# Landau's "Grundlagen der Analysis", formal specification in AUTOMATH # Copyright (C) 1977, L.S. van Benthem Jutting # 1992, revised by F. Wiedijk (http://www.cs.ru.nl/~freek/aut/) # 2008, revised by F. Guidi to remove the eta-reductions +l @[a:'prop'][b:'prop'] imp:=[x:a]b:'prop' [a1:a][i:imp(a,b)] mp:=i:b a@refimp:=[x:a]x:imp(a,a) b@[c:'prop'][i:imp(a,b)][j:imp(b,c)] trimp:=[x:a]<i>j:imp(a,c) @con:='prim':'prop' a@not:=imp(con):'prop' wel:=not(not(a)):'prop' [a1:a] weli:=[x:not(a)]x:wel(a) a@[w:wel(a)] et:='prim':a a@[c1:con] cone:=et([x:not(a)]c1):a +imp b@[i:imp(a,b)][j:imp(not(a),b)] th1:=et(b,[x:not(b)]<j>x):b b@[n:not(a)] th2:=trimp(con,b,n,[x:con]cone(b,x)):imp(a,b) b@[n:not(b)][i:imp(a,b)] th3:=trimp(con,i,n):not(a) b@[a1:a][n:not(b)] th4:=[x:imp(a,b)]th3(n,x):not(imp(a,b)) b@[n:not(imp(a,b))] th5:=et([x:not(a)]n):a th6:=[x:b]<[y:a]x>n:not(b) b@[n:not(b)][i:imp(not(a),b)] th7:=et(a,th3(not(a),b,n,i)):a -imp b@[i:imp(not(b),not(a))] cp:=[x:a]th7".imp"(b,not(a),weli(x),i):imp(a,b) @obvious:=imp(con,con):'prop' obviousi:=refimp(con):obvious b@ec:=imp(a,not(b)):'prop' [n:not(a)] eci1:=th2".imp"(not(b),n):ec(a,b) b@[n:not(b)] eci2:=[x:a]n:ec(a,b) +ec b@[i:imp(a,not(b))] th1:=i:ec(a,b) b@[i:imp(b,not(a))] th2:=[x:a][y:b]i:ec(a,b) -ec b@[e:ec(a,b)] comec:=th2".ec"(b,a,e):ec(b,a) [a1:a] ece1:=e:not(b) e@[b1:b] ece2:=th3".imp"(not(b),weli(b,b1),e):not(a) +*ec c@[e:ec(a,b)][i:imp(c,a)] th3:=trimp(c,a,not(b),i,e):ec(c,b) e@[i:imp(c,b)] th4:=comec(c,a,th3(b,a,c,comec(e),i)):ec(a,c) -ec b@and:=not(ec(a,b)):'prop' [a1:a][b1:b] andi:=th4".imp"(not(b),a1,weli(b,b1)):and(a,b) b@[a1:and(a,b)] ande1:=th5".imp"(not(b),a1):a ande2:=et(b,th6".imp"(not(b),a1)):b comand:=andi(b,a,ande2,ande1):and(b,a) +and b@[n:not(a)] th1:=weli(ec,eci1(n)):not(and) b@[n:not(b)] th2:=weli(ec,eci2(n)):not(and) b@[n:not(and)][a1:a] th3:=ece1(et(ec,n),a1):not(b) n@[b1:b] th4:=ece2(et(ec,n),b1):not(a) n@th5:=th3"l.imp"(and(b,a),and(a,b),n,[x:and(b,a)]comand(b,a,x)):not(and(b,a)) c@[a1:and(a,b)][i:imp(a,c)] th6:=andi(c,b,i,ande2(a1)):and(c,b) a1@[i:imp(b,c)] th7:=andi(a,c,ande1(a1),i):and(a,c) -and b@or:=imp(not(a),b):'prop' [a1:a] ori1:=th2".imp"(not(a),b,weli(a1)):or(a,b) b@[b1:b] ori2:=[x:not(a)]b1:or(a,b) +or b@[i:imp(not(a),b)] th1:=i:or(a,b) b@[i:imp(not(b),a)] th2:=[x:not]et(b,th3"l.imp"(not(b),a,x,i)):or(a,b) -or b@[o:or(a,b)][n:not(a)] ore2:=o:b o@[n:not(b)] ore1:=et(th3".imp"(not(a),b,n,o)):a o@comor:=[x:not(b)]ore1(x):or(b,a) +*or b@[n:not(a)][m:not(b)] th3:=th4"l.imp"(not(a),b,n,m):not(or(a,b)) b@[n:not(or(a,b))] th4:=th5"l.imp"(not(a),b,n):not(a) th5:=th6"l.imp"(not(a),b,n):not(b) a@th6:=refimp(not(a)):or(a,not(a)) -or c@[o:or(a,b)][i:imp(a,c)][j:imp(b,c)] orapp:=th1".imp"(c,i,trimp(not,b,c,o,j)):c c@[d:'prop'] +*or o@[i:imp(a,c)] th7:=trimp(not(c),not,b,[x:not(c)]th3"l.imp"(a,c,x,i),o):or(c,b) o@[i:imp(b,c)] th8:=trimp(not(a),b,c,o,i):or(a,c) d@[o:or(a,b)][i:imp(a,c)][j:imp(b,d)] th9:=th7(a,d,c,th8(a,b,d,o,j),i):or(c,d) b@[o:or(a,b)] th10:=o:imp(not(a),b) th11:=comor(o):imp(not(b),a) b@[o:or(not(a),b)] th12:=trimp(a,wel(a),b,[x:a]weli(x),o):imp(a,b) b@[i:imp(a,b)] th13:=trimp(wel(a),a,b,[x:wel(a)]et(x),i):or(not(a),b) b@[o:or(not(a),not(b))] th14:=weli(ec,th12(not(b),o)):not(and) b@[n:not(and)] th15:=th13(not(b),et(ec,n)):or(not(a),not(b)) b@[a1:and(not(a),not(b))] th16:=th3(ande1(not(a),not(b),a1),ande2(not(a),not(b),a1)):not(or(a,b)) b@[n:not(or(a,b))] th17:=andi(not(a),not(b),th4(n),th5(n)):and(not(a),not(b)) -or b@orec:=and(or(a,b),ec(a,b)):'prop' [o:or(a,b)][e:ec(a,b)] oreci:=andi(or(a,b),ec(a,b),o,e):orec(a,b) +orec b@[a1:a][n:not(b)] th1:=oreci(ori1(a1),eci2(n)):orec(a,b) b@[n:not(a)][b1:b] th2:=oreci(ori2(b1),eci1(n)):orec(a,b) -orec b@[o:orec(a,b)] orece1:=ande1(or(a,b),ec,o):or(a,b) orece2:=ande2(or(a,b),ec,o):ec(a,b) comorec:=oreci(b,a,comor(orece1),comec(orece2)):orec(b,a) +*orec o@[a1:a] th3:=ece1(orece2,a1):not(b) o@[b1:b] th4:=ece2(orece2,b1):not(a) o@[n:not(a)] th5:=ore2(orece1,n):b o@[n:not(b)] th6:=ore1(orece1,n):a -orec b@iff:=and(imp(a,b),imp(b,a)):'prop' [i:imp(a,b)][j:imp(b,a)] iffi:=andi(imp(a,b),imp(b,a),i,j):iff(a,b) +iff b@[a1:a][b1:b] th1:=iffi([x:a]b1,[x:b]a1):iff(a,b) b@[n:not(a)][m:not(b)] th2:=iffi(th2"l.imp"(n),th2"l.imp"(b,a,m)):iff(a,b) -iff b@[i:iff(a,b)] iffe1:=ande1(imp(a,b),imp(b,a),i):imp(a,b) iffe2:=ande2(imp(a,b),imp(b,a),i):imp(b,a) comiff:=iffi(b,a,iffe2,iffe1):iff(b,a) +*iff i@[a1:a] th3:=iffe1:b i@[b1:b] th4:=iffe2:a i@[n:not(a)] th5:=th3"l.imp"(b,a,n,iffe2):not(b) i@[n:not(b)] th6:=th3"l.imp"(n,iffe1):not(a) b@[a1:a][n:not(b)] th7:=th1"l.and"(imp(a,b),imp(b,a),th4"l.imp"(a1,n)):not(iff(a,b)) b@[n:not(a)][b1:b] th8:=th2"l.and"(imp(a,b),imp(b,a),th4"l.imp"(b,a,b1,n)):not(iff(a,b)) -iff a@refiff:=iffi(a,refimp,refimp):iff(a,a) b@[i:iff(a,b)] symiff:=comiff(i):iff(b,a) c@[i:iff(a,b)][j:iff(b,c)] triff:=iffi(a,c,trimp(iffe1(i),iffe1(b,c,j)),trimp(c,b,a,iffe2(b,c,j),iffe2(i))):iff(a,c) +*iff b@[i:iff(a,b)] th9:=[x:not(a)]th5(i,x):imp(not(a),not(b)) th10:=[x:not(b)]th6(i,x):imp(not(b),not(a)) th11:=iffi(not(a),not(b),th9,th10):iff(not(a),not(b)) b@[i:imp(not(a),not(b))][j:imp(not(b),not(a))] th12:=iffi(cp(j),cp(b,a,i)):iff(a,b) b@[o:orec(a,b)] th13:=iffi(not(b),orece2(o),comor(orece1(o))):iff(a,not(b)) th14:=th13(b,a,comorec(o)):iff(b,not(a)) b@[i:iff(a,not(b))] th15:=oreci(comor(b,a,iffe2(not(b),i)),iffe1(not(b),i)):orec(a,b) b@[i:iff(b,not(a))] th16:=comorec(b,a,th15(b,a,i)):orec(a,b) c@[i:iff(a,b)][j:imp(a,c)] thimp1:=trimp(b,a,c,iffe2(i),j):imp(b,c) i@[j:imp(c,a)] thimp2:=trimp(c,a,b,j,iffe1(i)):imp(c,b) i@[e:ec(a,c)] thec1:=th3"l.ec"(c,b,e,iffe2(i)):ec(b,c) i@[e:ec(c,a)] thec2:=th4"l.ec"(c,a,b,e,iffe2(i)):ec(c,b) i@[a1:and(a,c)] thand1:=th6"l.and"(c,b,a1,iffe1(i)):and(b,c) i@[a1:and(c,a)] thand2:=th7"l.and"(c,a,b,a1,iffe1(i)):and(c,b) i@[o:or(a,c)] thor1:=th7"l.or"(c,b,o,iffe1(i)):or(b,c) i@[o:or(c,a)] thor2:=th8"l.or"(c,a,b,o,iffe1(i)):or(c,b) i@[o:orec(a,c)] thorec1:=oreci(b,c,thor1(orece1(a,c,o)),thec1(orece2(a,c,o))):orec(b,c) i@[o:orec(c,a)] thorec2:=oreci(c,b,thor2(orece1(c,a,o)),thec2(orece2(c,a,o))):orec(c,b) -iff @[sigma:'type'][p:[x:sigma]'prop'] %suggestion by van Daalen to remove the first eta-reduction %all:=p:'prop' %original line all:=[x:sigma]p:'prop' %end of suggestion [a1:all(sigma,p)][s:sigma] alle:=a1:p +all p@[s:sigma][n:not(p)] th1:=[x:all(sigma,p)]<x>n:not(all(sigma,p)) -all p@non:=[x:sigma]not(p):[x:sigma]'prop' some:=not(non(p)):'prop' [s:sigma][sp:p] somei:=th1".all"(non(p),s,weli(p,sp)):some(sigma,p) +some p@[n:not(all(sigma,p))][m:non(non(p))][s:sigma] t1:=et(p,m):p %set etared m@t2:=<[x:sigma]t1(x)>n:con %reset etared n@th1:=[x:non(non(p))]t2(x):some(non(p)) p@[s:some(non(p))][a1:all(sigma,p)][t:sigma] t3:=weli(p,a1):not(not(p)) a1@t4:=<[x:sigma]t3(x)>s:con s@th2:=[x:all(sigma,p)]t4(x):not(all(sigma,p)) p@[n:not(some(sigma,p))] th3:=et(non(p),n):non(p) [s:sigma] th4:=th3:not(p) p@[n:non(p)] th5:=weli(non(p),n):not(some(sigma,p)) -some p@[s:some(sigma,p)][x:'prop'][i:[y:sigma]imp(p,x)] +*some i@[n:not(x)][t:sigma] t5:=th3"l.imp"(p,x,n,i):not(p) n@t6:=mp(some(sigma,p),con,s,th5([y:sigma]t5(y))):con -some i@someapp:=et(x,[y:not(x)]t6".some"(y)):x +*some p@[q:[x:sigma]'prop'][s:some(sigma,p)][i:[x:sigma]imp(p,q)] th6:=someapp(s,some(q),[x:sigma][y:p]somei(q,x,mp(p,q,y,i))):some(q) -some c@or3:=or(a,or(b,c)):'prop' [o:or3(a,b,c)][n:not(a)] +or3 th1:=ore2(or(b,c),o,n):or(b,c) -or3 [m:not(b)] or3e3:=ore2(b,c,th1".or3",m):c o@[n:not(b)] +*or3 