(m0))
10151 t30:=isrlnt2(m,t28):is"r"(nr(m0),m)
10152 t31:=ispw2(ts(x,y),m,nr(m0),t29,mi,t4(ts(x,y),m0),lemma292c,t5(ts(x,y),m0)):is(pw(ts(x,y),m,mi,lemma292c),p0(ts(x,y),m0))
10153 t32:=ists12(p0(x,m0),pw(x,m,mi,lemma292a),p0(y,m0),pw(y,m,mi,lemma292b),ispw2(x,nr(m0),m,t30,t4(x,m0),mi,t5(x,m0),lemma292a),ispw2(y,nr(m0),m,t30,t4(y,m0),mi,t5(y,m0),lemma292b)):is(ts(p0(x,m0),p0(y,m0)),ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)))
10154 t33:=tr3is(cx,pw(ts(x,y),m,mi,lemma292c),p0(ts(x,y),m0),ts(p0(x,m0),p0(y,m0)),ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)),t31,t26(m0),t32):prop2
10155 o@[i:is"r"(m,0)]
10156 t34:=tris(cx,ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)),ts(1c,1c),1c,ists12(pw(x,m,mi,lemma292a),1c,pw(y,m,mi,lemma292b),1c,0exp(x,m,mi,lemma292a,i),0exp(y,m,mi,lemma292b,i)),satz222(1c)):is(ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)),1c)
10157 t35:=tris2(cx,pw(ts(x,y),m,mi,lemma292c),ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)),1c,0exp(ts(x,y),m,mi,lemma292c,i),t34):prop2
10158 o@[n:neg(m)]
10159 t36:=intabs(m,mi):intrl(abs(m))
10160 t37:=ori2(and(nis(x,0c),nis(y,0c)),pos(abs(m)),satz166b(m,n)):or(and(nis(x,0c),nis(y,0c)),pos(abs(m)))
10161 t38:=lemma292a(abs(m),t36,t37):or(nis(x,0c),pos(abs(m)))
10162 t39:=lemma292b(abs(m),t36,t37):or(nis(y,0c),pos(abs(m)))
10163 t40:=lemma292c(abs(m),t36,t37):or(nis(ts(x,y),0c),pos(abs(m)))
10164 t41:=lemmapw3(x,m,mi,lemma292a,n):or(nis(x,0c),pos(abs(m)))
10165 t42:=lemmapw3(y,m,mi,lemma292b,n):or(nis(y,0c),pos(abs(m)))
10166 t43:=lemmapw3(ts(x,y),m,mi,lemma292c,n):or(nis(ts(x,y),0c),pos(abs(m)))
10167 t44:=ispw2(ts(x,y),abs(m),abs(m),refis(real,abs(m)),t36,t36,t43,t40):is(pw(ts(x,y),abs(m),t36,t43),pw(ts(x,y),abs(m),t36,t40))
10168 t45:=t33(abs(m),t36,t37,satz166b(m,n)):is(pw(ts(x,y),abs(m),t36,t40),ts(pw(x,abs(m),t36,t38),pw(y,abs(m),t36,t39)))
10169 t46:=ists12(pw(x,abs(m),t36,t38),pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t39),pw(y,abs(m),t36,t42),ispw2(x,abs(m),abs(m),refis(real,abs(m)),t36,t36,t38,t41),ispw2(y,abs(m),abs(m),refis(real,abs(m)),t36,t36,t39,t42)):is(ts(pw(x,abs(m),t36,t38),pw(y,abs(m),t36,t39)),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)))
10170 t47:=tr3is(cx,pw(ts(x,y),abs(m),t36,t43),pw(ts(x,y),abs(m),t36,t40),ts(pw(x,abs(m),t36,t38),pw(y,abs(m),t36,t39)),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),t44,t45,t46):is(pw(ts(x,y),abs(m),t36,t43),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)))
10171 t48:=lemmapw1(x,abs(m),t36,t41,satz166b(m,n),lemmapw2(x,m,mi,lemma292a,n)):nis(pw(x,abs(m),t36,t41),0c)
10172 t49:=lemmapw1(y,abs(m),t36,t42,satz166b(m,n),lemmapw2(y,m,mi,lemma292b,n)):nis(pw(y,abs(m),t36,t42),0c)
10173 t50:=lemmapw1(ts(x,y),abs(m),t36,t43,satz166b(m,n),lemmapw2(ts(x,y),m,mi,lemma292c,n)):nis(pw(ts(x,y),abs(m),t36,t43),0c)
10174 t51:=satz221d(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42),t48,t49):nis(ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),0c)
10175 t52:=negexp(ts(x,y),m,mi,lemma292c,n):is(pw(ts(x,y),m,mi,lemma292c),ov(1c,pw(ts(x,y),abs(m),t36,t43),t50))
10176 t53:=isov12(1c,ts(1c,1c),pw(ts(x,y),abs(m),t36,t43),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),satz222a(1c),t47,t50,t51):is(ov(1c,pw(ts(x,y),abs(m),t36,t43),t50),ov(ts(1c,1c),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),t51))
10177 t54:=tris(cx,pw(ts(x,y),m,mi,lemma292c),ov(1c,pw(ts(x,y),abs(m),t36,t43),t50),ov(ts(1c,1c),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),t51),t52,t53):is(pw(ts(x,y),m,mi,lemma292c),ov(ts(1c,1c),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),t51))
10178 t55:=ists12(pw(x,m,mi,lemma292a),ov(1c,pw(x,abs(m),t36,t41),t48),pw(y,m,mi,lemma292b),ov(1c,pw(y,abs(m),t36,t42),t49),negexp(x,m,mi,lemma292a,n),negexp(y,m,mi,lemma292b,n)):is(ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)),ts(ov(1c,pw(x,abs(m),t36,t41),t48),ov(1c,pw(y,abs(m),t36,t42),t49)))
10179 t56:=satz247(1c,pw(x,abs(m),t36,t41),1c,pw(y,abs(m),t36,t42),t48,t49):is(ts(ov(1c,pw(x,abs(m),t36,t41),t48),ov(1c,pw(y,abs(m),t36,t42),t49)),ov(ts(1c,1c),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),t51))
10180 t57:=tris(cx,ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)),ts(ov(1c,pw(x,abs(m),t36,t41),t48),ov(1c,pw(y,abs(m),t36,t42),t49)),ov(ts(1c,1c),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),t51),t55,t56):is(ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)),ov(ts(1c,1c),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),t51))
10181 t58:=tris2(cx,pw(ts(x,y),m,mi,lemma292c),ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)),ov(ts(1c,1c),ts(pw(x,abs(m),t36,t41),pw(y,abs(m),t36,t42)),t51),t54,t57):prop2
10182 -9292
10183 o@satz292:=rapp(m,prop2".9292",[t:pos(m)]t33".9292"(t),[t:is"r"(m,0)]t35".9292"(t),[t:neg(m)]t58".9292"(t)):is(pw(ts(x,y),m,mi,lemma292c),ts(pw(x,m,mi,lemma292a),pw(y,m,mi,lemma292b)))
10184 @[m:real]
10185 lemma293:=ori1(nis(1c,0c),pos(m),1not0):or(nis(1c,0c),pos(m))
10186 [mi:intrl(m)]
10187 +9293
10188 t1:=ori1(and(nis(1c,0c),nis(1c,0c)),pos(m),andi(nis(1c,0c),nis(1c,0c),1not0,1not0)):or(and(nis(1c,0c),nis(1c,0c)),pos(m))
10189 1m:=pw(1c,m,mi,lemma293):cx
10190 t2:=satz222(1m):is(ts(1m,1c),1m)
10191 t3:=ispw1(1c,ts(1c,1c),m,satz222a(1c),mi,lemma293,lemma292c(1c,1c,m,mi,t1)):is(1m,pw(ts(1c,1c),m,mi,lemma292c(1c,1c,m,mi,t1)))
10192 t4:=satz292(1c,1c,m,mi,t1):is(pw(ts(1c,1c),m,mi,lemma292c(1c,1c,m,mi,t1)),ts(pw(1c,m,mi,lemma292a(1c,1c,m,mi,t1)),pw(1c,m,mi,lemma292b(1c,1c,m,mi,t1))))
10193 t5:=ists12(pw(1c,m,mi,lemma292a(1c,1c,m,mi,t1)),1m,pw(1c,m,mi,lemma292b(1c,1c,m,mi,t1)),1m,ispw1(1c,1c,m,refis(cx,1c),mi,lemma292a(1c,1c,m,mi,t1),lemma293),ispw1(1c,1c,m,refis(cx,1c),mi,lemma292b(1c,1c,m,mi,t1),lemma293)):is(ts(pw(1c,m,mi,lemma292a(1c,1c,m,mi,t1)),pw(1c,m,mi,lemma292b(1c,1c,m,mi,t1))),ts(1m,1m))
10194 t6:=tr4is(cx,ts(1m,1c),1m,pw(ts(1c,1c),m,mi,lemma292c(1c,1c,m,mi,t1)),ts(pw(1c,m,mi,lemma292a(1c,1c,m,mi,t1)),pw(1c,m,mi,lemma292b(1c,1c,m,mi,t1))),ts(1m,1m),t2,t3,t4,t5):is(ts(1m,1c),ts(1m,1m))
10195 t7:=tris(cx,ts(1m,mn(1m,1c)),mn(ts(1m,1m),ts(1m,1c)),0c,disttm2(1m,1m,1c),satz213b(ts(1m,1m),ts(1m,1c),symis(cx,ts(1m,1c),ts(1m,1m),t6))):is(ts(1m,mn(1m,1c)),0c)
10196 t8:=ore2(is(1m,0c),is(mn(1m,1c),0c),satz221c(1m,mn(1m,1c),t7),satz290(1c,m,mi,lemma293,1not0)):is(mn(1m,1c),0c)
10197 -9293
10198 satz293:=satz213a(1m".9293",1c,t8".9293"):is(pw(1c,m,mi,lemma293),1c)
10199 x@[m:real][n:real][mi:intrl(m)][ni:intrl(n)][o:or(nis(x,0c),and(pos(m),pos(n)))]
10200 +9294
10201 [a:and(pos(m),pos(n))]
10202 t1:=ande1(pos(m),pos(n),a):pos(m)
10203 t2:=ande2(pos(m),pos(n),a):pos(n)
10204 t3:=pospl(m,n,t1,t2):pos(pl"r"(m,n))