n@th2:=th2"l.or"(c,a,[x:not(a)]or3e3(x,n)):or(c,a) -or3 n@[m:not(c)] or3e1:=ore2(c,a,th2".or3",m):a o@[n:not(c)] +*or3 n@th3:=th2"l.or"([x:not(b)]or3e1(x,n)):or(a,b) -or3 n@[m:not(a)] or3e2:=ore2(th3".or3",m):b +*or3 o@th4:=th1"l.or"(b,or(c,a),[x:not(b)]th2(x)):or3(b,c,a) th5:=th4(b,c,a,th4):or3(c,a,b) -or3 c@[a1:a] or3i1:=ori1(a,or(b,c),a1):or3(a,b,c) c@[b1:b] or3i2:=ori2(a,or(b,c),ori1(b,c,b1)):or3(a,b,c) c@[c1:c] or3i3:=ori2(a,or(b,c),ori2(b,c,c1)):or3(a,b,c) +*or3 c@[o:or(a,b)] th6:=th4"or3"(c,a,b,ori2(c,or(a,b),o)):or3(a,b,c) c@[o:or(b,c)] th7:=ori2(or(b,c),o):or3(a,b,c) c@[o:or(c,a)] th8:=th4"or3"(c,a,b,th6(c,a,b,o)):or3(a,b,c) -or3 d@[o:or3(a,b,c)][i:imp(a,d)][j:imp(b,d)][k:imp(c,d)] or3app:=orapp(or(b,c),d,o,i,[x:or(b,c)]orapp(b,c,d,x,j,k)):d c@and3:=and(a,and(b,c)):'prop' [a1:and3(a,b,c)] and3e1:=ande1(and(b,c),a1):a and3e2:=ande1(b,c,ande2(and(b,c),a1)):b and3e3:=ande2(b,c,ande2(and(b,c),a1)):c c@[a1:a][b1:b][c1:c] and3i:=andi(a,and(b,c),a1,andi(b,c,b1,c1)):and3(a,b,c) +and3 c@[a1:and3(a,b,c)] th1:=and3i(b,c,a,and3e2(a1),and3e3(a1),and3e1(a1)):and3(b,c,a) th2:=th1(b,c,a,th1):and3(c,a,b) th3:=andi(and3e1(a1),and3e2(a1)):and(a,b) th4:=ande2(and(b,c),a1):and(b,c) th5:=th3(c,a,b,th2):and(c,a) th6:=and3i(c,b,a,and3e3(a1),and3e2(a1),and3e1(a1)):and3(c,b,a) -and3 c@ec3:=and3(ec,ec(b,c),ec(c,a)):'prop' [e:ec3(a,b,c)] +ec3 th1:=and3e1(ec,ec(b,c),ec(c,a),e):ec(a,b) th2:=and3e2(ec,ec(b,c),ec(c,a),e):ec(b,c) th3:=and3e3(ec,ec(b,c),ec(c,a),e):ec(c,a) th4:=th1"l.and3"(ec,ec(b,c),ec(c,a),e):ec3(b,c,a) th5:=th4(b,c,a,th4):ec3(c,a,b) 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) -ec3 [a1:a] ec3e12:=ece1(th1".ec3",a1):not(b) ec3e13:=ece2(c,a,th3".ec3",a1):not(c) e@[b1:b] ec3e23:=ec3e12(b,c,a,th4".ec3",b1):not(c) ec3e21:=ec3e13(b,c,a,th4".ec3",b1):not(a) e@[c1:c] ec3e31:=ec3e12(c,a,b,th5".ec3",c1):not(a) ec3e32:=ec3e13(c,a,b,th5".ec3",c1):not(b) +*ec3 c@[e:ec(a,b)][f:ec(b,c)][g:ec(c,a)] th6:=and3i(ec,ec(b,c),ec(c,a),e,f,g):ec3(a,b,c) c@[e:ec3(a,b,c)][o:or(a,b)] 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) e@[o:or(b,c)] th8:=th7(b,c,a,th4"ec3"(e),o):not(a) e@[o:or(c,a)] th9:=th7(c,a,b,th5"ec3"(e),o):not(b) -ec3 c@[n:not(a)][m:not(b)] ec3i1:=th6".ec3"(eci1(n),eci1(b,c,m),eci2(c,a,n)):ec3(a,b,c) c@[n:not(b)][m:not(c)] ec3i2:=th6".ec3"(eci2(n),eci1(b,c,n),eci1(c,a,m)):ec3(a,b,c) c@[n:not(c)][m:not(a)] ec3i3:=th6".ec3"(eci1(m),eci2(b,c,n),eci1(c,a,n)):ec3(a,b,c) +*ec3 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] t1:=[x:b]mp(e,con,j,ec3e12(d,e,f,p1,d1)):not(b) t2:=[x:c]mp(f,con,k,ec3e13(d,e,f,p1,d1)):not(c) th10:=or3e1(o1,t1,t2):a k@[e1:e] th11:=th10(b,c,a,e,f,d,th4"l.or3"(o1),th4"ec3"(d,e,f,p1),j,k,i,e1):b k@[f1:f] th12:=th10(c,a,b,f,d,e,th5"l.or3"(o1),th5"ec3"(d,e,f,p1),k,i,j,f1):c -ec3 c@orec3:=and(or3(a,b,c),ec3(a,b,c)):'prop' [o:orec3(a,b,c)] orec3e1:=ande1(or3(a,b,c),ec3(a,b,c),o):or3(a,b,c) orec3e2:=ande2(or3(a,b,c),ec3(a,b,c),o):ec3(a,b,c) c@[o:or3(a,b,c)][e:ec3(a,b,c)] orec3i:=andi(or3(a,b,c),ec3(a,b,c),o,e):orec3(a,b,c) +orec3 c@[o:orec3(a,b,c)] th1:=orec3i(b,c,a,th4"l.or3"(orec3e1(o)),th4"l.ec3"(orec3e2(o))):orec3(b,c,a) th2:=orec3i(c,a,b,th5"l.or3"(orec3e1(o)),th5"l.ec3"(orec3e2(o))):orec3(c,a,b) -orec3 +e sigma@[s:sigma][t:sigma] is:='prim':'prop' s@refis:='prim':is(s,s) p@[s:sigma][t:sigma][sp:p][i:is(s,t)] isp:='prim':p sigma@[s:sigma][t:sigma][i:is(s,t)] symis:=isp([x:sigma]is(x,s),s,t,refis(s),i):is(t,s) t@[u:sigma][i:is(s,t)][j:is(t,u)] tris:=isp([x:sigma]is(x,u),t,s,j,symis(i)):is(s,u) u@[i:is(u,s)][j:is(u,t)] tris1:=tris(s,u,t,symis(u,s,i),j):is(s,t) u@[i:is(s,u)][j:is(t,u)] tris2:=tris(s,u,t,i,symis(t,u,j)):is(s,t) sp@[i:is(t,s)] isp1:=isp(symis(t,s,i)):p t@[n:not(is(s,t))] symnotis:=th3"l.imp"(is(t,s),is(s,t),n,[x:is(t,s)]symis(t,s,x)):not(is(t,s)) +notis u@[n:not(is(s,t))][i:is(s,u)] th1:=isp([x:sigma]not(is(x,t)),s,u,n,i):not(is(u,t)) n@[i:is(u,s)] th2:=th1(symis(u,s,i)):not(is(u,t)) n@[i:is(t,u)] th3:=isp([x:sigma]not(is(s,x)),t,u,n,i):not(is(s,u)) n@[i:is(u,t)] th4:=th3(symis(u,t,i)):not(is(s,u)) u@[v:sigma][n:not(is(s,t))][i:is(s,u)][j:is(t,v)] th5:=th3(u,t,v,th1(n,i),j):not(is(u,v)) -notis u@[v:sigma][i:is(s,t)][j:is(t,u)][k:is(u,v)] tr3is:=tris(s,u,v,tris(i,j),k):is(s,v) v@[w:sigma][i:is(s,t)][j:is(t,u)][k:is(u,v)][l:is(v,w)] tr4is:=tris(s,v,w,tr3is(i,j,k),l):is(s,w) p@amone:=[x:sigma][y:sigma][u:p][v:p]is(x,y):'prop' [a1:amone(sigma,p)][s:sigma][t:sigma][sp:p][tp:p] amonee:=a1:is(s,t) p@one:=and(amone(sigma,p),some(sigma,p)):'prop' [a1:amone(sigma,p)][s:some(sigma,p)] onei:=andi(amone(sigma,p),some(sigma,p),a1,s):one(sigma,p) p@[o1:one(sigma,p)] onee1:=ande1(amone(sigma,p),some(sigma,p),o1):amone(sigma,p) onee2:=ande2(amone(sigma,p),some(sigma,p),o1):some(sigma,p) ind:='prim':sigma oneax:='prim':p +one [s:sigma][sp:p] th1:=amonee(onee1,ind,s,oneax,sp):is(ind,s) -one sigma@[tau:'type'][f:[x:sigma]tau][s:sigma][t:sigma][i:is(s,t)] isf:=isp(sigma,[x:sigma]is(tau,f,f),s,t,refis(tau,f),i):is(tau,f,f) f@injective:=all([x:sigma]all([y:sigma]imp(is(tau,f,f),is(x,y)))):'prop' [i:injective(f)][s:sigma][t:sigma][j:is(tau,f,f)] isfe:=i:is(s,t) f@[t0:tau] image:=some([x:sigma]is(tau,t0,f)):'prop' f@[s:sigma] tofs:=f:tau imagei:=somei([x:sigma]is(tau,tofs,f),s,refis(tau,tofs)):image(tofs) +inj 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))] t1:=tr3is(tau,tofs(sa),ta,tb,tofs(sb),symis(tau,ta,tofs(sa),ja),j,jb):is(tau,tofs(sa),tofs(sb)) th1:=isfe(sa,sb,t1):is(sa,sb) i@[t0:tau] th2:=[x:sigma][y:sigma][u:is(tau,t0,f)][v:is(tau,t0,f)]th1(t0,t0,refis(tau,t0),x,y,u,v):amone([x:sigma]is(tau,t0,f)) [j:image(f,t0)] th3:=onei([x:sigma]is(tau,t0,f),th2,j):one([x:sigma]is(tau,t0,f)) -inj i@[t0:tau][j:image(f,t0)] soft:=ind([x:sigma]is(tau,t0,f),th3".inj"(t0,j)):sigma i@inverse:=[x:tau][u:image(f,x)]soft(x,u):[x:tau][u:image(f,x)]sigma j@ists1:=oneax([x:sigma]is(tau,t0,f),th3".inj"(t0,j)):is(tau,t0,tofs(soft(t0,j))) ists2:=symis(tau,t0,tofs(soft(t0,j)),ists1):is(tau,tofs(soft(t0,j)),t0) i@[ta:tau][ja:image(ta)][tb:tau][jb:image(tb)][j:is(tau,ta,tb)] 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)) jb@[j:is(soft(ta,ja),soft(tb,jb))] 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) i@[s:sigma] isst1:=isfe(s,soft(tofs(s),imagei(s)),ists1(tofs(s),imagei(s))):is(s,soft(tofs(s),imagei(s))) isst2:=symis(s,soft(tofs(s),imagei(s)),isst1):is(soft(tofs(s),imagei(s)),s) f@surjective:=all(tau,[x:tau]image(x)):'prop' bijective:=and(injective,surjective):'prop' [b:bijective(f)] +*inj b@t2:=ande1(injective,surjective,b):injective(f) t3:=ande2(injective,surjective,b):surjective(f) [t:tau] so:=soft(t2,t,t3):sigma -inj b@invf:=[x:tau]so".inj"(x):[x:tau]sigma [s:sigma] thinvf1:=tris(s,soft(t2".inj",tofs(s),imagei(s)),<f>invf,isst1(t2".inj",s),isinv(t2".inj",tofs(s),imagei(s),tofs(s),t3".inj",refis(tau,tofs(s)))):is(s,<f>invf) b@[t:tau] thinvf2:=ists1(t2".inj",t,t3".inj"):is(tau,t,<invf>f) tau@[upsilon:'type'][f:[x:sigma]tau][g:[x:tau]upsilon] +*inj g@[if:injective(sigma,tau,f)][ig:injective(tau,upsilon,g)] h:=[x:sigma]<f>g:[x:sigma]upsilon [s:sigma][t:sigma][i:is(upsilon,h,h)] t4:=isfe(tau,upsilon,g,ig,f,f,i):is(tau,f,f) t5:=isfe(f,if,s,t,t4):is(s,t) ig@th4:=[x:sigma][y:sigma][v:is(upsilon,h,h)]t5(x,y,v):injective(sigma,upsilon,[x:sigma]<f>g) -inj +surj g@[sf:surjective(sigma,tau,f)][sg:surjective(tau,upsilon,g)] h:=[x:sigma]<f>g:[x:sigma]upsilon [u:upsilon] t1:=sg:image(tau,upsilon,g,u) [t:tau][i:is(upsilon,u,g)] t2:=sf:image(sigma,tau,f,t) [s:sigma][j:is(tau,t,f)] t3:=tris(upsilon,u,g,h,i,isf(tau,upsilon,g,t,f,j)):is(upsilon,u,h) t4:=somei([x:sigma]is(upsilon,u,h),s,t3):image(sigma,upsilon,h,u) i@t5:=someapp([x:sigma]is(tau,t,f),t2,image(sigma,upsilon,h,u),[x:sigma][v:is(tau,t,f)]t4(x,v)):image(sigma,upsilon,h,u) u@t6:=someapp(tau,[x:tau]is(upsilon,u,g),t1,image(sigma,upsilon,h,u),[x:tau][v:is(upsilon,u,g)]t5(x,v)):image(sigma,upsilon,h,u) sg@th1:=[x:upsilon]t6(x):surjective(sigma,upsilon,[x:sigma]<f>g) -surj +bij g@[bf:bijective(sigma,tau,f)][bg:bijective(tau,upsilon,g)] h:=[x:sigma]<f>g:[x:sigma]upsilon 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) 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) th1:=andi(injective(sigma,upsilon,h),surjective(sigma,upsilon,h),t1,t2):bijective(sigma,upsilon,[x:sigma]<f>g) -bij tau@[f:[x:sigma]tau][g:[x:sigma]tau][i:is([x:sigma]tau,f,g)][s:sigma] fise:=isp([x:sigma]tau,[y:[x:sigma]tau]is(tau,f,y),f,g,refis(tau,f),i):is(tau,f,g) g@[i:[x:sigma]is(tau,f,g)] fisi:='prim':is([x:sigma]tau,f,g) +fis g@[i:is([x:sigma]tau,f,g)][s:sigma][t:sigma][j:is(s,t)] th1:=tris(tau,f,f,g,isf(f,s,t,j),fise(i,t)):is(tau,f,g) -fis p@ot:='prim':'type' [o1:ot] in:='prim':sigma inp:='prim':p p@otax1:='prim':injective(ot,sigma,[x:ot]in(x)) [s:sigma][sp:p] otax2:='prim':image(ot,sigma,[x:ot]in(x),s) o1@[o2:ot][i:is(ot,o1,o2)] isini:=isf(ot,sigma,[x:ot]in(x),o1,o2,i):is(in(o1),in(o2)) o2@[i:is(in(o1),in(o2))] isine:=isfe(ot,sigma,[x:ot]in(x),otax1,o1,o2,i):is(ot,o1,o2) sp@out:=soft(ot,sigma,[x:ot]in(x),otax1,s,otax2):ot [t:sigma][tp:p][i:is(s,t)] isouti:=isinv(ot,sigma,[x:ot]in(x),otax1,s,otax2,t,otax2(t,tp),i):is(ot,out(s,sp),out(t,tp)) tp@[i:is(ot,out(s,sp),out(t,tp))] isoute:=isinve(ot,sigma,[x:ot]in(x),otax1,s,otax2,t,otax2(t,tp),i):is(s,t) 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))) sp@isinout:=ists1(ot,sigma,[x:ot]in(x),otax1,s,otax2):is(s,in(out(s,sp))) tau@pairtype:='prim':'type' [s:sigma][t:tau] pair:='prim':pairtype tau@[p1:pairtype] first:='prim':sigma second:='prim':tau pairis1:='prim':is(pairtype,pair(first,second),p1) pairis2:=symis(pairtype,pair(first,second),p1,pairis1):is(pairtype,p1,pair(first,second)) t@firstis1:='prim':is(sigma,first(pair),s) firstis2:=symis(sigma,first(pair),s,firstis1):is(sigma,s,first(pair)) secondis1:='prim':is(tau,second(pair),t) secondis2:=symis(tau,second(pair),t,secondis1):is(tau,t,second(pair)) a@[ksi:'type'][x:ksi][y:ksi] +ite [z:ksi] prop1:=and(imp(a,is(ksi,z,x)),imp(not(a),is(ksi,z,y))):'prop' y@[a1:a][x1:ksi][y1:ksi][px1:prop1(x1)][py1:prop1(y1)] t1:=ande1(imp(a,is(ksi,x1,x)),imp(not(a),is(ksi,x1,y)),px1):imp(a,is(ksi,x1,x)) t2:=mp(a,is(ksi,x1,x),a1,t1):is(ksi,x1,x) t3:=t2(y1,x1,py1,px1):is(ksi,y1,x) t4:=tris2(ksi,x1,y1,x,t2,t3):is(ksi,x1,y1) a1@t5:=[s:ksi][t:ksi][ps:prop1(s)][pt:prop1(t)]t4(s,t,ps,pt):amone(ksi,[t:ksi]prop1(t)) t6:=[x1:a]refis(ksi,x):imp(a,is(ksi,x,x)) t7:=th2"l.imp"(not(a),is(ksi,x,y),weli(a,a1)):imp(not(a),is(ksi,x,y)) t8:=andi(imp(a,is(ksi,x,x)),imp(not(a),is(ksi,x,y)),t6,t7):prop1(x) t9:=somei(ksi,[t:ksi]prop1(t),x,t8):some(ksi,[t:ksi]prop1(t)) t10:=onei(ksi,[t:ksi]prop1(t),t5,t9):one(ksi,[t:ksi]prop1(t)) y@[n:not(a)][x1:ksi][y1:ksi][px1:prop1(x1)][py1:prop1(y1)] t11:=ande2(imp(a,is(ksi,x1,x)),imp(not(a),is(ksi,x1,y)),px1):imp(not(a),is(ksi,x1,y)) t12:=mp(not(a),is(ksi,x1,y),n,t11):is(ksi,x1,y) t13:=t12(y1,x1,py1,px1):is(ksi,y1,y) t14:=tris2(ksi,x1,y1,y,t12,t13):is(ksi,x1,y1) n@t15:=[s:ksi][t:ksi][ps:prop1(s)][pt:prop1(t)]t14(s,t,ps,pt):amone(ksi,[t:ksi]prop1(t)) t16:=[x1:not(a)]refis(ksi,y):imp(not(a),is(ksi,y,y)) t17:=th2"l.imp"(a,is(ksi,y,x),n):imp(a,is(ksi,y,x)) t18:=andi(imp(a,is(ksi,y,x)),imp(not(a),is(ksi,y,y)),t17,t16):prop1(y) t19:=somei(ksi,[t:ksi]prop1(t),y,t18):some(ksi,[t:ksi]prop1(t)) t20:=onei(ksi,[t:ksi]prop1(t),t15,t19):one(ksi,[t:ksi]prop1(t)) 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)) -ite ite:=ind(ksi,[t:ksi]prop1".ite"(t),t21".ite"):ksi +*ite y@t22:=oneax(ksi,[t:ksi]prop1(t),t21):prop1(ite) t23:=ande1(imp(a,is(ksi,ite,x)),imp(not(a),is(ksi,ite,y)),t22):imp(a,is(ksi,ite,x)) t24:=ande2(imp(a,is(ksi,ite,x)),imp(not(a),is(ksi,ite,y)),t22):imp(not(a),is(ksi,ite,y)) -ite y@[a1:a] itet:=mp(a,is(ksi,ite,x),a1,t23".ite"):is(ksi,ite,x) y@[n:not(a)] itef:=mp(not(a),is(ksi,ite,y),n,t24".ite"):is(ksi,ite,y) sigma@[s0:sigma][t0:sigma] +wissel [s:sigma] wa:=ite(is(s,s0),sigma,t0,s):sigma [i:is(s,s0)] t1:=itet(is(s,s0),sigma,t0,s,i):is(wa,t0) s@[n:not(is(s,s0))] t2:=itef(is(s,s0),sigma,t0,s,n):is(wa,s) s@wb:=ite(is(s,t0),sigma,s0,wa):sigma [i:is(s,t0)] t3:=itet(is(s,t0),sigma,s0,wa,i):is(wb,s0) s@[n:not(is(s,t0))] t4:=itef(is(s,t0),sigma,s0,wa,n):is(wb,wa) s@[i:is(s,s0)][j:is(s0,t0)] t5:=tris(wb,s0,t0,t3(tris(s,s0,t0,i,j)),j):is(wb,t0) i@[n:not(is(s0,t0))] t6:=tris(wb,wa,t0,t4(th2"e.notis"(s0,t0,s,n,i)),t1(i)):is(wb,t0) 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) s@[n:not(is(s,s0))][o:not(is(s,t0))] t8:=tris(wb,wa,s,t4(o),t2(n)):is(wb,s) -wissel wissel:=[x:sigma]wb".wissel"(x):[x:sigma]sigma [s:sigma][i:is(s,s0)] iswissel1:=t7".wissel"(s,i):is(wissel,t0) s@[i:is(s,t0)] iswissel2:=t3".wissel"(s,i):is(wissel,s0) s@[n:not(is(s,s0))][o:not(is(s,t0))] iswissel3:=t8".wissel"(s,n,o):is(wissel,s) +*wissel s@[t:sigma][i:is(wb(s),wb(t))][n:not(is(s,t))][j:is(s,s0)] t9:=symnotis(s0,t,th1"e.notis"(s,t,s0,n,j)):not(is(t,s0)) [k:is(s0,t0)] t10:=th3"e.notis"(t,s0,t0,t9,k):not(is(t,t0)) t11:=tris(wb(s),wb(t),t,i,t8(t,t9,t10)):is(wb(s),t) t12:=t10:con j@t13:=[v:is(s0,t0)]t12(v):not(is(s0,t0)) [k:is(t,t0)] t14:=tris(wb(s),wb(t),s0,i,t3(t,k)):is(wb(s),s0) t15:=t12(tris1(s0,t0,wb(s),t14,t7(j))):con j@t16:=[v:is(t,t0)]t15(v):not(is(t,t0)) t17:=tris(wb(s),wb(t),t,i,t8(t,t9,t16)):is(wb(s),t) t18:=t15(tris1(t,t0,wb(s),t17,t7(j))):con n@t19:=[v:is(s,s0)]t18(v):not(is(s,s0)) t20:=t19(t,s,symis(wb(s),wb(t),i),symnotis(s,t,n)):not(is(t,s0)) [j:is(s,t0)] t21:=symnotis(t0,t,th1"e.notis"(s,t,t0,n,j)):not(is(t,t0)) t22:=tris(wb(s),wb(t),t,i,t8(t,t20,t21)):is(wb(s),t) t23:=t20:con n@t24:=[v:is(s,t0)]t23(v):not(is(s,t0)) t25:=t24(t,s,symis(wb(s),wb(t),i),symnotis(s,t,n)):not(is(t,t0)) t26:=tris(wb(s),wb(t),t,i,t8(t,t20,t25)):is(wb(s),t) t27:=n:con i@t28:=et(is(s,t),[v:not(is(s,t))]t27(v)):is(s,t) t0@th1:=[x:sigma][y:sigma][v:is(wb(x),wb(y))]t28(x,y,v):injective(sigma,sigma,wissel) s@[i:is(s,s0)] t29:=tris2(s,wb(t0),s0,i,t3(t0,refis(t0))):is(s,wb(t0)) t30:=somei(sigma,[x:sigma]is(s,wb(x)),t0,t29):image(sigma,sigma,wissel,s) s@[i:is(s,t0)] t31:=tris2(s,wb(s0),t0,i,t7(s0,refis(s0))):is(s,wb(s0)) t32:=somei(sigma,[x:sigma]is(s,wb(x)),s0,t31):image(sigma,sigma,wissel,s) s@[n:not(is(s,s0))][o:not(is(s,t0))] t33:=symis(wb(s),s,t8(n,o)):is(s,wb(s)) t34:=somei(sigma,[x:sigma]is(s,wb(x)),s,t33):image(sigma,sigma,wissel,s) 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) 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) t0@th2:=[x:sigma]t36(x):surjective(sigma,sigma,wissel) th3:=andi(injective(sigma,sigma,wissel),surjective(sigma,sigma,wissel),th1,th2):bijective(sigma,sigma,wissel) -wissel tau@[f:[x:sigma]tau][s0:sigma][t0:sigma] changef:=[x:sigma]<wissel(s0,t0)>f:[x:sigma]tau [s:sigma][i:is(s,s0)] changef1:=isf(sigma,tau,f,wissel(s0,t0),t0,iswissel1(s0,t0,s,i)):is(tau,changef,f) s@[i:is(s,t0)] changef2:=isf(sigma,tau,f,wissel(s0,t0),s0,iswissel2(s0,t0,s,i)):is(tau,changef,f) s@[n:not(is(s,s0))][o:not(is(s,t0))] changef3:=isf(sigma,tau,f,wissel(s0,t0),s,iswissel3(s0,t0,s,n,o)):is(tau,changef,f) +*wissel