10205 -9294
10206 lemma294a:=th8"l.or"(nis(x,0c),and(pos(m),pos(n)),pos(m),o,[t:and(pos(m),pos(n))]t1".9294"(t)):or(nis(x,0c),pos(m))
10207 lemma294b:=th8"l.or"(nis(x,0c),and(pos(m),pos(n)),pos(n),o,[t:and(pos(m),pos(n))]t2".9294"(t)):or(nis(x,0c),pos(n))
10208 lemma294c:=th8"l.or"(nis(x,0c),and(pos(m),pos(n)),pos(pl"r"(m,n)),o,[t:and(pos(m),pos(n))]t3".9294"(t)):or(nis(x,0c),pos(pl"r"(m,n)))
10209 +*9294
10210 o@prop1:=is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c)):'prop'
10211 a@m1:=ntofrl(m,posintnatrl(m,t1,mi)):nat
10212 n1:=ntofrl(n,posintnatrl(n,t2,ni)):nat
10213 t4:=ists12(pw(x,m,mi,lemma294a),prod(m1,[t:1to(m1)]x),pw(x,n,ni,lemma294b),prod(n1,[t:1to(n1)]x),posexp(x,m,mi,lemma294a,t1),posexp(x,n,ni,lemma294b,t2)):is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(prod(m1,[t:1to(m1)]x),prod(n1,[t:1to(n1)]x)))
10214 p1:=ntofrl(pl"r"(m,n),posintnatrl(pl"r"(m,n),t3,intpl(m,mi,n,ni))):nat
10215 t5:=posexp(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c,t3):is(pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),prod(p1,[t:1to(p1)]x))
10216 t6:=tris(real,pl"r"(m,n),pl"r"(rlofnt(m1),rlofnt(n1)),rlofnt(pl"n"(m1,n1)),ispl12"r"(m,rlofnt(m1),n,rlofnt(n1),isrlnt1(m,posintnatrl(m,t1,mi)),isrlnt1(n,posintnatrl(n,t2,ni))),satzr155b(m1,n1)):is"r"(pl"r"(m,n),rlofnt(pl"n"(m1,n1)))
10217 t7:=tris2(nat,pl"n"(m1,n1),p1,ntofrl(rlofnt(pl"n"(m1,n1)),natrli(pl"n"(m1,n1))),isntrl1(pl"n"(m1,n1)),isrlent(pl"r"(m,n),posintnatrl(pl"r"(m,n),t3,intpl(m,mi,n,ni)),rlofnt(pl"n"(m1,n1)),natrli(pl"n"(m1,n1)),t6)):is"n"(pl"n"(m1,n1),p1)
10218 t8:=lessisi2"n"(pl"n"(m1,n1),p1,t7):lessis"n"(pl"n"(m1,n1),p1)
10219 t9:=issmpr([t:cx][u:cx]ts(t,u),p1,[t:1to(p1)]x,pl"n"(m1,n1),t7):is(prod(pl"n"(m1,n1),left(cx,p1,pl"n"(m1,n1),t8,[t:1to(p1)]x)),prod(p1,[t:1to(p1)]x))
10220 t10:=tris2(cx,pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),prod(pl"n"(m1,n1),[t:1to(pl"n"(m1,n1))]x),prod(p1,[t:1to(p1)]x),t5,t9):is(pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),prod(pl"n"(m1,n1),[t:1to(pl"n"(m1,n1))]x))
10221 t11:=lessisi1"n"(m1,pl"n"(m1,n1),satz18a(m1,n1)):lessis"n"(m1,pl"n"(m1,n1))
10222 t12:=satz281([t:cx][u:cx]ts(t,u),assocts,m1,n1,[t:1to(pl"n"(m1,n1))]x):is(prod(pl"n"(m1,n1),[t:1to(pl"n"(m1,n1))]x),ts(prod(m1,left(cx,pl"n"(m1,n1),m1,t11,[t:1to(pl"n"(m1,n1))]x)),prod(n1,right(cx,m1,n1,[t:1to(pl"n"(m1,n1))]x))))
10223 t13:=tris(cx,pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),prod(pl"n"(m1,n1),[t:1to(pl"n"(m1,n1))]x),ts(prod(m1,[t:1to(m1)]x),prod(n1,[t:1to(n1)]x)),t10,t12):is(pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),ts(prod(m1,[t:1to(m1)]x),prod(n1,[t:1to(n1)]x)))
10224 t14:=tris2(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),ts(prod(m1,[t:1to(m1)]x),prod(n1,[t:1to(n1)]x)),t4,t13):prop1
10225 o@[na:not(and(pos(m),pos(n)))]
10226 t15:=ore1(nis(x,0c),and(pos(m),pos(n)),o,na):nis(x,0c)
10227 t16:=th15"l.or"(pos(m),pos(n),na):or(not(pos(m)),not(pos(n)))
10228 o@am:=abs(m):real
10229 an:=abs(n):real
10230 ap:=abs(pl"r"(m,n)):real
10231 t17:=intabs(m,mi):intrl(am)
10232 t18:=intabs(n,ni):intrl(an)
10233 t19:=intabs(pl"r"(m,n),intpl(m,mi,n,ni)):intrl(ap)
10234 na@[nm:neg(m)][nn:neg(n)]
10235 t20:=andi(pos(am),pos(an),satz166e(m,nnot0(m,nm)),satz166e(n,nnot0(n,nn))):and(pos(am),pos(an))
10236 t21:=ori2(nis(x,0c),and(pos(am),pos(an)),t20):or(nis(x,0c),and(pos(am),pos(an)))
10237 t22:=lemmapw3(x,m,mi,lemma294a,nm):or(nis(x,0c),pos(am))
10238 t23:=lemmapw3(x,n,ni,lemma294b,nn):or(nis(x,0c),pos(an))
10239 t24:=lemma294a(x,am,an,t17,t18,t21):or(nis(x,0c),pos(am))
10240 t25:=lemma294b(x,am,an,t17,t18,t21):or(nis(x,0c),pos(an))
10241 t26:=ists12(pw(x,am,t17,t22),pw(x,am,t17,t24),pw(x,an,t18,t23),pw(x,an,t18,t25),ispw1(x,x,am,refis(cx,x),t17,t22,t24),ispw1(x,x,an,refis(cx,x),t18,t23,t25)):is(ts(pw(x,am,t17,t22),pw(x,an,t18,t23)),ts(pw(x,am,t17,t24),pw(x,an,t18,t25)))
10242 t27:=lemma294c(x,am,an,t17,t18,t21):or(nis(x,0c),pos(pl"r"(am,an)))
10243 t28:=t14(x,am,an,t17,t18,t21,t20):is(ts(pw(x,am,t17,t24),pw(x,an,t18,t25)),pw(x,pl"r"(am,an),intpl(am,t17,an,t18),t27))
10244 t29:=tr3is(real,pl"r"(am,an),pl"r"(m0"r"(m),m0"r"(n)),m0"r"(pl"r"(m,n)),ap,ispl12"r"(am,m0"r"(m),an,m0"r"(n),absn(m,nm),absn(n,nn)),satz180a(m,n),symis(real,ap,m0"r"(pl"r"(m,n)),absn(pl"r"(m,n),negpl(m,n,nm,nn)))):is"r"(pl"r"(am,an),ap)
10245 t30:=lemmapw3(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c,negpl(m,n,nm,nn)):or(nis(x,0c),pos(ap))
10246 t31:=ispw2(x,pl"r"(am,an),ap,t29,intpl(am,t17,an,t18),t19,t27,t30):is(pw(x,pl"r"(am,an),intpl(am,t17,an,t18),t27),pw(x,ap,t19,t30))
10247 t32:=tr3is(cx,ts(pw(x,am,t17,t22),pw(x,an,t18,t23)),ts(pw(x,am,t17,t24),pw(x,an,t18,t25)),pw(x,pl"r"(am,an),intpl(am,t17,an,t18),t27),pw(x,ap,t19,t30),t26,t28,t31):is(ts(pw(x,am,t17,t22),pw(x,an,t18,t23)),pw(x,ap,t19,t30))
10248 t33:=lemmapw1(x,am,t17,t22,satz166b(m,nm),lemmapw2(x,m,mi,lemma294a,nm)):nis(pw(x,am,t17,t22),0c)
10249 t34:=lemmapw1(x,an,t18,t23,satz166b(n,nn),lemmapw2(x,n,ni,lemma294b,nn)):nis(pw(x,an,t18,t23),0c)
10250 t35:=lemmapw1(x,ap,t19,t30,satz166b(pl"r"(m,n),negpl(m,n,nm,nn)),lemmapw2(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c,negpl(m,n,nm,nn))):nis(pw(x,ap,t19,t30),0c)
10251 t36:=ists12(pw(x,m,mi,lemma294a),ov(1c,pw(x,am,t17,t22),t33),pw(x,n,ni,lemma294b),ov(1c,pw(x,an,t18,t23),t34),negexp(x,m,mi,lemma294a,nm),negexp(x,n,ni,lemma294b,nn)):is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(ov(1c,pw(x,am,t17,t22),t33),ov(1c,pw(x,an,t18,t23),t34)))
10252 t37:=satz221d(pw(x,am,t17,t22),pw(x,an,t18,t23),t33,t34):nis(ts(pw(x,am,t17,t22),pw(x,an,t18,t23)),0c)
10253 t38:=satz247(1c,pw(x,am,t17,t22),1c,pw(x,an,t18,t23),t33,t34):is(ts(ov(1c,pw(x,am,t17,t22),t33),ov(1c,pw(x,an,t18,t23),t34)),ov(ts(1c,1c),ts(pw(x,am,t17,t22),pw(x,an,t18,t23)),t37))
10254 t39:=isov12(ts(1c,1c),1c,ts(pw(x,am,t17,t22),pw(x,an,t18,t23)),pw(x,ap,t19,t30),satz222(1c),t32,t37,t35):is(ov(ts(1c,1c),ts(pw(x,am,t17,t22),pw(x,an,t18,t23)),t37),ov(1c,pw(x,ap,t19,t30),t35))
10255 t40:=symis(cx,pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),ov(1c,pw(x,ap,t19,t30),t35),negexp(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c,negpl(m,n,nm,nn))):is(ov(1c,pw(x,ap,t19,t30),t35),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c))
10256 t41:=tr4is(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(ov(1c,pw(x,am,t17,t22),t33),ov(1c,pw(x,an,t18,t23),t34)),ov(ts(1c,1c),ts(pw(x,am,t17,t22),pw(x,an,t18,t23)),t37),ov(1c,pw(x,ap,t19,t30),t35),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),t36,t38,t39,t40):prop1
10257 na@[pm:pos(m)][nn:neg(n)]
10258 t42:=lemmapw3(x,n,ni,lemma294b,nn):or(nis(x,0c),pos(an))