t0@[i:injective(f)] th4:=th4"e.inj"(sigma,sigma,tau,wissel(s0,t0),f,th1(s0,t0),i):injective(changef) t0@[s:surjective(f)] th5:=th1"e.surj"(sigma,sigma,tau,wissel(s0,t0),f,th2(s0,t0),s):surjective(changef) t0@[b:bijective(f)] th6:=th1"e.bij"(sigma,sigma,tau,wissel(s0,t0),f,th3(s0,t0),b):bijective(changef) -wissel -e +r a@[b:[x:a]'prop'] %suggestion by van Daalen to remove the second eta-reduction %imp:=b:'prop' %original line imp:=[x:a]b:'prop' %end of suggestion [a1:a][i:imp(a,b)] mp:=i:b +imp b@[n:not(a)] %set etared th2:=[x:a]cone(b,mp"l"(a,con,x,n)):imp(a,b) %reset etared -imp b@ec:=[x:a]not(b):'prop' [n:not(a)] eci1:=[x:a]cone(not(b),mp"l"(a,con,x,n)):ec(a,b) b@[a1:and(a,b)] ande2:=ande2"l"(a,b,a1):b a@[ksi:'type'] +ite [x1:ksi][y1:ksi] is:=is"l.e"(ksi,x1,y1):'prop' -ite [x:[t:a]ksi][y:[t:not(a)]ksi][i:[t:a][u:a]is".ite"(x,x)][j:[t:not(a)][u:not(a)]is".ite"(y,y)] +*ite j@[z:ksi] prop1:=and(imp(a,[t:a]is(z,x)),imp(not(a),[t:not(a)]is(z,y))):'prop' j@[a1:a][x1:ksi][y1:ksi][px1:prop1(x1)][py1:prop1(y1)] t1:=ande1"l"(imp(a,[t:a]is(x1,x)),imp(not(a),[t:not(a)]is(x1,y)),px1):imp(a,[t:a]is(x1,x)) t2:=mp(a,[t:a]is(x1,x),a1,t1):is(x1,x) t3:=t2(y1,x1,py1,px1):is(y1,x) t4:=tris2"l.e"(ksi,x1,y1,x,t2,t3):is(x1,y1) 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)) t6:=i:imp(a,[t:a]is(x,x)) t7:=th2"r.imp"(not(a),[t:not(a)]is(x,y),weli(a,a1)):imp(not(a),[t:not(a)]is(x,y)) t8:=andi(imp(a,[t:a]is(x,x)),imp(not(a),[t:not(a)]is(x,y)),t6,t7):prop1(x) t9:=somei(ksi,[t:ksi]prop1(t),x,t8):some(ksi,[t:ksi]prop1(t)) t10:=onei"l.e"(ksi,[t:ksi]prop1(t),t5,t9):one"l.e"(ksi,[t:ksi]prop1(t)) j@[n:not(a)][x1:ksi][y1:ksi][px1:prop1(x1)][py1:prop1(y1)] t11:=ande2"l"(imp(a,[t:a]is(x1,x)),imp(not(a),[t:not(a)]is(x1,y)),px1):imp(not(a),[t:not(a)]is(x1,y)) t12:=mp(not(a),[t:not(a)]is(x1,y),n,t11):is(x1,y) t13:=t12(y1,x1,py1,px1):is(y1,y) t14:=tris2"l.e"(ksi,x1,y1,y,t12,t13):is(x1,y1) 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)) t16:=j:imp(not(a),[t:not(a)]is(y,y)) t17:=th2"r.imp"(a,[t:a]is(y,x),n):imp(a,[t:a]is(y,x)) t18:=andi"l"(imp(a,[t:a]is(y,x)),imp(not(a),[t:not(a)]is(y,y)),t17,t16):prop1(y) t19:=somei(ksi,[t:ksi]prop1(t),y,t18):some(ksi,[t:ksi]prop1(t)) t20:=onei"l.e"(ksi,[t:ksi]prop1(t),t15,t19):one"l.e"(ksi,[t:ksi]prop1(t)) 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)) -ite j@ite:=ind"l.e"(ksi,[t:ksi]prop1".ite"(t),t21".ite"):ksi +*ite j@t22:=oneax"l.e"(ksi,[t:ksi]prop1(t),t21):prop1(ite) t23:=ande1"l"(imp(a,[t:a]is(ite,x)),imp(not(a),[t:not(a)]is(ite,y)),t22):imp(a,[t:a]is(ite,x)) t24:=ande2"l"(imp(a,[t:a]is(ite,x)),imp(not(a),[t:not(a)]is(ite,y)),t22):imp(not(a),[t:not(a)]is(ite,y)) -ite j@[a1:a] itet:=mp(a,[t:a]is".ite"(ite,x),a1,t23".ite"):is".ite"(ksi,ite,x) j@[n:not(a)] itef:=mp(not(a),[t:not(a)]is".ite"(ite,y),n,t24".ite"):is".ite"(ksi,ite,y) -r +*e +st sigma@set:='prim':'type' [s:sigma][s0:set] esti:='prim':'prop' p@setof:='prim':set [s:sigma][sp:p] estii:='prim':esti(s,setof(p)) s@[e:esti(s,setof(p))] estie:='prim':p sigma@[s0:set] empty:=non([x:sigma]esti(x,s0)):'prop' nonempty:=some([x:sigma]esti(x,s0)):'prop' [n:[x:sigma]not(esti(x,s0))] emptyi:=n:empty(s0) s0@[e:empty(s0)][s:sigma] emptye:=e:not(esti(s,s0)) s0@[s:sigma][ses0:esti(s,s0)] nonemptyi:=somei([x:sigma]esti(x,s0),s,ses0):nonempty(s0) s0@[n:nonempty(s0)][x:'prop'][x1:[y:sigma][z:esti(y,s0)]x] nonemptyapp:=someapp([y:sigma]esti(y,s0),n,x,x1):x s0@[t0:set] incl:=all([x:sigma]imp(esti(x,s0),esti(x,t0))):'prop' [e:[x:sigma][y:esti(x,s0)]esti(x,t0)] incli:=e:incl(s0,t0) t0@[i:incl(s0,t0)][s:sigma][ses0:esti(s,s0)] incle:=i:esti(s,t0) s0@refincl:=[x:sigma][y:esti(x,s0)]y:incl(s0,s0) t0@disj:=all([x:sigma]ec(esti(x,s0),esti(x,t0))):'prop' [n:[x:sigma][y:esti(x,s0)]not(esti(x,t0))] disji1:=n:disj(s0,t0) t0@[n:[x:sigma][y:esti(x,t0)]not(esti(x,s0))] disji2:=[x:sigma]th2"l.ec"(esti(x,s0),esti(x,t0),n):disj(s0,t0) t0@[d:disj(s0,t0)][s:sigma][ses0:esti(s,s0)] disje1:=ece1(esti(s,s0),esti(s,t0),d,ses0):not(esti(s,t0)) s@[set0:esti(s,t0)] disje2:=ece2(esti(s,s0),esti(s,t0),d,set0):not(esti(s,s0)) t0@[d:disj(s0,t0)] symdisj:=[x:sigma][y:esti(x,t0)]disje2(d,x,y):disj(t0,s0) +disj t0@[s:sigma][ses0:esti(s,s0)][set0:esti(s,t0)] 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)) th2:=th1(t0,s0,s,set0,ses0):not(disj(t0,s0)) -disj t0@[i:is(set,s0,t0)][s:sigma][ses0:esti(s,s0)] issete1:=isp(set,[x:set]esti(s,x),s0,t0,ses0,i):esti(s,t0) s@[set0:esti(s,t0)] issete2:=isp1(set,[x:set]esti(s,x),t0,s0,set0,i):esti(s,s0) +isset i@th1:=[x:sigma][y:esti(x,s0)]issete1(x,y):incl(s0,t0) th2:=[x:sigma][y:esti(x,t0)]issete2(x,y):incl(t0,s0) -isset t0@[i:incl(s0,t0)][j:incl(t0,s0)] isseti:='prim':is(set,s0,t0) +*isset t0@[s:sigma][ses0:esti(s,s0)][n:not(esti(s,t0))] 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)) th4:=symnotis(set,s0,t0,th3):not(is(set,t0,s0)) s@nissetprop:=and(esti(s,s0),not(esti(s,t0))):'prop' [n:not(nissetprop(s0,t0,s))][e:esti(s,s0)] t1:=et(esti(s,t0),th3"l.and"(esti(s,s0),not(esti(s,t0)),n,e)):esti(s,t0) 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] t2:=th4"l.some"([x:sigma]nissetprop(s0,t0,x),m,s):not(nissetprop(s0,t0,s)) t3:=l:not(nissetprop(t0,s0,s)) 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) 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)) 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))) -isset sigma@[alpha:'type'][sa:[x:alpha]set] unmore:=setof([x:sigma]some(alpha,[y:alpha]esti(x,sa))):set [s:sigma][a:alpha][seasa:esti(s,sa)] eunmore1:=estii([x:sigma]some(alpha,[y:alpha]esti(x,sa)),s,somei(alpha,[y:alpha]esti(s,sa),a,seasa)):esti(s,unmore(sa)) s@[seun:esti(s,unmore(sa))][x:'prop'][x1:[y:alpha][z:esti(s,sa)]x] unmoreapp:=someapp(alpha,[y:alpha]esti(s,sa),estie([z:sigma]some(alpha,[y:alpha]esti(z,sa)),s,seun),x,x1):x +eq sigma@[r:[x:sigma][y:sigma]'prop'][refr1:[x:sigma]r][symr1:[x:sigma][y:sigma][t:r]r][trr1:[x:sigma][y:sigma][z:sigma][t:r][u:r]r][s:sigma] refr:=refr1:r [t:sigma][tsr:r] symr:=symr1:r t@[u:sigma][tsr:r][utr:r] trr:=trr1:r u@[sur:r][tur:r] tr1r:=trr(s,u,t,symr(u,s,sur),tur):r u@[usr:r][utr:r] tr2r:=trr(s,u,t,usr,symr(t,u,utr)):r s@ecelt:=setof(r):set +1 th1:=estii(r,s,refr):esti(s,ecelt(s)) t@[tsr:r] th2:=estii(r,t,tsr):esti(t,ecelt(s)) t@[e:esti(t,ecelt(s))] th3:=estie(r,t,e):r tsr@[u:sigma][e:esti(u,ecelt(s))] t1:=th2(t,u,tr1r(t,u,s,tsr,th3(u,e))):esti(u,ecelt(t)) 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)) t@[n:not(r)][u:sigma][e:esti(u,ecelt(s))] t2:=th3"l.imp"(esti(u,ecelt(t)),r,n,[x:esti(u,ecelt(t))]tr2r(s,t,u,th3(u,e),th3(t,u,x))):not(esti(u,ecelt(t))) n@th5:=[x:sigma][y:esti(x,ecelt(s))]t2(x,y):disj(ecelt(s),ecelt(t)) s@th6:=nonemptyi(ecelt(s),s,th1):nonempty(ecelt(s)) -1 trr1@[s0:set][s:sigma] ecp:=is(set,s0,ecelt(s)):'prop' s0@anec:=some([x:sigma]ecp(x)):'prop' +2 trr1@[s:sigma] th1:=somei([x:sigma]ecp(ecelt(s),x),s,refis(set,ecelt(s))):anec(ecelt(s)) -2 [ecs0:anec(s0)] +*2 ecs0@[s:sigma][ses0:esti(s,s0)][t:sigma][e:ecp(s0,t)] t1:=issete1(s0,ecelt(t),e,s,ses0):esti(s,ecelt(t)) t2:=th4"eq.1"(t,s,th3"eq.1"(t,s,t1)):is(set,ecelt(t),ecelt(s)) t3:=tris(set,s0,ecelt(t),ecelt(s),e,t2):is(set,s0,ecelt(s)) 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)) [t:sigma][tes0:esti(t,s0)] th3:=th3"eq.1"(s,t,issete1(s0,ecelt(s),th2,t,tes0)):r t@[tsr:r] th4:=issete2(s0,ecelt(s),th2,t,th2"eq.1"(s,t,tsr)):esti(t,s0) ecs0@[s:sigma][e:ecp(s0,s)] t4:=isp(set,[x:set]nonempty(x),ecelt(s),s0,th6"eq.1"(s),symis(set,s0,ecelt(s),e)):nonempty(s0) ecs0@th5:=someapp([x:sigma]ecp(x),ecs0,nonempty(s0),[x:sigma][y:ecp(x)]t4(x,y)):nonempty(s0) -2 +3 ecs0@[t0:set][ect0:anec(t0)][s:sigma][ses0:esti(s,s0)][t:sigma][tet0:esti(t,t0)][tsr:r] 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) tet0@[n:not(r)] 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)) th2:=isp1(set,[x:set]disj(s0,x),ecelt(t),t0,t1,th2"eq.2"(t0,ect0,t,tet0)):disj(s0,t0) t0@[i:is(set,s0,t0)][s:sigma][ses0:esti(s,s0)] t2:=issete1(s0,t0,i,s,ses0):esti(s,t0) t3:=th1"st.disj"(s0,t0,s,ses0,t2):not(disj(s0,t0)) 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)) -3 trr1@ect:=ot(set,[x:set]anec(x)):'type' ecs0@ectset:=out(set,[x:set]anec(x),s0,ecs0):ect trr1@[s:sigma] ectelt:=ectset(ecelt(s),th1".2"(s)):ect trr1@[e:ect] ecect:=in(set,[x:set]anec(x),e):set +4 th1:=inp(set,[x:set]anec(x),e):anec(ecect(e)) th2:=th5"eq.2"(ecect(e),th1):nonempty(ecect(e)) [x:'prop'][x1:[y:sigma][z:esti(y,ecect(e))]x] th3:=nonemptyapp(ecect(e),th2,x,x1):x s@th4:=isinout(set,[x:set]anec(x),ecelt(s),th1"eq.2"(s)):is(set,ecelt(s),ecect(ectelt(s))) th5:=issete1(ecelt(s),ecect(ectelt(s)),th4,s,th1"eq.1"(s)):esti(s,ecect(ectelt(s))) th6:=eunmore1(ect,[x:ect]ecect(x),s,ectelt(s),th5):esti(s,unmore(ect,[x:ect]ecect(x))) e@[s:sigma][see:esti(s,ecect(e))][t:sigma][tee:esti(t,ecect(e))] th7:=th3"eq.2"(ecect(e),th1,s,see,t,tee):r t@[tsr:r] th8:=th4"eq.2"(ecect(e),th1,s,see,t,tsr):esti(t,ecect(e)) -4 +5 [f:ect][i:is(ect,e,f)] th1:=isini(set,[x:set]anec(x),e,f,i):is(set,ecect(e),ecect(f)) f@[i:is(set,ecect(e),ecect(f))] th2:=isine(set,[x:set]anec(x),e,f,i):is(ect,e,f) f@[s:sigma][see:esti(s,ecect(e))][t:sigma][tef:esti(t,ecect(f))][tsr:r] 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) see@[i:is(ect,e,f)] th4:=issete1(ecect(e),ecect(f),th1(i),s,see):esti(s,ecect(f)) tef@[i:is(ect,e,f)] th5:=th3"eq.2"(ecect(f),th1"eq.4"(f),s,th4(i),t,tef):r trr1@[s:sigma][t:sigma][tsr:r] 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)) -5 trr1@[alpha:'type'][fu:[x:sigma]alpha] fixfu:=[x:sigma][y:sigma][z:r]is(alpha,fu,fu):'prop' +10 [ff:fixfu][e:ect][a1:alpha][s:sigma] prop1:=and(esti(s,ecect(e)),is(alpha,fu,a1)):'prop' a1@prop2:=some([x:sigma]prop1(x)):'prop' e@[s:sigma][see:esti(s,ecect(e))] t1:=andi(esti(s,ecect(e)),is(alpha,fu,fu),see,refis(alpha,fu)):prop1(fu,s) t2:=somei([x:sigma]prop1(fu,x),s,t1):prop2(fu) t3:=somei(alpha,[x:alpha]prop2(x),fu,t2):some(alpha,[x:alpha]prop2(x)) 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)) a1@[b1:alpha][pa1:prop2(a1)][pb1:prop2(b1)][s:sigma][pa1s:prop1(a1,s)][t:sigma][pb1t:prop1(b1,t)] t5:=ande1(esti(s,ecect(e)),is(alpha,fu,a1),pa1s):esti(s,ecect(e)) t6:=ande1(esti(t,ecect(e)),is(alpha,fu,b1),pb1t):esti(t,ecect(e)) t7:=th7"eq.4"(e,s,t5,t,t6):r t8:=ande2(esti(s,ecect(e)),is(alpha,fu,a1),pa1s):is(alpha,fu,a1) t9:=ande2(esti(t,ecect(e)),is(alpha,fu,b1),pb1t):is(alpha,fu,b1) t10:=tr3is(alpha,a1,fu,fu,b1,symis(alpha,fu,a1,t8),ff,t9):is(alpha,a1,b1) 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) 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) e@t13:=[x:alpha][y:alpha][u:prop2(x)][v:prop2(y)]t12(x,y,u,v):amone(alpha,[x:alpha]prop2(x)) t14:=onei(alpha,[x:alpha]prop2(x),t13,t4):one(alpha,[x:alpha]prop2(x)) -10 e".10"@indeq:=ind(alpha,[x:alpha]prop2".10"(x),t14".10"):alpha +*10 e@th1:=oneax(alpha,[x:alpha]prop2(x),t14):some([x:sigma]and(esti(x,ecect(e)),is(alpha,fu,indeq))) [s:sigma][see:esti(s,ecect(e))] th2:=t12(fu,indeq,t2(s,see),th1):is(alpha,fu,indeq) ff@[s:sigma] th3:=th2(ectelt(s),s,th5"eq.4"(s)):is(alpha,fu,indeq(ectelt(s))) -10 alpha@[fu2:[x:sigma][y:sigma]alpha] fixfu2:=[x:sigma][y:sigma][z:sigma][u:sigma][v:r][w:r]is(alpha,fu2,fu2):'prop' +11 [ff2:fixfu2][s:sigma][t:sigma][tsr:r][u:sigma] t1:=ff2:is(alpha,fu2,fu2) tsr@t2:=fisi(sigma,alpha,fu2,fu2,[x:sigma]t1(x)):is([x:sigma]alpha,fu2,fu2) ff2@[e:ect] i:=indeq([x:sigma]alpha,fu2,[x:sigma][y:sigma][z:r]t2(x,y,z),e):[x:sigma]alpha [s:sigma][t:sigma][tsr:r][u:sigma][uee:esti(u,ecect(e))] t3:=th2"eq.10"([x:sigma]alpha,fu2,[x:sigma][y:sigma][z:r]t2(x,y,z),e,u,uee):is([x:sigma]alpha,fu2,i) t4:=fise(alpha,fu2,i,t3,s):is(alpha,fu2,i) t5:=fise(alpha,fu2,i,t3,t):is(alpha,fu2,i) t6:=ff2:is(alpha,fu2,fu2) t7:=tr3is(alpha,i,fu2,fu2,i,symis(alpha,fu2,i,t4),t6,t5):is(alpha,i,i) tsr@t8:=th3"eq.4"(e,is(alpha,i,i),[x:sigma][y:esti(x,ecect(e))]t7(x,y)):is(alpha,i,i) -11 e".11"@[f:ect] indeq2:=indeq(i".11",[x:sigma][y:sigma][z:r]t8".11"(x,y,z),f):alpha +*11 f@[s:sigma][see:esti(s,ecect(e))][t:sigma][tef:esti(t,ecect(f))] t9:=th2"eq.10"(i,[x:sigma][y:sigma][z:r]t8(x,y,z),f,t,tef):is(alpha,i,indeq2(e,f)) t10:=th2"eq.10"([x:sigma]alpha,fu2,[x:sigma][y:sigma][z:r]t2(x,y,z),e,s,see):is([x:sigma]alpha,fu2,i) t11:=fise(alpha,fu2,i,t10,t):is(alpha,fu2,i) th1:=tris(alpha,fu2,i,indeq2,t11,t9):is(alpha,fu2,indeq2(e,f)) ff2@[s:sigma][t:sigma] th2:=th1(ectelt(s),ectelt(t),s,th5"eq.4"(s),t,th5"eq.4"(t)):is(alpha,fu2,indeq2(ectelt(s),ectelt(t))) -11 +landau +n @nat:='prim':'type' [x:nat][y:nat] is:=is"e"(nat,x,y):'prop' nis:=not(is(x,y)):'prop' x@[s:set(nat)] in:=esti(nat,x,s):'prop' @[p:[x:nat]'prop'] some:=some"l"(nat,p):'prop' all:=all"l"(nat,p):'prop' one:=one"e"(nat,p):'prop' @1:='prim':nat suc:='prim':[x:nat]nat [x:nat][y:nat][i:is(x,y)] ax2:=isf(nat,nat,suc,x,y,i):is(suc,suc) @ax3:='prim':[x:nat]nis(suc,1) ax4:='prim':[x:nat][y:nat][u:is(suc,suc)]is(x,y) [s:set(nat)] cond1:=in(1,s):'prop' cond2:=all([x:nat]imp(in(x,s),in(suc,s))):'prop' @ax5:='prim':[s:set(nat)][u:cond1(s)][v:cond2(s)][x:nat]in(x,s) [p:[x:nat]'prop'][1p:<1>p][xsp:[x:nat][y:p]<suc>p][x:nat] +i1 s:=setof(nat,p):set(nat) t1:=estii(nat,p,1,1p):cond1(s) [y:nat][yes:in(y,s)] t2:=estie(nat,p,y,yes):p t3:=estii(nat,p,suc,xsp):in(suc,s) x@t4:=<[y:nat][u:in(y,s)]t3(y,u)>ax5:in(x,s) -i1 induction:=estie(nat,p,x,t4".i1"):p @[x:nat][y:nat][n:nis(x,y)] +21 [i:is(suc,suc)] t1:=ax4:is(x,y) -21 satz1:=th3"l.imp"(is(suc,suc),is(x,y),n,[u:is(suc,suc)]t1".21"(u)):nis(suc,suc) +22 x@prop1:=nis(suc,x):'prop' @t1:=<1>ax3:prop1(1) x@[p:prop1(x)] t2:=satz1(suc,x,p):prop1(suc) -22 x@satz2:=induction([y:nat]prop1".22"(y),t1".22",[y:nat][u:prop1".22"(y)]t2".22"(y,u),x):nis(suc,x) +23 prop1:=or(is(x,1),some([u:nat]is(x,suc))):'prop' @t1:=ori1(is(1,1),some([u:nat]is(1,suc)),refis(nat,1)):prop1(1) x@t2:=somei(nat,[u:nat]is(suc,suc),x,refis(nat,suc)):some([u:nat]is(suc,suc)) t3:=ori2(is(suc,1),some([u:nat]is(suc,suc)),t2):prop1(suc) t4:=induction([y:nat]prop1(y),t1,[y:nat][u:prop1(y)]t3(y),x):prop1(x) -23 [n:nis(x,1)] satz3:=ore2(is(x,1),some([u:nat]is(x,suc)),t4".23",n):some([u:nat]is(x,suc)) y@[z:nat][i:is(x,suc)][j:is(x,suc)] +*23 j@t5:=suc,suc,x,i,j)>ax4:is(y,z) x@t6:=[y:nat][z:nat][u:is(x,suc)][v:is(x,suc)]t5(y,z,u,v):amone(nat,[u:nat]is(x,suc)) -23 n@satz3a:=onei(nat,[u:nat]is(x,suc),t6".23",satz3):one([u:nat]is(x,suc)) +24 x@[f:[y:nat]nat] prop1:=all([y:nat]is(<suc>f,<f>suc)):'prop' prop2:=and(is(<1>f,suc),prop1):'prop' x@[a:[y:nat]nat][b:[y:nat]nat][pa:prop2(a)][pb:prop2(b)][y:nat] prop3:=is(a,b):'prop' pb@t1:=ande1(is(<1>a,suc),prop1(a),pa):is(<1>a,suc) t2:=ande1(is(<1>b,suc),prop1(b),pb):is(<1>b,suc) t3:=tris2(nat,<1>a,<1>b,suc,t1,t2):prop3(1) y@[p:prop3(y)] t4:=ax2(a,b,p):is(<a>suc,<b>suc) t5:=ande2(is(<1>a,suc),prop1(a),pa):prop1(a) t6:=ande2(is(<1>b,suc),prop1(b),pb):prop1(b) t7:=t5:is(<suc>a,<a>suc) t8:=t6:is(<suc>b,<b>suc) t9:=tr3is(nat,<suc>a,<a>suc,<b>suc,<suc>b,t7,t4,symis"e"(nat,<suc>b,<b>suc,t8)):prop3(suc) y@t10:=induction([z:nat]prop3(z),t3,[z:nat][u:prop3(z)]t9(z,u),y):prop3(y) pb@t11:=fisi(nat,nat,a,b,[y:nat]t10(y)):is"e"([y:nat]nat,a,b) 