10259 t43:=lemmapw1(x,an,t18,t42,satz166b(n,nn),lemmapw2(x,n,ni,lemma294b,nn)):nis(pw(x,an,t18,t42),0c)
10260 t44:=ists2(pw(x,n,ni,lemma294b),ov(1c,pw(x,an,t18,t42),t43),pw(x,m,mi,lemma294a),negexp(x,n,ni,lemma294b,nn)):is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(pw(x,m,mi,lemma294a),ov(1c,pw(x,an,t18,t42),t43)))
10261 t45:=satz244a(pw(x,m,mi,lemma294a),1c,pw(x,an,t18,t42),t43):is(ts(pw(x,m,mi,lemma294a),ov(1c,pw(x,an,t18,t42),t43)),ov(ts(pw(x,m,mi,lemma294a),1c),pw(x,an,t18,t42),t43))
10262 t46:=isov1(ts(pw(x,m,mi,lemma294a),1c),pw(x,m,mi,lemma294a),pw(x,an,t18,t42),satz222(pw(x,m,mi,lemma294a)),t43):is(ov(ts(pw(x,m,mi,lemma294a),1c),pw(x,an,t18,t42),t43),ov(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43))
10263 t47:=tr3is(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(pw(x,m,mi,lemma294a),ov(1c,pw(x,an,t18,t42),t43)),ov(ts(pw(x,m,mi,lemma294a),1c),pw(x,an,t18,t42),t43),ov(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43),t44,t45,t46):is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ov(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43))
10264 [casea:more(m,an)]
10265 t48:=satz182d(m,an,casea):pos(mn"r"(m,an))
10266 t49:=satz166e(n,nnot0(n,nn)):pos(an)
10267 t50:=andi(pos(an),pos(mn"r"(m,an)),t49,t48):and(pos(an),pos(mn"r"(m,an)))
10268 t51:=ori2(nis(x,0c),and(pos(an),pos(mn"r"(m,an))),t50):or(nis(x,0c),and(pos(an),pos(mn"r"(m,an))))
10269 t52:=intmn(m,mi,an,t18):intrl(mn"r"(m,an))
10270 t53:=lemma294a(x,an,mn"r"(m,an),t18,t52,t51):or(nis(x,0c),pos(an))
10271 t54:=lemma294b(x,an,mn"r"(m,an),t18,t52,t51):or(nis(x,0c),pos(mn"r"(m,an)))
10272 t55:=lemma294c(x,an,mn"r"(m,an),t18,t52,t51):or(nis(x,0c),pos(pl"r"(an,mn"r"(m,an))))
10273 t56:=intpl(an,t18,mn"r"(m,an),t52):intrl(pl"r"(an,mn"r"(m,an)))
10274 t57:=t14(x,an,mn"r"(m,an),t18,t52,t51,t50):is(ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,pl"r"(an,mn"r"(m,an)),t56,t55))
10275 t58:=satz187a(m,an):is"r"(pl"r"(an,mn"r"(m,an)),m)
10276 t59:=ispw2(x,pl"r"(an,mn"r"(m,an)),m,t58,t56,mi,t55,lemma294a):is(pw(x,pl"r"(an,mn"r"(m,an)),t56,t55),pw(x,m,mi,lemma294a))
10277 t60:=tris(cx,ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,pl"r"(an,mn"r"(m,an)),t56,t55),pw(x,m,mi,lemma294a),t57,t59):is(ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,m,mi,lemma294a))
10278 t61:=ispw1(x,x,an,refis(cx,x),t18,t53,t42):is(pw(x,an,t18,t53),pw(x,an,t18,t42))
10279 t62:=isp1(cx,[t:cx]nis(t,0c),pw(x,an,t18,t42),pw(x,an,t18,t53),t43,t61):nis(pw(x,an,t18,t53),0c)
10280 t63:=isov12(ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,m,mi,lemma294a),pw(x,an,t18,t53),pw(x,an,t18,t42),t60,t61,t62,t43):is(ov(ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,an,t18,t53),t62),ov(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43))
10281 t64:=tris2(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ov(ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,an,t18,t53),t62),ov(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43),t47,t63):is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ov(ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,an,t18,t53),t62))
10282 t65:=satz229h(ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54),t62,refis(cx,ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)))):is(ov(ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,an,t18,t53),t62),pw(x,mn"r"(m,an),t52,t54))
10283 t66:=tris(real,mn"r"(m,an),mn"r"(m,m0"r"(n)),pl"r"(m,n),ismn2"r"(an,m0"r"(n),m,absn(n,nn)),ispl2"r"(m0"r"(m0"r"(n)),n,m,satz177(n))):is"r"(mn"r"(m,an),pl"r"(m,n))
10284 t67:=ispw2(x,mn"r"(m,an),pl"r"(m,n),t66,t52,intpl(m,mi,n,ni),t54,lemma294c):is(pw(x,mn"r"(m,an),t52,t54),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c))
10285 t68:=tr3is(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ov(ts(pw(x,an,t18,t53),pw(x,mn"r"(m,an),t52,t54)),pw(x,an,t18,t53),t62),pw(x,mn"r"(m,an),t52,t54),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),t64,t65,t67):prop1
10286 nn@[caseb:is"r"(m,an)]
10287 t69:=ispw2(x,m,an,caseb,mi,t18,lemma294a,t42):is(pw(x,m,mi,lemma294a),pw(x,an,t18,t42))
10288 t70:=satz251a(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43,t69):is(ov(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43),1c)
10289 t71:=tr3is(real,pl"r"(m,n),mn"r"(m,m0"r"(n)),mn"r"(m,an),0,ispl2"r"(n,m0"r"(m0"r"(n)),m,satz177a(n)),ismn2"r"(m0"r"(n),an,m,symis(real,an,m0"r"(n),absn(n,nn))),satz182e(m,an,caseb)):is"r"(pl"r"(m,n),0)
10290 t72:=symis(cx,pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),1c,0exp(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c,t71)):is(1c,pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c))
10291 t73:=tr3is(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ov(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43),1c,pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),t47,t70,t72):prop1
10292 nn@[casec:less(m,an)]
10293 t74:=satz182d(an,m,lemma2"r"(m,an,casec)):pos(mn"r"(an,m))
10294 t75:=andi(pos(m),pos(mn"r"(an,m)),pm,t74):and(pos(m),pos(mn"r"(an,m)))
10295 t76:=ori2(nis(x,0c),and(pos(m),pos(mn"r"(an,m))),t75):or(nis(x,0c),and(pos(m),pos(mn"r"(an,m))))
10296 t77:=intmn(an,t18,m,mi):intrl(mn"r"(an,m))
10297 t78:=lemma294a(x,m,mn"r"(an,m),mi,t77,t76):or(nis(x,0c),pos(m))
10298 t79:=lemma294b(x,m,mn"r"(an,m),mi,t77,t76):or(nis(x,0c),pos(mn"r"(an,m)))
10299 t80:=lemma294c(x,m,mn"r"(an,m),mi,t77,t76):or(nis(x,0c),pos(pl"r"(m,mn"r"(an,m))))
10300 t81:=intpl(m,mi,mn"r"(an,m),t77):intrl(pl"r"(m,mn"r"(an,m)))
10301 t81a:=t14(x,m,mn"r"(an,m),mi,t77,t76,t75):is(ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),pw(x,pl"r"(m,mn"r"(an,m)),t81,t80))
10302 t82:=satz187a(an,m):is"r"(pl"r"(m,mn"r"(an,m)),an)
10303 t83:=ispw2(x,pl"r"(m,mn"r"(an,m)),an,t82,t81,t18,t80,t42):is(pw(x,pl"r"(m,mn"r"(an,m)),t81,t80),pw(x,an,t18,t42))
10304 t84:=tris(cx,ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),pw(x,pl"r"(m,mn"r"(an,m)),t81,t80),pw(x,an,t18,t42),t81a,t83):is(ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),pw(x,an,t18,t42))
10305 t85:=satz290(x,m,mi,t78,t15):nis(pw(x,m,mi,t78),0c)
10306 t86:=satz290(x,mn"r"(an,m),t77,t79,t15):nis(pw(x,mn"r"(an,m),t77,t79),0c)
10307 t87:=satz221d(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79),t85,t86):nis(ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),0c)
10308 t88:=satz222(pw(x,m,mi,t78)):is(ts(pw(x,m,mi,t78),1c),pw(x,m,mi,t78))
10309 t89:=ispw1(x,x,m,refis(cx,x),mi,t78,lemma294a):is(pw(x,m,mi,t78),pw(x,m,mi,lemma294a))
10310 t90:=tris(cx,ts(pw(x,m,mi,t78),1c),pw(x,m,mi,t78),pw(x,m,mi,lemma294a),t88,t89):is(ts(pw(x,m,mi,t78),1c),pw(x,m,mi,lemma294a))