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)) prop4:=some"l"([y:nat]nat,[z:[y:nat]nat]prop2(z)):'prop' @t12:=[x:nat]refis(nat,<suc>suc):prop1(1,suc) t13:=andi(is(<1>suc,<1>suc),prop1(1,suc),refis(nat,<1>suc),t12):prop2(1,suc) t14:=somei([y:nat]nat,[z:[y:nat]nat]prop2(1,z),suc,t13):prop4(1) x@[p:prop4(x)][f:[y:nat]nat][pf:prop2(f)] g:=[y:nat]<f>suc:[y:nat]nat [y:nat] t15:=refis(nat,g):is(g,<f>suc) pf@t16:=ande1(is(<1>f,suc),prop1(f),pf):is(<1>f,suc) t17:=tris(nat,<1>g,<<1>f>suc,<suc>suc,t15(1),ax2(<1>f,suc,t16)):is(<1>g,<suc>suc) y@t18:=ande2(is(<1>f,suc),prop1(f),pf):prop1(f) t19:=t18:is(<suc>f,<f>suc) t20:=tris2(nat,<suc>f,g,<f>suc,t19,t15):is(<suc>f,g) t21:=tris(nat,<suc>g,<<suc>f>suc,<g>suc,t15(suc),ax2(<suc>f,g,t20)):is(<suc>g,<g>suc) pf@t22:=[y:nat]t21(y):prop1(suc,g) t23:=andi(is(<1>g,<suc>suc),prop1(suc,g),t17,t22):prop2(suc,g) t24:=somei([y:nat]nat,[z:[y:nat]nat]prop2(suc,z),g,t23):prop4(suc) p@t25:=someapp([y:nat]nat,[z:[y:nat]nat]prop2(z),p,prop4(suc),[z:[y:nat]nat][u:prop2(z)]t24(z,u)):prop4(suc) x@bb:=induction([y:nat]prop4(y),t14,[y:nat][u:prop4(y)]t25(y,u),x):prop4(x) -24 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,suc),all([y:nat]is(<suc>z,<z>suc)))) plus:=ind([y:nat]nat,[z:[y:nat]nat]prop2".24"(z),satz4):[y:nat]nat y@pl:=plus:nat +*24 x@t26:=oneax([y:nat]nat,[z:[y:nat]nat]prop2(z),satz4):prop2(plus) -24 x@satz4a:=ande1(is(<1>plus,suc),prop1".24"(plus),t26".24"):is(pl(x,1),suc) +*24 x@t27:=ande2(is(<1>plus,suc),prop1(plus),t26):prop1(plus) -24 y@satz4b:=t27".24":is(pl(x,suc),suc) +*24 @t28:=t11(1,plus(1),suc,t26(1),t13):is"e"([y:nat]nat,plus(1),suc) -24 x@satz4c:=fise(nat,nat,plus(1),suc,t28".24",x):is(pl(1,x),suc) +*24 x@t29:=t11(suc,plus(suc),[y:nat]<plus>suc,t26(suc),t23(bb,plus,t26)):is"e"([y:nat]nat,plus(suc),[y:nat]<plus>suc) -24 y@satz4d:=fise(nat,nat,plus(suc),[z:nat]<plus>suc,t29".24",y):is(pl(suc,y),suc) x@satz4e:=symis(nat,pl(x,1),suc,satz4a):is(suc,pl(x,1)) y@satz4f:=symis(nat,pl(x,suc),suc,satz4b):is(suc,pl(x,suc)) x@satz4g:=symis(nat,pl(1,x),suc,satz4c):is(suc,pl(1,x)) y@satz4h:=symis(nat,pl(suc,y),suc,satz4d):is(suc,pl(suc,y)) z@[i:is(x,y)] ispl1:=isf(nat,nat,[u:nat]pl(u,z),x,y,i):is(pl(x,z),pl(y,z)) ispl2:=isf(nat,nat,[u:nat]pl(z,u),x,y,i):is(pl(z,x),pl(z,y)) z@[u:nat][i:is(x,y)][j:is(z,u)] 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)) +25 z@prop1:=is(pl(pl(x,y),z),pl(x,pl(y,z))):'prop' y@t1:=tr3is(nat,pl(pl(x,y),1),suc,pl(x,suc),pl(x,pl(y,1)),satz4a(pl(x,y)),satz4f,ispl2(suc,pl(y,1),x,satz4e(y))):prop1(1) z@[p:prop1(z)] t2:=ax2(pl(pl(x,y),z),pl(x,pl(y,z)),p):is(suc,suc) t3:=tr4is(nat,pl(pl(x,y),suc),suc,suc,pl(x,suc),pl(x,pl(y,suc)),satz4b(pl(x,y),z),t2,satz4f(x,pl(y,z)),ispl2(suc,pl(y,suc),x,satz4f(y,z))):prop1(suc) -25 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))) asspl1:=satz5:is(pl(pl(x,y),z),pl(x,pl(y,z))) 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)) +26 y@prop1:=is(pl(x,y),pl(y,x)):'prop' t1:=satz4a(y):is(pl(y,1),suc) t2:=satz4c(y):is(pl(1,y),suc) t3:=tris2(nat,pl(1,y),pl(y,1),suc,t2,t1):prop1(1,y) [p:prop1(x,y)] t4:=tris(nat,suc,suc,pl(y,suc),ax2(pl(x,y),pl(y,x),p),satz4f(y,x)):is(suc,pl(y,suc)) t5:=satz4d:is(pl(suc,y),suc) t6:=tris(nat,pl(suc,y),suc,pl(y,suc),t5,t4):prop1(suc,y) -26 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)) compl:=satz6:is(pl(x,y),pl(y,x)) +*26 x@t7:=tris(nat,pl(x,1),suc,pl(1,x),satz4a,satz4g):prop1(1) y@[p:prop1(y)] t8:=tr3is(nat,pl(x,suc),suc,suc,pl(suc,x),satz4b,ax2(pl(x,y),pl(y,x),p),satz4h(y,x)):prop1(suc) y@anders:=induction([z:nat]prop1(z),t7,[z:nat][u:prop1(z)]t8(z,u),y):is(pl(x,y),pl(y,x)) -26 +27 y@prop1:=nis(y,pl(x,y)):'prop' x@t1:=symnotis(nat,suc,1,ax3):nis(1,suc) t2:=th4"e.notis"(nat,1,suc,pl(x,1),t1,satz4a):prop1(1) y@[p:prop1(y)] t3:=satz1(y,pl(x,y),p):nis(suc,suc) t4:=th4"e.notis"(nat,suc,suc,pl(x,suc),t3,satz4b):prop1(suc) -27 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)) z@[n:nis(y,z)] +28 prop1:=nis(pl(x,y),pl(x,z)):'prop' t1:=satz1(y,z,n):nis(suc,suc) t2:=th5"e.notis"(nat,suc,suc,pl(1,y),pl(1,z),t1,satz4g(y),satz4g(z)):prop1(1,y,z,n) [p:prop1(x,y,z,n)] t3:=satz1(pl(x,y),pl(x,z),p):nis(suc,suc) t4:=th5"e.notis"(nat,suc,suc,pl(suc,y),pl(suc,z),t3,satz4h,satz4h(z)):prop1(suc,y,z,n) -28 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)) z@[i:is(pl(x,y),pl(x,z))] 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) z@diffprop:=is(x,pl(y,z)):'prop' +*28 y@[u:nat][v:nat][du:diffprop(u)][dv:diffprop(v)] t5:=satz8a(y,u,v,tris1(nat,pl(y,u),pl(y,v),x,du,dv)):is(u,v) -28 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))) +29 i:=is(x,y):'prop' ii:=some([u:nat]diffprop(x,y,u)):'prop' iii:=some([v:nat]diffprop(y,x,v)):'prop' [one1:i][u:nat] 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)) t2:=th3"e.notis"(nat,x,pl(u,x),pl(y,u),satz7(u,x),t1):nis(x,pl(y,u)) one1@t3:=th5"l.some"(nat,[u:nat]diffprop(u),[u:nat]t2(u)):not(ii) y@t4:=th1"l.ec"(i,ii,[z:i]t3(z)):ec(i,ii) one1@t5:=t3(y,x,symis(nat,x,y,one1)):not(iii) y@t6:=th2"l.ec"(iii,i,[z:i]t5(z)):ec(iii,i) [two1:ii][three1:iii][u:nat][du:diffprop(x,y,u)][v:nat][dv:diffprop(y,x,v)] 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)) t7:=mp(is(x,pl(pl(v,u),x)),con,t6a,satz7(pl(v,u),x)):con du@t8:=someapp(nat,[v:nat]diffprop(y,x,v),three1,con,[v:nat][dv:diffprop(y,x,v)]t7(v,dv)):con three1@t9:=someapp(nat,[u:nat]diffprop(u),two1,con,[u:nat][du:diffprop(u)]t8(u,du)):con two1@t10:=[z:iii]t9(z):not(iii) y@t11:=th1"l.ec"(ii,iii,[z:ii]t10(z)):ec(ii,iii) a:=th6"l.ec3"(i,ii,iii,t4,t11,t6):ec3(i,ii,iii) prop1:=or3(i,ii,iii):'prop' x@[j:is(x,1)] t12:=or3i1(i(1),ii(1),iii(1),j):prop1(1) x@[n:nis(x,1)][u:nat][j:is(x,suc)] t13:=tris(nat,x,suc,pl(1,u),j,satz4g(u)):is(x,pl(1,u)) t14:=somei(nat,[z:nat]diffprop(x,1,z),u,t13):ii(1) t15:=someapp(nat,[u1:nat]is(x,suc),satz3(x,n),ii(1),[u1:nat][z:is(x,suc)]t14(u1,z)):ii(1) t16:=or3i2(i(1),ii(1),iii(1),t15):prop1(1) n@t16a:=someapp(nat,[u:nat]is(x,suc),satz3(x,n),prop1(1),[u:nat][v:is(x,suc)]t16(u,v)):prop1(1) x@t17:=th1"l.imp"(is(x,1),prop1(1),[z:is(x,1)]t12(z),[z:nis(x,1)]t16a(z)):prop1(1) y@[p:prop1(y)][one1:i(y)] t18:=tris(nat,suc,pl(y,1),pl(x,1),satz4e(y),ispl1(y,x,1,symis(nat,x,y,one1))):is(suc,pl(x,1)) t19:=somei(nat,[z:nat]diffprop(suc,x,z),1,t18):iii(suc) t20:=or3i3(i(suc),ii(suc),iii(suc),t19):prop1(suc) p@[two1:ii(y)][u:nat][du:diffprop(x,y,u)][j:is(u,1)] t21:=tr3is(nat,x,pl(y,u),pl(y,1),suc,du,ispl2(u,1,y,j),satz4a(y)):is(x,suc) t22:=or3i1(i(suc),ii(suc),iii(suc),t21):prop1(suc) du@[n:nis(u,1)][w:nat][j:is(u,suc)] t23:=tris(nat,u,suc,pl(1,w),j,satz4g(w)):is(u,pl(1,w)) t24:=tr4is(nat,x,pl(y,u),pl(y,pl(1,w)),pl(pl(y,1),w),pl(suc,w),du,ispl2(u,pl(1,w),y,t23),asspl2(y,1,w),ispl1(pl(y,1),suc,w,satz4a(y))):is(x,pl(suc,w)) t25:=somei(nat,[z:nat]diffprop(x,suc,z),w,t24):ii(suc) n@t26:=someapp(nat,[z:nat]is(u,suc),satz3(u,n),ii(suc),[z:nat][v:is(u,suc)]t25(z,v)):ii(suc) t27:=or3i2(i(suc),ii(suc),iii(suc),t26):prop1(suc) du@t28:=th1"l.imp"(is(u,1),prop1(suc),[z:is(u,1)]t22(z),[z:nis(u,1)]t27(z)):prop1(suc) two1@t28a:=someapp(nat,[u:nat]diffprop(u),two1,prop1(suc),[u:nat][du:diffprop(u)]t28(u,du)):prop1(suc) p@[three1:iii(y)][v:nat][dv:diffprop(y,x,v)] t29:=tris(nat,suc,suc,pl(x,suc),ax2(y,pl(x,v),dv),satz4f(x,v)):is(suc,pl(x,suc)) t30:=somei(nat,[z:nat]diffprop(suc,x,z),suc,t29):iii(suc) three1@t31:=someapp(nat,[v:nat]diffprop(y,x,v),three1,iii(suc),[v:nat][dv:diffprop(y,x,v)]t30(v,dv)):iii(suc) t32:=or3i3(i(suc),ii(suc),iii(suc),t31):prop1(suc) p@t33:=or3app(i(y),ii(y),iii(y),prop1(suc),p,[z:i(y)]t20(z),[z:ii(y)]t28a(z),[z:iii(y)]t32(z)):prop1(suc) y@b:=induction([z:nat]prop1(z),t17,[z:nat][u:prop1(z)]t33(z,u),y):or3(i,ii,iii) -29 