10311 t91:=isov12(ts(pw(x,m,mi,t78),1c),pw(x,m,mi,lemma294a),ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),pw(x,an,t18,t42),t90,t84,t87,t43):is(ov(ts(pw(x,m,mi,t78),1c),ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),t87),ov(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43))
10312 t92:=tris2(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ov(ts(pw(x,m,mi,t78),1c),ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),t87),ov(pw(x,m,mi,lemma294a),pw(x,an,t18,t42),t43),t47,t91):is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ov(ts(pw(x,m,mi,t78),1c),ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),t87))
10313 t93:=satz246a(1c,pw(x,mn"r"(an,m),t77,t79),pw(x,m,mi,t78),t86,t85):is(ov(ts(pw(x,m,mi,t78),1c),ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),t87),ov(1c,pw(x,mn"r"(an,m),t77,t79),t86))
10314 t94:=satz182f(m,an,casec):neg(mn"r"(m,an))
10315 t94a:=tris(real,mn"r"(m,an),mn"r"(m,m0"r"(n)),pl"r"(m,n),ismn2"r"(an,m0"r"(n),m,absn(n,nn)),ispl2"r"(m0"r"(m0"r"(n)),n,m,satz177(n))):is"r"(mn"r"(m,an),pl"r"(m,n))
10316 t95:=tr3is(real,mn"r"(an,m),m0"r"(mn"r"(m,an)),abs(mn"r"(m,an)),ap,satz181a(an,m),symis(real,abs(mn"r"(m,an)),m0"r"(mn"r"(m,an)),absn(mn"r"(m,an),t94)),isabs(mn"r"(m,an),pl"r"(m,n),t94a)):is"r"(mn"r"(an,m),ap)
10317 t96:=isp(real,[t:real]neg(t),mn"r"(m,an),pl"r"(m,n),t94,t94a):neg(pl"r"(m,n))
10318 t97:=lemmapw3(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c,t96):or(nis(x,0c),pos(ap))
10319 t98:=lemmapw1(x,ap,t19,t97,satz166b(pl"r"(m,n),t96),lemmapw2(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c,t96)):nis(pw(x,ap,t19,t97),0c)
10320 t99:=ispw2(x,mn"r"(an,m),ap,t95,t77,t19,t79,t97):is(pw(x,mn"r"(an,m),t77,t79),pw(x,ap,t19,t97))
10321 t100:=isov2(pw(x,mn"r"(an,m),t77,t79),pw(x,ap,t19,t97),1c,t99,t86,t98):is(ov(1c,pw(x,mn"r"(an,m),t77,t79),t86),ov(1c,pw(x,ap,t19,t97),t98))
10322 t101:=symis(cx,pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),ov(1c,pw(x,ap,t19,t97),t98),negexp(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c,t96)):is(ov(1c,pw(x,ap,t19,t97),t98),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c))
10323 t102:=tr4is(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ov(ts(pw(x,m,mi,t78),1c),ts(pw(x,m,mi,t78),pw(x,mn"r"(an,m),t77,t79)),t87),ov(1c,pw(x,mn"r"(an,m),t77,t79),t86),ov(1c,pw(x,ap,t19,t97),t98),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),t92,t93,t100,t101):prop1
10324 nn@t103:=or3app(is"r"(m,an),more(m,an),less(m,an),prop1,satz167a(m,an),[t:is"r"(m,an)]t73(t),[t:more(m,an)]t68(t),[t:less(m,an)]t102(t)):prop1
10325 na@[nm:neg(m)][qn:pos(n)]
10326 na@t104:=ori1(nis(x,0c),and(pos(n),pos(m)),t15):or(nis(x,0c),and(pos(n),pos(m)))
10327 t104a:=th5"l.and"(pos(m),pos(n),na):not(and(pos(n),pos(m)))
10328 t105:=lemma294a(x,n,m,ni,mi,t104):or(nis(x,0c),pos(n))
10329 t106:=lemma294b(x,n,m,ni,mi,t104):or(nis(x,0c),pos(m))
10330 t107:=lemma294c(x,n,m,ni,mi,t104):or(nis(x,0c),pos(pl"r"(n,m)))
10331 t108:=ists12(pw(x,m,mi,lemma294a),pw(x,m,mi,t106),pw(x,n,ni,lemma294b),pw(x,n,ni,t105),ispw1(x,x,m,refis(cx,x),mi,lemma294a,t106),ispw1(x,x,n,refis(cx,x),ni,lemma294b,t105)):is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(pw(x,m,mi,t106),pw(x,n,ni,t105)))
10332 t109:=comts(pw(x,m,mi,t106),pw(x,n,ni,t105)):is(ts(pw(x,m,mi,t106),pw(x,n,ni,t105)),ts(pw(x,n,ni,t105),pw(x,m,mi,t106)))
10333 qn@t110:=t103(x,n,m,ni,mi,t104,t104a,qn,nm):is(ts(pw(x,n,ni,t105),pw(x,m,mi,t106)),pw(x,pl"r"(n,m),intpl(n,ni,m,mi),t107))
10334 na@t111:=ispw2(x,pl"r"(n,m),pl"r"(m,n),compl"r"(n,m),intpl(n,ni,m,mi),intpl(m,mi,n,ni),t107,lemma294c):is(pw(x,pl"r"(n,m),intpl(n,ni,m,mi),t107),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c))
10335 qn@t112:=tr4is(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(pw(x,m,mi,t106),pw(x,n,ni,t105)),ts(pw(x,n,ni,t105),pw(x,m,mi,t106)),pw(x,pl"r"(n,m),intpl(n,ni,m,mi),t107),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),t108,t109,t110,t111):prop1
10336 na@[i:is"r"(m,0)]
10337 t113:=ists1(pw(x,m,mi,lemma294a),1c,pw(x,n,ni,lemma294b),0exp(x,m,mi,lemma294a,i)):is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(1c,pw(x,n,ni,lemma294b)))
10338 t114:=satz222b(pw(x,n,ni,lemma294b)):is(ts(1c,pw(x,n,ni,lemma294b)),pw(x,n,ni,lemma294b))
10339 t115:=symis(real,pl"r"(m,n),n,pl01"r"(m,n,i)):is"r"(n,pl"r"(m,n))
10340 t116:=ispw2(x,n,pl"r"(m,n),t115,ni,intpl(m,mi,n,ni),lemma294b,lemma294c):is(pw(x,n,ni,lemma294b),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c))
10341 t117:=tr3is(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(1c,pw(x,n,ni,lemma294b)),pw(x,n,ni,lemma294b),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),t113,t114,t116):prop1
10342 na@[i:is"r"(n,0)]
10343 t118:=t117(x,n,m,ni,mi,t104,t104a,i):is(ts(pw(x,n,ni,t105),pw(x,m,mi,t106)),pw(x,pl"r"(n,m),intpl(n,ni,m,mi),t107))
10344 t119:=tr4is(cx,ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),ts(pw(x,m,mi,t106),pw(x,n,ni,t105)),ts(pw(x,n,ni,t105),pw(x,m,mi,t106)),pw(x,pl"r"(n,m),intpl(n,ni,m,mi),t107),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c),t108,t109,t118,t111):prop1
10345 na@[pm:pos(m)]
10346 t120:=ore2(not(pos(m)),not(pos(n)),t16,weli(pos(m),pm)):not(pos(n))
10347 t121:=rapp(n,prop1,th2"l.imp"(pos(n),prop1,t120),[t:is"r"(n,0)]t119(t),[t:neg(n)]t103(pm,t)):prop1
10348 na@[nm:neg(m)]
10349 t122:=rapp(n,prop1,[t:pos(n)]t112(nm,t),[t:is"r"(n,0)]t119(t),[t:neg(n)]t41(nm,t)):prop1
10350 na@t123:=rapp(m,prop1,[t:pos(m)]t121(t),[t:is"r"(m,0)]t117(t),[t:neg(m)]t122(t)):prop1
10351 -9294
10352 o@satz294:=th1"l.imp"(and(pos(m),pos(n)),prop1".9294",[t:and(pos(m),pos(n))]t14".9294"(t),[t:not(and(pos(m),pos(n)))]t123".9294"(t)):is(ts(pw(x,m,mi,lemma294a),pw(x,n,ni,lemma294b)),pw(x,pl"r"(m,n),intpl(m,mi,n,ni),lemma294c))
10353 ni@[o:nis(x,0c)]
10354 lemma295a:=ori1(nis(x,0c),pos(m),o):or(nis(x,0c),pos(m))
10355 lemma295b:=ori1(nis(x,0c),pos(n),o):or(nis(x,0c),pos(n))
10356 lemma295c:=ori1(nis(x,0c),pos(mn"r"(m,n)),o):or(nis(x,0c),pos(mn"r"(m,n)))
10357 +9295
10358 t1:=ori1(nis(x,0c),and(pos(mn"r"(m,n)),pos(n)),o):or(nis(x,0c),and(pos(mn"r"(m,n)),pos(n)))
10359 t2:=intmn(m,mi,n,ni):intrl(mn"r"(m,n))
10360 t3:=lemma294a(x,mn"r"(m,n),n,t2,ni,t1):or(nis(x,0c),pos(mn"r"(m,n)))
10361 t4:=lemma294b(x,mn"r"(m,n),n,t2,ni,t1):or(nis(x,0c),pos(n))
10362 t5:=lemma294c(x,mn"r"(m,n),n,t2,ni,t1):or(nis(x,0c),pos(pl"r"(mn"r"(m,n),n)))
10363 t6:=ists12(pw(x,mn"r"(m,n),t2,lemma295c),pw(x,mn"r"(m,n),t2,t3),pw(x,n,ni,lemma295b),pw(x,n,ni,t4),ispw1(x,x,mn"r"(m,n),refis(cx,x),t2,lemma295c,t3),ispw1(x,x,n,refis(cx,x),ni,lemma295b,t4)):is(ts(pw(x,mn"r"(m,n),t2,lemma295c),pw(x,n,ni,lemma295b)),ts(pw(x,mn"r"(m,n),t2,t3),pw(x,n,ni,t4)))
10364 t7:=satz294(x,mn"r"(m,n),n,t2,ni,t1):is(ts(pw(x,mn"r"(m,n),t2,t3),pw(x,n,ni,t4)),pw(x,pl"r"(mn"r"(m,n),n),intpl(mn"r"(m,n),t2,n,ni),t5))
10365 t8:=plmn(m,n):is"r"(pl"r"(mn"r"(m,n),n),m)
10366 t9:=ispw2(x,pl"r"(mn"r"(m,n),n),m,t8,intpl(mn"r"(m,n),t2,n,ni),mi,t5,lemma295a):is(pw(x,pl"r"(mn"r"(m,n),n),intpl(mn"r"(m,n),t2,n,ni),t5),pw(x,m,mi,lemma295a))
10367 t10:=tr3is(cx,ts(pw(x,mn"r"(m,n),t2,lemma295c),pw(x,n,ni,lemma295b)),ts(pw(x,mn"r"(m,n),t2,t3),pw(x,n,ni,t4)),pw(x,pl"r"(mn"r"(m,n),n),intpl(mn"r"(m,n),t2,n,ni),t5),pw(x,m,mi,lemma295a),t6,t7,t9):is(ts(pw(x,mn"r"(m,n),t2,lemma295c),pw(x,n,ni,lemma295b)),pw(x,m,mi,lemma295a))
10368 t11:=satz290(x,n,ni,lemma295b,o):nis(pw(x,n,ni,lemma295b),0c)
10369 -9295
10370 satz295:=satz229k(pw(x,m,mi,lemma295a),pw(x,n,ni,lemma295b),pw(x,mn"r"(m,n),t2".9295",lemma295c),t11".9295",t10".9295"):is(ov(pw(x,m,mi,lemma295a),pw(x,n,ni,lemma295b),satz290(x,n,ni,lemma295b,o)),pw(x,mn"r"(m,n),intmn(m,mi,n,ni),lemma295c))
10371 m@[mi:intrl(m)][n:nis(x,0c)]
10372 lemma296:=ori1(nis(x,0c),pos(m),n):or(nis(x,0c),pos(m))
10373 +9296
10374 t1:=intrli0(0,refis(real,0)):intrl(0)
10375 t2:=lemma295a(x,0,m,t1,mi,n):or(nis(x,0c),pos(0))
10376 t3:=lemma295b(x,0,m,t1,mi,n):or(nis(x,0c),pos(m))
10377 t4:=lemma295c(x,0,m,t1,mi,n):or(nis(x,0c),pos(mn"r"(0,m)))
10378 t5:=satz290(x,m,mi,lemma296,n):nis(pw(x,m,mi,lemma296),0c)
10379 t6:=satz290(x,m,mi,t3,n):nis(pw(x,m,mi,t3),0c)
10380 t7:=symis(cx,pw(x,0,t1,t2),1c,0exp(x,0,t1,t2,refis(real,0))):is(1c,pw(x,0,t1,t2))
10381 t8:=ispw1(x,x,m,refis(cx,x),mi,lemma296,t3):is(pw(x,m,mi,lemma296),pw(x,m,mi,t3))
10382 t9:=isov12(1c,pw(x,0,t1,t2),pw(x,m,mi,lemma296),pw(x,m,mi,t3),t7,t8,t5,t6):is(ov(1c,pw(x,m,mi,lemma296),t5),ov(pw(x,0,t1,t2),pw(x,m,mi,t3),t6))
10383 t10:=satz295(x,0,m,t1,mi,n):is(ov(pw(x,0,t1,t2),pw(x,m,mi,t3),t6),pw(x,mn"r"(0,m),intmn(0,t1,m,mi),t4))
10384 t11:=pl01(0,m0"r"(m),refis(real,0)):is"r"(mn"r"(0,m),m0"r"(m))
10385 t12:=lemma296(x,m0"r"(m),intm0(m,mi),n):or(nis(x,0c),pos(m0"r"(m)))
10386 t13:=ispw2(x,mn"r"(0,m),m0"r"(m),t11,intmn(0,t1,m,mi),intm0(m,mi),t4,t12):is(pw(x,mn"r"(0,m),intmn(0,t1,m,mi),t4),pw(x,m0"r"(m),intm0(m,mi),t12))
10387 t14:=tr3is(cx,ov(1c,pw(x,m,mi,lemma296),t5),ov(pw(x,0,t1,t2),pw(x,m,mi,t3),t6),pw(x,mn"r"(0,m),intmn(0,t1,m,mi),t4),pw(x,m0"r"(m),intm0(m,mi),t12),t9,t10,t13):is(ov(1c,pw(x,m,mi,lemma296),t5),pw(x,m0"r"(m),intm0(m,mi),t12))
10388 -9296
10389 satz296:=t14".9296":is(ov(1c,pw(x,m,mi,lemma296),satz290(x,m,mi,lemma296,n)),pw(x,m0"r"(m),intm0(m,mi),lemma296(m0"r"(m),intm0(m,mi),n)))
10390 ni@[o:or(nis(x,0c),and(pos(m),pos(n)))]
10391 +9297
10392 [p:nis(x,0c)]
10393 t1:=satz290(x,m,mi,lemma294a(o),p):nis(pw(x,m,mi,lemma294a(o)),0c)
10394 o@[a:and(pos(m),pos(n))]
10395 t2:=ande2(pos(m),pos(n),a):pos(n)
10396 t3:=postspp(m,n,ande1(pos(m),pos(n),a),t2):pos(ts"r"(m,n))
10397 -9297
10398 lemma297a:=th9"l.or"(nis(x,0c),and(pos(m),pos(n)),nis(pw(x,m,mi,lemma294a(o)),0c),pos(n),o,[t:nis(x,0c)]t1".9297"(t),[t:and(pos(m),pos(n))]t2".9297"(t)):or(nis(pw(x,m,mi,lemma294a(o)),0c),pos(n))
10399 lemma297b:=th8"l.or"(nis(x,0c),and(pos(m),pos(n)),pos(ts"r"(m,n)),o,[t:and(pos(m),pos(n))]t3".9297"(t)):or(nis(x,0c),pos(ts"r"(m,n)))
10400 mi@[o:or(nis(x,0c),pos(m))][i:is(x,0c)]
10401 +*9297
10402 i@t4:=ore2(nis(x,0c),pos(m),o,weli(is(x,0c),i)):pos(m)
10403 m1:=ntofrl(m,posintnatrl(m,t4,mi)):nat
10404 t5:=posexp(x,m,mi,o,t4):is(pw(x,m,mi,o),prod(m1,[t:1to(m1)]x))
10405 t6:=satz289b(m1,[t:1to(m1)]x,xout(m1),i):is(prod(m1,[t:1to(m1)]x),0c)
10406 t7:=tris(cx,pw(x,m,mi,o),prod(m1,[t:1to(m1)]x),0c,t5,t6):is(pw(x,m,mi,o),0c)
10407 -9297
10408 i@pw0:=t7".9297":is(pw(x,m,mi,o),0c)
10409 ni@[o:or(nis(x,0c),and(pos(m),pos(n)))]
10410 +*9297
10411 ni@t8:=intts(m,mi,n,ni):intrl(ts"r"(m,n))
10412 o@prop1:=is(pw(pw(x,m,mi,lemma294a(o)),n,ni,lemma297a(o)),pw(x,ts"r"(m,n),t8,lemma297b(o))):'prop'
10413 [i:is(x,0c)]
10414 t9:=pw0(x,m,mi,lemma294a(o),i):is(pw(x,m,mi,lemma294a(o)),0c)
10415 t10:=pw0(pw(x,m,mi,lemma294a(o)),n,ni,lemma297a(o),t9):is(pw(pw(x,m,mi,lemma294a(o)),n,ni,lemma297a(o)),0c)
10416 t11:=pw0(x,ts"r"(m,n),t8,lemma297b(o),i):is(pw(x,ts"r"(m,n),t8,lemma297b(o)),0c)
10417 t12:=tris2(cx,pw(pw(x,m,mi,lemma294a(o)),n,ni,lemma297a(o)),pw(x,ts"r"(m,n),t8,lemma297b(o)),0c,t10,t11):prop1
10418 m@[mi:intrl(m)][p:nis(x,0c)]
10419 t13:=ori1(nis(x,0c),pos(m),p):or(nis(x,0c),pos(m))
10420 p0:=pw(x,m,mi,t13):cx
10421 [n:nat]
10422 nr:=rlofnt(n):real
10423 t14:=natintrl(nr,natrli(n)):intrl(nr)
10424 t15:=ori2(nis(p0,0c),pos(nr),natpos(nr,natrli(n))):or(nis(p0,0c),pos(nr))
10425 t16:=ori1(nis(x,0c),pos(ts"r"(m,nr)),p):or(nis(x,0c),pos(ts"r"(m,nr)))
10426 t17:=intts(m,mi,nr,t14):intrl(ts"r"(m,nr))
10427 prop2:=is(pw(p0,nr,t14,t15),pw(x,ts"r"(m,nr),t17,t16)):'prop'
10428 p@t18:=ispw1(p0,p0,1rl,refis(cx,p0),intrl1,t15(1),lemma291(p0)):is(pw(p0,1rl,t14(1),t15(1)),pw(p0,1rl,intrl1,lemma291(p0)))
10429 t19:=satz291(p0):is(pw(p0,1rl,intrl1,lemma291(p0)),p0)
10430 t20:=ispw2(x,m,ts"r"(m,1rl),satz195a(m),mi,t17(1),t13,t16(1)):is(p0,pw(x,ts"r"(m,1rl),t17(1),t16(1)))
10431 t21:=tr3is(cx,pw(p0,1rl,t14(1),t15(1)),pw(p0,1rl,intrl1,lemma291(p0)),p0,pw(x,ts"r"(m,1rl),t17(1),t16(1)),t18,t19,t20):prop2(1)
10432 n@[p2:prop2(n)]
10433 n1:=pl"n"(n,1):nat
10434 t22:=satz290(x,m,mi,t13,p):nis(p0,0c)
10435 t23:=ori1(nis(p0,0c),and(pos(nr),pos(1rl)),t22):or(nis(p0,0c),and(pos(nr),pos(1rl)))
10436 t24:=lemma294a(p0,nr,1rl,t14,intrl1,t23):or(nis(p0,0c),pos(nr))
10437 t25:=lemma294b(p0,nr,1rl,t14,intrl1,t23):or(nis(p0,0c),pos(1rl))
10438 t26:=lemma294c(p0,nr,1rl,t14,intrl1,t23):or(nis(p0,0c),pos(pl"r"(nr,1rl)))
10439 t27:=ispw2(p0,nr(n1),pl"r"(nr,1rl),satzr155a(n,1),t14(n1),intpl(nr,t14,1rl,intrl1),t15(n1),t26):is(pw(p0,nr(n1),t14(n1),t15(n1)),pw(p0,pl"r"(nr,1rl),intpl(nr,t14,1rl,intrl1),t26))
10440 t27a:=satz294(p0,nr,1rl,t14,intrl1,t23):is(ts(pw(p0,nr,t14,t24),pw(p0,1rl,intrl1,t25)),pw(p0,pl"r"(nr,1rl),intpl(nr,t14,1rl,intrl1),t26))
10441 t28:=tris2(cx,pw(p0,nr(n1),t14(n1),t15(n1)),ts(pw(p0,nr,t14,t24),pw(p0,1rl,intrl1,t25)),pw(p0,pl"r"(nr,1rl),intpl(nr,t14,1rl,intrl1),t26),t27,t27a):is(pw(p0,nr(n1),t14(n1),t15(n1)),ts(pw(p0,nr,t14,t24),pw(p0,1rl,intrl1,t25)))
10442 t29:=ori1(nis(x,0c),and(pos(ts"r"(m,nr)),pos(m)),p):or(nis(x,0c),and(pos(ts"r"(m,nr)),pos(m)))
10443 t30:=lemma294a(x,ts"r"(m,nr),m,t17,mi,t29):or(nis(x,0c),pos(ts"r"(m,nr)))
10444 t31:=lemma294b(x,ts"r"(m,nr),m,t17,mi,t29):or(nis(x,0c),pos(m))
10445 t32:=lemma294c(x,ts"r"(m,nr),m,t17,mi,t29):or(nis(x,0c),pos(pl"r"(ts"r"(m,nr),m)))
10446 t33:=tr3is(cx,pw(p0,nr,t14,t24),pw(p0,nr,t14,t15),pw(x,ts"r"(m,nr),t17,t16),pw(x,ts"r"(m,nr),t17,t30),ispw1(p0,p0,nr,refis(cx,p0),t14,t24,t15),p2,ispw1(x,x,ts"r"(m,nr),refis(cx,x),t17,t16,t30)):is(pw(p0,nr,t14,t24),pw(x,ts"r"(m,nr),t17,t30))
10447 t34:=tr3is(cx,pw(p0,1rl,intrl1,t25),pw(p0,1rl,intrl1,lemma291(p0)),p0,pw(x,m,mi,t31),ispw1(p0,p0,1rl,refis(cx,p0),intrl1,t25,lemma291(p0)),t19,ispw1(x,x,m,refis(cx,x),mi,t13,t31)):is(pw(p0,1rl,intrl1,t25),pw(x,m,mi,t31))
10448 t35:=ists12(pw(p0,nr,t14,t24),pw(x,ts"r"(m,nr),t17,t30),pw(p0,1rl,intrl1,t25),pw(x,m,mi,t31),t33,t34):is(ts(pw(p0,nr,t14,t24),pw(p0,1rl,intrl1,t25)),ts(pw(x,ts"r"(m,nr),t17,t30),pw(x,m,mi,t31)))
10449 t36:=satz294(x,ts"r"(m,nr),m,t17,mi,t29):is(ts(pw(x,ts"r"(m,nr),t17,t30),pw(x,m,mi,t31)),pw(x,pl"r"(ts"r"(m,nr),m),intpl(ts"r"(m,nr),t17,m,mi),t32))
10450 t37:=tr3is(real,pl"r"(ts"r"(m,nr),m),pl"r"(ts"r"(m,nr),ts"r"(m,1rl)),ts"r"(m,pl"r"(nr,1rl)),ts"r"(m,nr(n1)),ispl2"r"(m,ts"r"(m,1rl),ts"r"(m,nr),satz195a(m)),distpt2"r"(m,nr,1rl),ists2"r"(pl"r"(nr,1rl),nr(n1),m,satzr155b(n,1))):is"r"(pl"r"(ts"r"(m,nr),m),ts"r"(m,nr(n1)))
10451 t38:=ispw2(x,pl"r"(ts"r"(m,nr),m),ts"r"(m,nr(n1)),t37,intpl(ts"r"(m,nr),t17,m,mi),t17(n1),t32,t16(n1)):is(pw(x,pl"r"(ts"r"(m,nr),m),intpl(ts"r"(m,nr),t17,m,mi),t32),pw(x,ts"r"(m,nr(n1)),t17(n1),t16(n1)))
10452 t39:=tr4is(cx,pw(p0,nr(n1),t14(n1),t15(n1)),ts(pw(p0,nr,t14,t24),pw(p0,1rl,intrl1,t25)),ts(pw(x,ts"r"(m,nr),t17,t30),pw(x,m,mi,t31)),pw(x,pl"r"(ts"r"(m,nr),m),intpl(ts"r"(m,nr),t17,m,mi),t32),pw(x,ts"r"(m,nr(n1)),t17(n1),t16(n1)),t28,t35,t36,t38):prop2(n1)
10453 t40:=isp(nat,[t:nat]prop2(t),n1,<n>suc,t39,satz4a(n)):prop2(<n>suc)
10454 n@t41:=induction([t:nat]prop2(t),t21,[t:nat][u:prop2(t)]t40(t,u),n):prop2(n)
10455 o@[p:nis(x,0c)][q:pos(n)]
10456 t42:=posintnatrl(n,q,ni):natrl(n)
10457 n0:=ntofrl(n,t42):nat
10458 t43:=isrlnt1(n,t42):is"r"(n,rlofnt(n0))
10459 t44:=isrlnt2(n,t42):is"r"(rlofnt(n0),n)
10460 o@p1:=pw(x,m,mi,lemma294a(o)):cx
10461 q@t44a:=ispw2(x,m,m,refis(real,m),mi,mi,lemma294a(o),t13(mi,p)):is(p1,p0(mi,p))
10462 t45:=ispw12(p1,p0(mi,p),n,rlofnt(n0),t44a,t43,ni,t14(mi,p,n0),lemma297a(o),t15(mi,p,n0)):is(pw(p1,n,ni,lemma297a(o)),pw(p0(mi,p),rlofnt(n0),t14(mi,p,n0),t15(mi,p,n0)))
10463 t46:=t41(mi,p,n0):is(pw(p0(mi,p),rlofnt(n0),t14(mi,p,n0),t15(mi,p,n0)),pw(x,ts"r"(m,rlofnt(n0)),t17(mi,p,n0),t16(mi,p,n0)))
10464 t47:=ists2"r"(rlofnt(n0),n,m,t44):is"r"(ts"r"(m,rlofnt(n0)),ts"r"(m,n))
10465 t48:=ispw2(x,ts"r"(m,rlofnt(n0)),ts"r"(m,n),t47,t17(mi,p,n0),t8,t16(mi,p,n0),lemma297b(o)):is(pw(x,ts"r"(m,rlofnt(n0)),t17(mi,p,n0),t16(mi,p,n0)),pw(x,ts"r"(m,n),t8,lemma297b(o)))
10466 t49:=tr3is(cx,pw(p1,n,ni,lemma297a(o)),pw(p0(mi,p),rlofnt(n0),t14(mi,p,n0),t15(mi,p,n0)),pw(x,ts"r"(m,rlofnt(n0)),t17(mi,p,n0),t16(mi,p,n0)),pw(x,ts"r"(m,n),t8,lemma297b(o)),t45,t46,t48):prop1
10467 p@[i:is"r"(n,0)]
10468 t50:=0exp(p1,n,ni,lemma297a(o),i):is(pw(p1,n,ni,lemma297a(o)),1c)
10469 t51:=ts02"r"(m,n,i):is"r"(ts"r"(m,n),0)
10470 t52:=0exp(x,ts"r"(m,n),t8,lemma297b(o),t51):is(pw(x,ts"r"(m,n),t8,lemma297b(o)),1c)
10471 t53:=tris2(cx,pw(p1,n,ni,lemma297a(o)),pw(x,ts"r"(m,n),t8,lemma297b(o)),1c,t50,t52):prop1
10472 p@[q:neg(n)]
10473 an:=abs(n):real
10474 t54:=intabs(n,ni):intrl(an)
10475 t55:=ori1(nis(x,0c),and(pos(m),pos(an)),p):or(nis(x,0c),and(pos(m),pos(an)))
10476 p1t55:=p1(an,mi,t54,t55):cx
10477 t56:=satz166e(n,nnot0(n,q)):pos(an)
10478 t56a:=lemma294a(an,mi,t54,t55):or(nis(x,0c),pos(m))
10479 t57:=lemma297a(an,mi,t54,t55):or(nis(p1t55,0c),pos(an))
10480 t58:=lemma297b(an,mi,t54,t55):or(nis(x,0c),pos(ts"r"(m,an)))
10481 t59:=t49(an,mi,t54,t55,p,t56):is(pw(p1t55,an,t54,t57),pw(x,ts"r"(m,an),t8(an,mi,t54),t58))
10482 t60:=satz177c(an,n,absn(n,q)):is"r"(n,m0"r"(an))
10483 t61:=intm0(an,t54):intrl(m0"r"(an))
10484 t62:=satz290(x,m,mi,t56a,p):nis(p1t55,0c)
10485 t63:=lemma296(p1t55,an,t54,t62):or(nis(p1t55,0c),pos(an))
10486 t64:=satz290(p1t55,an,t54,t63,t62):nis(pw(p1t55,an,t54,t63),0c)
10487 t65:=lemma296(p1t55,m0"r"(an),t61,t62):or(nis(p1t55,0c),pos(m0"r"(an)))
10488 t66:=satz296(p1t55,an,t54,t62):is(ov(1c,pw(p1t55,an,t54,t63),t64),pw(p1t55,m0"r"(an),t61,t65))
10489 t67:=ispw1(x,x,m,refis(cx,x),mi,lemma294a(o),t56a):is(p1,p1t55)
10490 t68:=ispw12(p1,p1t55,n,m0"r"(an),t67,t60,ni,t61,lemma297a(o),t65):is(pw(p1,n,ni,lemma297a(o)),pw(p1t55,m0"r"(an),t61,t65))
10491 t69:=tris2(cx,pw(p1,n,ni,lemma297a(o)),ov(1c,pw(p1t55,an,t54,t63),t64),pw(p1t55,m0"r"(an),t61,t65),t68,t66):is(pw(p1,n,ni,lemma297a(o)),ov(1c,pw(p1t55,an,t54,t63),t64))
10492 t70:=tris(real,m0"r"(ts"r"(m,an)),ts"r"(m,m0"r"(an)),ts"r"(m,n),satz197f(m,an),ists2"r"(m0"r"(an),n,m,symis(real,n,m0"r"(an),t60))):is"r"(m0"r"(ts"r"(m,an)),ts"r"(m,n))
10493 t71:=intm0(ts"r"(m,an),t8(an,mi,t54)):intrl(m0"r"(ts"r"(m,an)))
10494 t72:=lemma296(x,ts"r"(m,an),t8(an,mi,t54),p):or(nis(x,0c),pos(ts"r"(m,an)))
10495 t73:=satz290(x,ts"r"(m,an),t8(an,mi,t54),t72,p):nis(pw(x,ts"r"(m,an),t8(an,mi,t54),t72),0c)
10496 t74:=lemma296(x,m0"r"(ts"r"(m,an)),t71,p):or(nis(x,0c),pos(m0"r"(ts"r"(m,an))))
10497 t75:=satz296(x,ts"r"(m,an),t8(an,mi,t54),p):is(ov(1c,pw(x,ts"r"(m,an),t8(an,mi,t54),t72),t73),pw(x,m0"r"(ts"r"(m,an)),t71,t74))
10498 t76:=ispw2(x,m0"r"(ts"r"(m,an)),ts"r"(m,n),t70,t71,t8,t74,lemma297b(o)):is(pw(x,m0"r"(ts"r"(m,an)),t71,t74),pw(x,ts"r"(m,n),t8,lemma297b(o)))
10499 t77:=ispw1(p1t55,p1t55,an,refis(cx,p1t55),t54,t63,t57):is(pw(p1t55,an,t54,t63),pw(p1t55,an,t54,t57))
10500 t78:=ispw1(x,x,ts"r"(m,an),refis(cx,x),t8(an,mi,t54),t58,t72):is(pw(x,ts"r"(m,an),t8(an,mi,t54),t58),pw(x,ts"r"(m,an),t8(an,mi,t54),t72))
10501 t79:=tr3is(cx,pw(p1t55,an,t54,t63),pw(p1t55,an,t54,t57),pw(x,ts"r"(m,an),t8(an,mi,t54),t58),pw(x,ts"r"(m,an),t8(an,mi,t54),t72),t77,t59,t78):is(pw(p1t55,an,t54,t63),pw(x,ts"r"(m,an),t8(an,mi,t54),t72))
10502 t80:=isov2(pw(p1t55,an,t54,t63),pw(x,ts"r"(m,an),t8(an,mi,t54),t72),1c,t79,t64,t73):is(ov(1c,pw(p1t55,an,t54,t63),t64),ov(1c,pw(x,ts"r"(m,an),t8(an,mi,t54),t72),t73))
10503 t81:=tr4is(cx,pw(p1,n,ni,lemma297a(o)),ov(1c,pw(p1t55,an,t54,t63),t64),ov(1c,pw(x,ts"r"(m,an),t8(an,mi,t54),t72),t73),pw(x,m0"r"(ts"r"(m,an)),t71,t74),pw(x,ts"r"(m,n),t8,lemma297b(o)),t69,t80,t75,t76):prop1
10504 p@t82:=rapp(n,prop1,[t:pos(n)]t49(t),[t:is"r"(n,0)]t53(t),[t:neg(n)]t81(t)):prop1
10505 -9297
10506 o@satz297:=th1"l.imp"(is(x,0c),prop1".9297",[t:is(x,0c)]t12".9297"(t),[t:nis(x,0c)]t82".9297"(t)):is(pw(pw(x,m,mi,lemma294a(o)),n,ni,lemma297a(o)),pw(x,ts"r"(m,n),intts(m,mi,n,ni),lemma297b(o)))
10507 @[r:real][s:real]
10508 +10298
10509 t1:=tris(cx,pl(pli(r,0),pli(s,0)),pli(pl"r"(r,s),pl"r"(0,0)),pli(pl"r"(r,s),0),plis12a(r,0,s,0),isrecx2(pl"r"(0,0),0,pl"r"(r,s),pl01(0,0,refis(real,0)))):is(pl(pli(r,0),pli(s,0)),pli(pl"r"(r,s),0))
10510 -10298
10511 satz298a:=symis(cx,pl(pli(r,0),pli(s,0)),pli(pl"r"(r,s),0),t1".10298"):is(pli(pl"r"(r,s),0),pl(pli(r,0),pli(s,0)))
10512 satz298b:=t1".10298":is(pl(pli(r,0),pli(s,0)),pli(pl"r"(r,s),0))
10513 +*10298
10514 s@t2:=tris(cx,mn(pli(r,0),pli(s,0)),pli(mn"r"(r,s),mn"r"(0,0)),pli(mn"r"(r,s),0),mnis12a(r,0,s,0),isrecx2(mn"r"(0,0),0,mn"r"(r,s),pl02(0,m0"r"(0),satz176b(0,refis(real,0))))):is(mn(pli(r,0),pli(s,0)),pli(mn"r"(r,s),0))
10515 -10298
10516 s@satz298c:=symis(cx,mn(pli(r,0),pli(s,0)),pli(mn"r"(r,s),0),t2".10298"):is(pli(mn"r"(r,s),0),mn(pli(r,0),pli(s,0)))
10517 satz298d:=t2".10298":is(mn(pli(r,0),pli(s,0)),pli(mn"r"(r,s),0))
10518 +*10298
10519 s@t3:=pl02(ts"r"(r,s),m0"r"(ts"r"(0,0)),satz176b(ts"r"(0,0),ts01(0,0,refis(real,0)))):is"r"(mn"r"(ts"r"(r,s),ts"r"(0,0)),ts"r"(r,s))
10520 t4:=tris(real,pl"r"(ts"r"(r,0),ts"r"(0,s)),ts"r"(r,0),0,pl02(ts"r"(r,0),ts"r"(0,s),ts01(0,s,refis(real,0))),ts02(r,0,refis(real,0))):is"r"(pl"r"(ts"r"(r,0),ts"r"(0,s)),0)
10521 t5:=tris(cx,ts(pli(r,0),pli(s,0)),pli(mn"r"(ts"r"(r,s),ts"r"(0,0)),pl"r"(ts"r"(r,0),ts"r"(0,s))),pli(ts"r"(r,s),0),tsis12a(r,0,s,0),isrecx12(mn"r"(ts"r"(r,s),ts"r"(0,0)),ts"r"(r,s),pl"r"(ts"r"(r,0),ts"r"(0,s)),0,t3,t4)):is(ts(pli(r,0),pli(s,0)),pli(ts"r"(r,s),0))
10522 -10298
10523 s@satz298e:=symis(cx,ts(pli(r,0),pli(s,0)),pli(ts"r"(r,s),0),t5".10298"):is(pli(ts"r"(r,s),0),ts(pli(r,0),pli(s,0)))
10524 satz298f:=t5".10298":is(ts(pli(r,0),pli(s,0)),pli(ts"r"(r,s),0))
10525 [n:nis"r"(s,0)]
10526 +*10298
10527 n@[i:is(pli(s,0),0c)]
10528 t6:=tr3is(real,s,re(pli(s,0)),re(0c),0,isre(s,0),iscere(pli(s,0),0c,i),reis(0,0)):is"r"(s,0)
10529 -10298
10530 n@lemma298:=th3"l.imp"(is(pli(s,0),0c),is"r"(s,0),n,[t:is(pli(s,0),0c)]t6".10298"(t)):nis(pli(s,0),0c)
10531 +*10298
10532 n@t7:=tris(cx,ts(pli(s,0),pli(ov"r"(r,s,n),0)),pli(ts"r"(s,ov"r"(r,s,n)),0),pli(r,0),t5(s,ov"r"(r,s,n)),isrecx1(ts"r"(s,ov"r"(r,s,n)),r,0,satz204c(r,s,n))):is(ts(pli(s,0),pli(ov"r"(r,s,n),0)),pli(r,0))
10533 -10298
10534 n@satz298g:=satz229g(pli(r,0),pli(s,0),pli(ov"r"(r,s,n),0),lemma298,t7".10298"):is(pli(ov"r"(r,s,n),0),ov(pli(r,0),pli(s,0),lemma298))
10535 satz298h:=satz229h(pli(r,0),pli(s,0),pli(ov"r"(r,s,n),0),lemma298,t7".10298"):is(ov(pli(r,0),pli(s,0),lemma298),pli(ov"r"(r,s,n),0))
10536 +*10298
10537 r@t8:=tris(cx,m0(pli(r,0)),pli(m0"r"(r),m0"r"(0)),pli(m0"r"(r),0),m0isa(r,0),isrecx2(m0"r"(0),0,m0"r"(r),satz176b(0,refis(real,0)))):is(m0(pli(r,0)),pli(m0"r"(r),0))
10538 -10298
10539 r@satz298j:=symis(cx,m0(pli(r,0)),pli(m0"r"(r),0),t8".10298"):is(pli(m0"r"(r),0),m0(pli(r,0)))
10540 satz298k:=t8".10298":is(m0(pli(r,0)),pli(m0"r"(r),0))
10541 +*10298
10542 r@t9:=tris(real,mod2(pli(r,0)),ts"r"(re(pli(r,0)),re(pli(r,0))),ts"r"(r,r),pl02(ts"r"(re(pli(r,0)),re(pli(r,0))),ts"r"(im(pli(r,0)),im(pli(r,0))),ts01(im(pli(r,0)),im(pli(r,0)),imis(r,0))),ists12"r"(re(pli(r,0)),r,re(pli(r,0)),r,reis(r,0),reis(r,0))):is"r"(mod2(pli(r,0)),ts"r"(r,r))
10543 ar:=abs(r):real
10544 [p:pos(r)]
10545 t10:=satz196a(r,r,p,p):is"r"(ts"r"(r,r),ts"r"(ar,ar))
10546 r@[i:is"r"(r,0)]
10547 t11:=tris2(real,ts"r"(r,r),ts"r"(ar,ar),0,ts01(r,r,i),ts01(ar,ar,abs0(r,i))):is"r"(ts"r"(r,r),ts"r"(ar,ar))
10548 r@[n:neg(r)]
10549 t12:=satz196b(r,r,n,n):is"r"(ts"r"(r,r),ts"r"(ar,ar))
10550 r@t13:=rapp(r,is"r"(ts"r"(r,r),ts"r"(ar,ar)),[t:pos(r)]t10(t),[t:is"r"(r,0)]t11(t),[t:neg(r)]t12(t)):is"r"(ts"r"(r,r),ts"r"(ar,ar))
10551 t14:=tris(real,mod2(pli(r,0)),ts"r"(r,r),ts"r"(ar,ar),t9,t13):is"r"(mod2(pli(r,0)),ts"r"(ar,ar))
10552 t15:=th1"l.imp"(is"r"(r,0),not(neg(ar)),[t:is"r"(r,0)]0notn(ar,abs0(r,t)),[t:nis"r"(r,0)]pnotn(ar,satz166e(r,t))):not(neg(ar))
10553 -10298
10554 r@satz298l:=thsqrt3(mod2(pli(r,0)),lemma5(pli(r,0)),abs(r),t15".10298",t14".10298"):is"r"(mod(pli(r,0)),abs(r))
10555 satz298m:=symis(real,mod(pli(r,0)),abs(r),satz298l):is"r"(abs(r),mod(pli(r,0)))
10556 cofrl:=pli(r,0):complex
10557 s@[i:is(cofrl(r),cofrl(s))]
10558 isrlic:=tr3is(real,r,re(cofrl(r)),re(cofrl(s)),s,isre(r,0),iscere(cofrl(r),cofrl(s),i),reis(s,0)):is"r"(r,s)
10559 s@[i:is"r"(r,s)]
10560 isrlec:=isrecx1(r,s,0,i):is(cofrl(r),cofrl(s))
10561 +v10
10562 @t1:=[t:real][u:real][v:is(cofrl(t),cofrl(u))]isrlic(t,u,v):injective(real,cx,[t:real]cofrl(t))
10563 -v10
10564 @[x:cx]
10565 realc:=image(real,cx,[t:real]cofrl(t),x):'prop'
10566 r@reali:=imagei(real,cx,[t:real]cofrl(t),r):realc(cofrl(r))
10567 x@[rx:realc(x)]
10568 rlofc:=soft(real,cx,[t:real]cofrl(t),t1".v10",x,rx):real
10569 [y:cx][ry:realc(y)][i:is"r"(rlofc(x,rx),rlofc(y,ry))]
10570 iscirl:=isinve(real,cx,[t:real]cofrl(t),t1".v10",x,rx,y,ry,i):is(x,y)
10571 ry@[i:is(x,y)]
10572 iscerl:=isinv(real,cx,[t:real]cofrl(t),t1".v10",x,rx,y,ry,i):is"r"(rlofc(x,rx),rlofc(y,ry))
10573 r@isrlc1:=isst1(real,cx,[t:real]cofrl(t),t1".v10",r):is"r"(r,rlofc(cofrl(r),reali(r)))
10574 isrlc2:=isst2(real,cx,[t:real]cofrl(t),t1".v10",r):is"r"(rlofc(cofrl(r),reali(r)),r)
10575 rx@iscrl1:=ists1"e"(real,cx,[t:real]cofrl(t),t1".v10",x,rx):is(x,cofrl(rlofc(x,rx)))
10576 iscrl2:=ists2"e"(real,cx,[t:real]cofrl(t),t1".v10",x,rx):is(cofrl(rlofc(x,rx)),x)
10577 @[n:nat]
10578 cofn:=cofrl(rlofnt(n)):complex
10579 [m:nat][i:is"n"(n,m)]
10580 isnec:=isrlec(rlofnt(n),rlofnt(m),isnterl(n,m,i)):is(cofn(n),cofn(m))
10581 m@[i:is(cofn(n),cofn(m))]
10582 isnic:=isntirl(n,m,isrlic(rlofnt(n),rlofnt(m),i)):is"n"(n,m)
10583 +*v10
10584 @t2:=[t:nat][u:nat][v:is(cofn(t),cofn(u))]isnic(t,u,v):injective(nat,cx,[t:nat]cofn(t))
10585 -v10
10586 x@natc:=image(nat,cx,[t:nat]cofn(t),x):'prop'
10587 n@nati:=imagei(nat,cx,[t:nat]cofn(t),n):natc(cofn(n))
10588 x@[nx:natc(x)]
10589 nofc:=soft(nat,cx,[t:nat]cofn(t),t2".v10",x,nx):nat
10590 [y:cx][ny:natc(y)][i:is(x,y)]
10591 iscen:=isinv(nat,cx,[t:nat]cofn(t),t2".v10",x,nx,y,ny,i):is"n"(nofc(x,nx),nofc(y,ny))
10592 ny@[i:is"n"(nofc(x,nx),nofc(y,ny))]
10593 iscin:=isinve(nat,cx,[t:nat]cofn(t),t2".v10",x,nx,y,ny,i):is(x,y)
10594 n@isnc1:=isst1(nat,cx,[t:nat]cofn(t),t2".v10",n):is"n"(n,nofc(cofn(n),nati(n)))
10595 isnc2:=isst2(nat,cx,[t:nat]cofn(t),t2".v10",n):is"n"(nofc(cofn(n),nati(n)),n)
10596 nx@iscn1:=ists1"e"(nat,cx,[t:nat]cofn(t),t2".v10",x,nx):is(x,cofn(nofc(x,nx)))
10597 iscn2:=ists2"e"(nat,cx,[t:nat]cofn(t),t2".v10",x,nx):is(cofn(nofc(x,nx)),x)
10598 @natt:=ot(cx,[t:cx]natc(t)):'type'
10599 [nt:natt]
10600 cofnt:=in"e"(cx,[t:cx]natc(t),nt):cx
10601 natti:=inp(cx,[t:cx]natc(t),nt):natc(cofnt(nt))
10602 [mt:natt][i:is"e"(natt,nt,mt)]
10603 isntec:=isini(cx,[t:cx]natc(t),nt,mt,i):is(cofnt(nt),cofnt(mt))
10604 mt@[i:is(cofnt(nt),cofnt(mt))]
10605 isntic:=isine(cx,[t:cx]natc(t),nt,mt,i):is"e"(natt,nt,mt)
10606 nx@ntofc:=out(cx,[t:cx]natc(t),x,nx):natt
10607 ny@[i:is(x,y)]
10608 iscent:=isouti(cx,[t:cx]natc(t),x,nx,y,ny,i):is"e"(natt,ntofc(x,nx),ntofc(y,ny))
10609 ny@[i:is"e"(natt,ntofc(x,nx),ntofc(y,ny))]
10610 iscint:=isoute(cx,[t:cx]natc(t),x,nx,y,ny,i):is(x,y)
10611 nt@isntc1:=isoutin(cx,[t:cx]natc(t),nt):is"e"(natt,nt,ntofc(cofnt(nt),natti(nt)))
10612 isntc2:=symis(natt,nt,ntofc(cofnt(nt),natti(nt)),isntc1):is"e"(natt,ntofc(cofnt(nt),natti(nt)),nt)
10613 nx@iscnt1:=isinout(cx,[t:cx]natc(t),x,nx):is(x,cofnt(ntofc(x,nx)))
10614 iscnt2:=symis(cx,x,cofnt(ntofc(x,nx)),iscnt1):is(cofnt(ntofc(x,nx)),x)
10615 n@ntofn:=ntofc(cofn(n),nati(n)):natt
10616 m@[i:is"n"(n,m)]
10617 isnent:=iscent(cofn(n),nati(n),cofn(m),nati(m),isnec(n,m,i)):is"e"(natt,ntofn(n),ntofn(m))
10618 m@[i:is"e"(natt,ntofn(n),ntofn(m))]
10619 isnint:=isnic(n,m,iscint(cofn(n),nati(n),cofn(m),nati(m),i)):is"n"(n,m)
10620 nt@nofnt:=nofc(cofnt(nt),natti(nt)):nat
10621 mt@[i:is"e"(natt,nt,mt)]
10622 isnter:=iscen(cofnt(nt),natti(nt),cofnt(mt),natti(mt),isntec(nt,mt,i)):is"n"(nofnt(nt),nofnt(mt))
10623 mt@[i:is"n"(nofnt(nt),nofnt(mt))]
10624 isntin:=isntic(nt,mt,iscin(cofnt(nt),natti(nt),cofnt(mt),natti(mt),i)):is"e"(natt,nt,mt)
10625 +*v10
10626 n@t3:=iscnt1(cofn(n),nati(n)):is(cofn(n),cofnt(ntofn(n)))
10627 -v10
10628 n@isnnt1:=tris(nat,n,nofc(cofn(n),nati(n)),nofnt(ntofn(n)),isnc1(n),iscen(cofn(n),nati(n),cofnt(ntofn(n)),natti(ntofn(n)),t3".v10")):is"n"(n,nofnt(ntofn(n)))
10629 isnnt2:=symis(nat,n,nofnt(ntofn(n)),isnnt1):is"n"(nofnt(ntofn(n)),n)
10630 +*v10
10631 nt@t4:=iscn1(cofnt(nt),natti(nt)):is(cofnt(nt),cofn(nofnt(nt)))
10632 -v10
10633 nt@isntn1:=tris(natt,nt,ntofc(cofnt(nt),natti(nt)),ntofn(nofnt(nt)),isntc1(nt),iscent(cofnt(nt),natti(nt),cofn(nofnt(nt)),nati(nofnt(nt)),t4".v10")):is"e"(natt,nt,ntofn(nofnt(nt)))
10634 isntn2:=symis(natt,nt,ntofn(nofnt(nt)),isntn1):is"e"(natt,ntofn(nofnt(nt)),nt)
10635 @1t:=ntofn(1):natt
10636 suct:=[t:natt]ntofn(<nofnt(t)>suc):[t:natt]natt
10637 +10299
10638 nt@[i:is"e"(natt,<nt>suct,1t)]
10639 t1:=isnint(<nofnt(nt)>suc,1,i):is"n"(<nofnt(nt)>suc,1)
10640 -10299
10641 nt@satz299a:=th3"l.imp"(is"e"(natt,<nt>suct,1t),is"n"(<nofnt(nt)>suc,1),<nofnt(nt)>ax3,[t:is"e"(natt,<nt>suct,1t)]t1".10299"(t)):not(is"e"(natt,<nt>suct,1t))
10642 @ax3t:=[t:natt]satz299a(t):[t:natt]not(is"e"(natt,<t>suct,1t))
10643 mt@[i:is"e"(natt,<nt>suct,<mt>suct)]
10644 +*10299
10645 i"c"@t2:=isnint(<nofnt(nt)>suc,<nofnt(mt)>suc,i):is"n"(<nofnt(nt)>suc,<nofnt(mt)>suc)
10646 t3:=<t2><nofnt(mt)><nofnt(nt)>ax4:is"n"(nofnt(nt),nofnt(mt))
10647 -10299
10648 i@satz299b:=isntin(nt,mt,t3".10299"):is"e"(natt,nt,mt)
10649 @ax4t:=[t:natt][u:natt][v:is"e"(natt,<t>suct,<u>suct)]satz299b(t,u,v):[t:natt][u:natt][v:is"e"(natt,<t>suct,<u>suct)]is"e"(natt,t,u)
10650 [s:set(natt)]
10651 cond1t:=esti(natt,1t,s):'prop'
10652 cond2t:=all"l"(natt,[t:natt]imp(esti(natt,t,s),esti(natt,<t>suct,s))):'prop'
10653 [c1:cond1t][c2:cond2t]
10654 +*10299
10655 c2@[n:nat]
10656 prop1:=esti(natt,ntofn(n),s):'prop'
10657 c2@t4:=c1:prop1(1)
10658 n@[p:prop1(n)]
10659 t5:=<p><ntofn(n)>c2:esti(natt,ntofn(<nofnt(ntofn(n))>suc),s)
10660 t6:=isp(nat,[t:nat]esti(natt,ntofn(<t>suc),s),nofnt(ntofn(n)),n,t5,isnnt2(n)):prop1(<n>suc)
10661 c2@[nt:natt]
10662 t7:=induction([t:nat]prop1(t),t4,[t:nat][u:prop1(t)]t6(t,u),nofnt(nt)):prop1(nofnt(nt))
10663 -10299
10664 c2@satz299c:=[t:natt]isp(natt,[u:natt]esti(natt,u,s),ntofn(nofnt(t)),t,t7".10299"(t),isntn2(t)):[t:natt]esti(natt,t,s)
10665 @ax5t:=[t:set(natt)][u:cond1t(t)][v:cond2t(t)]satz299c(t,u,v):[t:set(natt)][u:cond1t(t)][v:cond2t(t)][w:natt]esti(natt,w,t)
10666 ic:=pli(0,1rl):complex
10667 +10300
10668 t1:=tsis12a(0,1rl,0,1rl):is(ts(ic,ic),pli(mn"r"(ts"r"(0,0),ts"r"(1rl,1rl)),pl"r"(ts"r"(0,1rl),ts"r"(1rl,0))))
10669 t2:=tris(real,mn"r"(ts"r"(0,0),ts"r"(1rl,1rl)),m0"r"(ts"r"(1rl,1rl)),m0"r"(1rl),pl01(ts"r"(0,0),m0"r"(ts"r"(1rl,1rl)),ts01(0,0,refis(real,0))),ism0"r"(ts"r"(1rl,1rl),1rl,satz195(1rl))):is"r"(mn"r"(ts"r"(0,0),ts"r"(1rl,1rl)),m0"r"(1rl))
10670 t3:=tris(real,pl"r"(ts"r"(0,1rl),ts"r"(1rl,0)),ts"r"(1rl,0),0,pl01(ts"r"(0,1rl),ts"r"(1rl,0),ts01(0,1rl,refis(real,0))),ts02(1rl,0,refis(real,0))):is"r"(pl"r"(ts"r"(0,1rl),ts"r"(1rl,0)),0)
10671 t4:=isrecx12(mn"r"(ts"r"(0,0),ts"r"(1rl,1rl)),m0"r"(1rl),pl"r"(ts"r"(0,1rl),ts"r"(1rl,0)),0,t2,t3):is(pli(mn"r"(ts"r"(0,0),ts"r"(1rl,1rl)),pl"r"(ts"r"(0,1rl),ts"r"(1rl,0))),cofrl(m0"r"(1rl)))
10672 t5:=satz298j(1rl):is(cofrl(m0"r"(1rl)),m0(1c))
10673 -10300
10674 satz2300:=tr3is(cx,ts(ic,ic),pli(mn"r"(ts"r"(0,0),ts"r"(1rl,1rl)),pl"r"(ts"r"(0,1rl),ts"r"(1rl,0))),cofrl(m0"r"(1rl)),m0(1c),t1".10300",t4".10300",t5".10300"):is(ts(ic,ic),m0(1c))
10675 [r:real][s:real]
10676 +10301
10677 t1:=tsis12a(s,0,0,1rl):is(ts(cofrl(s),ic),pli(mn"r"(ts"r"(s,0),ts"r"(0,1rl)),pl"r"(ts"r"(s,1rl),ts"r"(0,0))))
10678 t2:=tris(real,mn"r"(ts"r"(s,0),ts"r"(0,1rl)),m0"r"(ts"r"(0,1rl)),0,pl01(ts"r"(s,0),m0"r"(ts"r"(0,1rl)),ts02(s,0,refis(real,0))),satz176b(ts"r"(0,1rl),ts01(0,1rl,refis(real,0)))):is"r"(mn"r"(ts"r"(s,0),ts"r"(0,1rl)),0)
10679 t3:=tris(real,pl"r"(ts"r"(s,1rl),ts"r"(0,0)),ts"r"(s,1rl),s,pl02(ts"r"(s,1rl),ts"r"(0,0),ts01(0,0,refis(real,0))),satz195(s)):is"r"(pl"r"(ts"r"(s,1rl),ts"r"(0,0)),s)
10680 t4:=isrecx12(mn"r"(ts"r"(s,0),ts"r"(0,1rl)),0,pl"r"(ts"r"(s,1rl),ts"r"(0,0)),s,t2,t3):is(pli(mn"r"(ts"r"(s,0),ts"r"(0,1rl)),pl"r"(ts"r"(s,1rl),ts"r"(0,0))),pli(0,s))
10681 t5:=tris(cx,ts(cofrl(s),ic),pli(mn"r"(ts"r"(s,0),ts"r"(0,1rl)),pl"r"(ts"r"(s,1rl),ts"r"(0,0))),pli(0,s),t1,t4):is(ts(cofrl(s),ic),pli(0,s))
10682 t6:=ispl2(ts(cofrl(s),ic),pli(0,s),cofrl(r),t5):is(pl(cofrl(r),ts(cofrl(s),ic)),pl(cofrl(r),pli(0,s)))
10683 t7:=plis12a(r,0,0,s):is(pl(cofrl(r),pli(0,s)),pli(pl"r"(r,0),pl"r"(0,s)))
10684 t8:=isrecx12(pl"r"(r,0),r,pl"r"(0,s),s,pl02(r,0,refis(real,0)),pl01(0,s,refis(real,0))):is(pli(pl"r"(r,0),pl"r"(0,s)),pli(r,s))
10685 -10301
10686 satz301a:=tr3is(cx,pl(cofrl(r),ts(cofrl(s),ic)),pl(cofrl(r),pli(0,s)),pli(pl"r"(r,0),pl"r"(0,s)),pli(r,s),t6".10301",t7".10301",t8".10301"):is"e"(cx,pl(cofrl(r),ts(cofrl(s),ic)),pli(r,s))
10687 satz301b:=symis(cx,pl(cofrl(r),ts(cofrl(s),ic)),pli(r,s),satz301a):is(pli(r,s),pl(cofrl(r),ts(cofrl(s),ic)))
10688 @[x:complex]
10689 satz301c:=tris(cx,x,pli(re(x),im(x)),pl(cofrl(re(x)),ts(cofrl(im(x)),ic)),ispli(x),satz301b(re(x),im(x))):is(x,pl(cofrl(re(x)),ts(cofrl(im(x)),ic)))
10690 satz301d:=symis(cx,x,pl(cofrl(re(x)),ts(cofrl(im(x)),ic)),satz301c):is(pl(cofrl(re(x)),ts(cofrl(im(x)),ic)),x)
10691 s@[t:real][u:real][i:is(pl(cofrl(r),ts(cofrl(s),ic)),pl(cofrl(t),ts(cofrl(u),ic)))]
10692 +*10301
10693 i@t9:=tr3is(cx,pli(r,s),pl(cofrl(r),ts(cofrl(s),ic)),pl(cofrl(t),ts(cofrl(u),ic)),pli(t,u),satz301b(r,s),i,satz301a(t,u)):is(pli(r,s),pli(t,u))
10694 -10301
10695 i@satz301e:=tr3is(real,r,re(pli(r,s)),re(pli(t,u)),t,isre(r,s),iscere(pli(r,s),pli(t,u),t9".10301"),reis(t,u)):is"r"(r,t)
10696 satz301f:=tr3is(real,s,im(pli(r,s)),im(pli(t,u)),u,isim(r,s),isceim(pli(r,s),pli(t,u),t9".10301"),imis(t,u)):is"r"(s,u)
10697 -c
10698 -r
10699 -rp
10700 -rt
10701 -n
10702 -landau
10703 -eq
10704 -st
10705 -e
10706 -l