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)))) satz9a:=b".29":or3(is(x,y),some([u:nat]diffprop(x,y,u)),some([v:nat]diffprop(y,x,v))) satz9b:=a".29":ec3(is(x,y),some([u:nat]diffprop(x,y,u)),some([v:nat]diffprop(y,x,v))) more:=some([u:nat]diffprop(x,y,u)):'prop' less:=some([v:nat]diffprop(y,x,v)):'prop' satz10:=satz9:orec3(is(x,y),more(x,y),less(x,y)) satz10a:=satz9a:or3(is(x,y),more(x,y),less(x,y)) satz10b:=satz9b:ec3(is(x,y),more(x,y),less(x,y)) [m:more(x,y)] satz11:=m:less(y,x) y@[l:less(x,y)] satz12:=l:more(y,x) y@moreis:=or(more,is(x,y)):'prop' lessis:=or(less,is(x,y)):'prop' [m:moreis(x,y)] 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) y@[l:lessis(x,y)] 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) z@[i:is(x,y)][m:more(x,z)] ismore1:=isp(nat,[u:nat]more(u,z),x,y,m,i):more(y,z) i@[m:more(z,x)] ismore2:=isp(nat,[u:nat]more(z,u),x,y,m,i):more(z,y) i@[l:less(x,z)] isless1:=isp(nat,[u:nat]less(u,z),x,y,l,i):less(y,z) i@[l:less(z,x)] isless2:=isp(nat,[u:nat]less(z,u),x,y,l,i):less(z,y) i@[m:moreis(x,z)] ismoreis1:=isp(nat,[u:nat]moreis(u,z),x,y,m,i):moreis(y,z) i@[m:moreis(z,x)] ismoreis2:=isp(nat,[u:nat]moreis(z,u),x,y,m,i):moreis(z,y) i@[l:lessis(x,z)] islessis1:=isp(nat,[u:nat]lessis(u,z),x,y,l,i):lessis(y,z) i@[l:lessis(z,x)] islessis2:=isp(nat,[u:nat]lessis(z,u),x,y,l,i):lessis(z,y) y@[i:is(x,y)] moreisi2:=ori2(more(x,y),is(x,y),i):moreis(x,y) lessisi2:=ori2(less(x,y),is(x,y),i):lessis(x,y) y@[m:more(x,y)] moreisi1:=ori1(more(x,y),is(x,y),m):moreis(x,y) y@[l:less(x,y)] lessisi1:=ori1(less(x,y),is(x,y),l):lessis(x,y) z@[u:nat][i:is(x,y)][j:is(z,u)][m:more(x,z)] ismore12:=ismore2(z,u,y,j,ismore1(x,y,z,i,m)):more(y,u) j@[l:less(x,z)] isless12:=isless2(z,u,y,j,isless1(x,y,z,i,l)):less(y,u) j@[m:moreis(x,z)] ismoreis12:=ismoreis2(z,u,y,j,ismoreis1(x,y,z,i,m)):moreis(y,u) j@[l:lessis(x,z)] islessis12:=islessis2(z,u,y,j,islessis1(x,y,z,i,l)):lessis(y,u) y@[m:moreis(x,y)] 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)) y@[l:lessis(x,y)] satz10d:=th9"l.ec3"(is(x,y),more(x,y),less(x,y),satz10b,l):not(more(x,y)) y@[n:not(more(x,y))] satz10e:=th2"l.or3"(is(x,y),more(x,y),less(x,y),satz10a,n):lessis(x,y) y@[n:not(less(x,y))] 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) y@[m:more(x,y)] 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)) y@[l:less(x,y)] 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)) y@[n:not(moreis(x,y))] 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) y@[n:not(lessis(x,y))] 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) z@[l:less(x,y)][k:less(y,z)] +315 [v:nat][dv:diffprop(y,x,v)][w:nat][dw:diffprop(z,y,w)] 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))) t2:=somei(nat,[u:nat]diffprop(z,x,u),pl(v,w),t1):less(x,z) 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) -315 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) trless:=satz15:less(x,z) z@[m:more(x,y)][n:more(y,z)] trmore:=satz15(z,y,x,n,m):more(x,z) +*315 n@anders:=satz12(z,x,satz15(z,y,x,satz11(y,z,n),satz11(m))):more(x,z) -315 z@[l:lessis(x,y)][k:less(y,z)] 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) z@[l:less(x,y)][k:lessis(y,z)] 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) z@[m:moreis(x,y)][n:more(y,z)] satz16c:=satz16b(z,y,x,n,satz13(m)):more(x,z) z@[m:more(x,y)][n:moreis(y,z)] satz16d:=satz16a(z,y,x,satz13(y,z,n),m):more(x,z) z@[l:lessis(x,y)][k:lessis(y,z)] +317 [i:is(x,y)][j:is(y,z)] t1:=lessisi2(x,z,tris(nat,x,y,z,i,j)):lessis(x,z) i@[j:less(y,z)] t2:=lessisi1(x,z,satz16a(l,j)):lessis(x,z) 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) k@[j:less(x,y)] t4:=lessisi1(x,z,satz16b(j,k)):lessis(x,z) -317 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) +*317 k@[j:less(x,y)] t5:=lessisi1(x,z,satz16b(j,k)):lessis(x,z) k@[i:is(x,y)] t6:=islessis1(y,x,z,symis(nat,x,y,i),k):lessis(x,z) 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) -317 k@trlessis:=satz17:lessis(x,z) z@[m:moreis(x,y)][n:moreis(y,z)] trmoreis:=satz14(z,x,satz17(z,y,x,satz13(y,z,n),satz13(m))):moreis(x,z) y@satz18:=somei(nat,[u:nat]diffprop(pl(x,y),x,u),y,refis(nat,pl(x,y))):more(pl(x,y),x) satz18a:=satz18:less(x,pl(x,y)) x@satz18b:=ismore1(pl(x,1),suc,x,satz4a,satz18(1)):more(suc,x) satz18c:=satz18b:less(x,suc) z@[m:more(x,y)] +319 [u:nat][du:diffprop(u)] t1:=tris(nat,x,pl(y,u),pl(u,y),du,compl(y,u)):is(x,pl(u,y)) 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)) t3:=somei(nat,[v:nat]diffprop(pl(x,z),pl(y,z),v),u,t2):more(pl(x,z),pl(y,z)) -319 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)) z@[i:is(x,y)] satz19b:=ispl1(x,y,z,i):is(pl(x,z),pl(y,z)) z@[l:less(x,y)] satz19c:=satz11(pl(y,z),pl(x,z),satz19a(y,x,z,satz12(x,y,l))):less(pl(x,z),pl(y,z)) +*319 l@anders1:=satz19a(y,x,z,l):less(pl(x,z),pl(y,z)) -319 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)) i@satz19e:=ispl2(x,y,z,i):is(pl(z,x),pl(z,y)) 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)) +*319 l@anders2:=satz19d(y,x,z,l):less(pl(z,x),pl(z,y)) -319 z@[u:nat][i:is(x,y)][m:more(z,u)] 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)) 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)) i@[l:less(z,u)] 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)) 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)) z@[m:moreis(x,y)] +*319 m@[n:more(x,y)] t4:=moreisi1(pl(x,z),pl(y,z),satz19a(n)):moreis(pl(x,z),pl(y,z)) m@[i:is(x,y)] t5:=moreisi2(pl(x,z),pl(y,z),ispl1(x,y,z,i)):moreis(pl(x,z),pl(y,z)) -319 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)) 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)) z@[l:lessis(x,y)] satz19n:=satz13(pl(y,z),pl(x,z),satz19l(y,x,z,satz14(l))):lessis(pl(x,z),pl(y,z)) satz19o:=satz13(pl(z,y),pl(z,x),satz19m(y,x,z,satz14(l))):lessis(pl(z,x),pl(z,y)) +320 z@t1:=satz10a(x,y):or3(is(x,y),more(x,y),less(x,y)) 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))) -320 z@[m:more(pl(x,z),pl(y,z))] 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) z@[i:is(pl(x,z),pl(y,z))] 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) z@[l:less(pl(x,z),pl(y,z))] 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) +*320 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)) andersb:=satz8a(z,x,y,t3):is(x,y) l@andersc:=satz20a(y,x,z,l):less(x,y) -320 z@[m:more(pl(z,x),pl(z,y))] 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) z@[i:is(pl(z,x),pl(z,y))] 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) z@[l:less(pl(z,x),pl(z,y))] 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) u@[m:more(x,y)][n:more(z,u)] +321 t1:=satz19a(x,y,z,m):more(pl(x,z),pl(y,z)) 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)) -321 satz21:=trmore(pl(x,z),pl(y,z),pl(y,u),t1".321",t2".321"):more(pl(x,z),pl(y,u)) +*321 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)) -321 u@[l:less(x,y)][k:less(z,u)] satz21a:=satz21(y,x,u,z,l,k):less(pl(x,z),pl(y,u)) +*321 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)) -321 u@[m:moreis(x,y)][n:more(z,u)] 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)) u@[m:more(x,y)][n:moreis(z,u)] 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)) u@[l:lessis(x,y)][k:less(z,u)] satz22c:=satz22a(y,x,u,z,satz14(x,y,l),k):less(pl(x,z),pl(y,u)) u@[l:less(x,y)][k:lessis(z,u)] satz22d:=satz22b(y,x,u,z,l,satz14(z,u,k)):less(pl(x,z),pl(y,u)) u@[m:moreis(x,y)][n:moreis(z,u)] +323 [i:is(x,y)][j:is(z,u)] 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)) i@[o:more(z,u)] t2:=moreisi1(pl(x,z),pl(y,u),satz22a(m,o)):moreis(pl(x,z),pl(y,u)) 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)) n@[o:more(x,y)] t4:=moreisi1(pl(x,z),pl(y,u),satz22b(o,n)):moreis(pl(x,z),pl(y,u)) -323 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)) +*323 n@[o:more(x,y)] t5:=moreisi1(pl(x,z),pl(y,u),satz22b(o,n)):moreis(pl(x,z),pl(y,u)) n@[i:is(x,y)] 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)) 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)) -323 u@[l:lessis(x,y)][k:lessis(z,u)] 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)) +324 x@[n:nis(x,1)][u:nat][i:is(x,suc)] t1:=tris(nat,x,suc,pl(1,u),i,satz4g(u)):is(x,pl(1,u)) t2:=ismore1(pl(1,u),x,1,symis(nat,x,pl(1,u),t1),satz18(1,u)):more(x,1) n@t3:=someapp(nat,[u:nat]is(x,suc),satz3(x,n),more(x,1),[u:nat][v:is(x,suc)]t2(u,v)):more(x,1) -324 x@satz24:=th2"l.or"(more(x,1),is(x,1),[u:nis(x,1)]t3".324"(u)):moreis(x,1) satz24a:=satz13(x,1,satz24):lessis(1,x) satz24b:=t3".324"(suc,ax3):more(suc,1) satz24c:=satz24b:less(1,suc) y@[m:more(y,x)] +325 [u:nat][du:diffprop(y,x,u)] t1:=satz19m(u,1,x,satz24(u)):moreis(pl(x,u),pl(x,1)) t2:=ismoreis1(pl(x,u),y,pl(x,1),symis(nat,y,pl(x,u),du),t1):moreis(y,pl(x,1)) -325 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)) satz25a:=ismoreis2(pl(x,1),suc,y,satz4a,satz25):moreis(y,suc) y@[l:less(y,x)] satz25b:=satz13(x,pl(y,1),satz25(y,x,l)):lessis(pl(y,1),x) satz25c:=islessis1(pl(y,1),suc,x,satz4a(y),satz25b):lessis(suc,x) y@[l:less(y,pl(x,1))] +326 [m:more(y,x)] t1:=satz25(m):moreis(y,pl(x,1)) 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)) -326 satz26:=satz10e(y,x,t2".326"):lessis(y,x) y@[l:less(y,suc)] satz26a:=satz26(isless2(suc,pl(x,1),y,satz4e,l)):lessis(y,x) y@[m:more(pl(y,1),x)] satz26b:=satz14(x,y,satz26(y,x,m)):moreis(y,x) y@[m:more(suc,x)] satz26c:=satz26b(ismore1(suc,pl(y,1),x,satz4e(y),m)):moreis(y,x) @[p:[x:nat]'prop'][n:nat] +327 [m:nat] lbprop:=imp(p,lessis(n,m)):'prop' -327 lb:=all([x:nat]lbprop".327"(x)):'prop' min:=and(lb,p):'prop' p@[s:some(p)] +*327 s@[n:nat] t1:=[x:p]satz24a(n):lbprop(1,n) s@t2:=[x:nat]t1(x):lb(1) [l:[x:nat]lb(x)][y:nat][yp:p] t3:=satz18(y,1):more(pl(y,1),y) t4:=satz10g(pl(y,1),y,t3):not(lessis(pl(y,1),y)) t5:=th4"l.imp"(p,lessis(pl(y,1),y),yp,t4):not(lbprop(pl(y,1),y)) t6:=th1"l.all"(nat,[x:nat]lbprop(pl(y,1),x),y,t5):not(lb(pl(y,1))) t7:=mp(lb(pl(y,1)),con,l,t6):con l@t8:=someapp(nat,p,s,con,[x:nat][y:p]t7(x,y)):con s@[n:non(nat,[x:nat]and(lb(x),not(lb(pl(x,1)))))][m:nat][l:lb(m)] t9:=n:not(and(lb(m),not(lb(pl(m,1))))) t10:=et(lb(pl(m,1)),th3"l.and"(lb(m),not(lb(pl(m,1))),t9,l)):lb(pl(m,1)) t11:=isp(nat,[x:nat]lb(x),pl(m,1),suc,t10,satz4a(m)):lb(suc) n@t12:=[x:nat]induction([y:nat]lb(y),t2,[y:nat][z:lb(y)]t11(y,z),x):[x:nat]lb(x) 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))))) [m:nat][a:and(lb(m),not(lb(pl(m,1))))] t14:=ande1(lb(m),not(lb(pl(m,1))),a):lb(m) t15:=ande2(lb(m),not(lb(pl(m,1))),a):not(lb(pl(m,1))) [nmp:not(p)][n:nat][np:p] t16:=mp(p,lessis(m,n),np,t14):lessis(m,n) t17:=th3"l.imp"(is(m,n),p,nmp,[x:is(m,n)]isp(nat,p,n,m,np,symis(nat,m,n,x))):not(is(m,n)) t18:=ore1(less(m,n),is(m,n),t16,t17):less(m,n) t19:=satz25b(n,m,t18):lessis(pl(m,1),n) nmp@t20:=[x:nat][y:p]t19(x,y):lb(pl(m,1)) t21:=mp(lb(pl(m,1)),con,t20,t15):con a@t22:=et(p,[x:not(p)]t21(x)):p t23:=andi(lb(m),p,t14,t22):min(m) -327 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)) +*327 p@[n:non(nat,[x:nat]min(x))][u:nat] t24:=[x:p]satz24a(u):lbprop(1,u) n@t25:=[x:nat]t24(x):lb(1) u@[l:lb(u)] t26:=n:not(min(u)) t27:=th3"l.and"(lb(u),p,t26,l):not(p) [v:nat][vp:p] t28:=th3"l.imp"(is(u,v),p,t27,[x:is(u,v)]isp1(nat,p,v,u,vp,x)):nis(u,v) t29:=mp(p,lessis(u,v),vp,l):lessis(u,v) t30:=ore1(less(u,v),is(u,v),t29,t28):less(u,v) t31:=satz25c(v,u,t30):lessis(suc,v) v@t32:=[x:p]t31(x):lbprop(suc,v) l@t33:=[x:nat]t32(x):lb(suc) u@t34:=induction([x:nat]lb(x),t25,[x:nat][y:lb(x)]t33(x,y),u):lb(u) p@[s:some(p)][u:nat][up:p] t35:=satz10g(suc,u,satz18b(u)):not(lessis(suc,u)) t36:=th4"l.imp"(p,lessis(suc,u),up,t35):not(lbprop(suc,u)) t37:=th1"l.all"(nat,[x:nat]lbprop(suc,x),u,t36):not(lb(suc)) t38:=[y:non(nat,[x:nat]min(x))]mp(lb(suc),con,t34(y,suc),t37):some([x:nat]min(x)) s@anders:=someapp(nat,p,s,some([x:nat]min(x)),[x:nat][y:p]t38(x,y)):some([x:nat]min(x)) -327 +*327 p@[n:nat][m:nat][mn:min(p,n)][mm:min(p,m)] t39:=ande1(lb(n),p,mn):lb(n) t40:=ande1(lb(m),p,mm):lb(m) t41:=ande2(lb(n),p,mn):p t42:=ande2(lb(m),p,mm):p t43:=t39:lbprop(n,m) t44:=t40:lbprop(m,n) t45:=mp(p,lessis(n,m),t42,t43):lessis(n,m) t46:=mp(p,lessis(m,n),t41,t44):lessis(m,n) t47:=ore2(more(n,m),is(n,m),satz14(m,n,t46),satz10d(n,m,t45)):is(n,m) p@t48:=[x:nat][y:nat][u:min(x)][v:min(y)]t47(x,y,u,v):amone(nat,[x:nat]min(p,x)) -327 s@satz27a:=onei(nat,[x:nat]min(p,x),t48".327",satz27):one([x:nat]min(p,x)) +428 x@[f:[y:nat]nat] prop1:=all([y:nat]is(<suc>f,pl(f,x))):'prop' prop2:=and(is(<1>f,x),prop1):'prop' x@[a:[y:nat]nat][b:[y:nat]nat][pa:prop2(a)][pb:prop2(b)][y:nat] prop3:=is(a,b):'prop' pb@t1:=ande1(is(<1>a,x),prop1(a),pa):is(<1>a,x) t2:=ande1(is(<1>b,x),prop1(b),pb):is(<1>b,x) t3:=tris2(nat,<1>a,<1>b,x,t1,t2):prop3(1) y@[p:prop3(y)] t4:=ispl1(a,b,x,p):is(pl(a,x),pl(b,x)) t5:=ande2(is(<1>a,x),prop1(a),pa):prop1(a) t6:=ande2(is(<1>b,x),prop1(b),pb):prop1(b) t7:=t5:is(<suc>a,pl(a,x)) t8:=t6:is(<suc>b,pl(b,x)) t9:=tr3is(nat,<suc>a,pl(a,x),pl(b,x),<suc>b,t7,t4,symis(nat,<suc>b,pl(b,x),t8)):prop3(suc) y@t10:=induction([z:nat]prop3(z),t3,[z:nat][u:prop3(z)]t9(z,u),y):prop3(y) pb@t11:=fisi(nat,nat,a,b,[y:nat]t10(y)):is"e"([y:nat]nat,a,b) 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)) prop4:=some"l"([y:nat]nat,[z:[y:nat]nat]prop2(z)):'prop' @id:=[y:nat]y:[y:nat]nat t12:=[x:nat]satz4e(x):prop1(1,id) t13:=andi(is(<1>id,1),prop1(1,id),refis(nat,1),t12):prop2(1,id) t14:=somei([y:nat]nat,[z:[y:nat]nat]prop2(1,z),id,t13):prop4(1) x@[p:prop4(x)][f:[y:nat]nat][pf:prop2(f)] g:=[y:nat]pl(f,y):[y:nat]nat t15:=ande1(is(<1>f,x),prop1(f),pf):is(<1>f,x) t16:=tris(nat,<1>g,pl(x,1),suc,ispl1(<1>f,x,1,t15),satz4a(x)):is(<1>g,suc) [y:nat] t17:=ande2(is(<1>f,x),prop1(f),pf):prop1(f) t18:=t17:is(<suc>f,pl(f,x)) t19:=tris(nat,<suc>g,pl(pl(f,x),suc),pl(f,pl(x,suc)),ispl1(<suc>f,pl(f,x),suc,t18),asspl1(f,x,suc)):is(<suc>g,pl(f,pl(x,suc))) t20:=tr3is(nat,pl(x,suc),suc,pl(suc,y),pl(y,suc),satz4b(x,y),satz4h(x,y),compl(suc,y)):is(pl(x,suc),pl(y,suc)) t21:=tr3is(nat,<suc>g,pl(f,pl(x,suc)),pl(f,pl(y,suc)),pl(g,suc),t19,ispl2(pl(x,suc),pl(y,suc),f,t20),asspl2(f,y,suc)):is(<suc>g,pl(g,suc)) pf@t22:=[y:nat]t21(y):prop1(suc,g) t23:=andi(is(<1>g,suc),prop1(suc,g),t16,t22):prop2(suc,g) t24:=somei([y:nat]nat,[z:[y:nat]nat]prop2(suc,z),g,t23):prop4(suc) p@t25:=someapp([y:nat]nat,[z:[y:nat]nat]prop2(z),p,prop4(suc),[z:[y:nat]nat][u:prop2(z)]t24(z,u)):prop4(suc) x@b1:=induction([y:nat]prop4(y),t14,[y:nat][u:prop4(y)]t25(y,u),x):prop4(x) -428 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(<suc>z,pl(z,x))))) times:=ind([y:nat]nat,[z:[y:nat]nat]prop2".428"(z),satz28):[y:nat]nat y@ts:=times:nat +*428 x@t26:=oneax([y:nat]nat,[z:[y:nat]nat]prop2(z),satz28):prop2(times) -428 x@satz28a:=ande1(is(<1>times,x),prop1".428"(times),t26".428"):is(ts(x,1),x) +*428 x@t27:=ande2(is(<1>times,x),prop1(times),t26):prop1(times) -428 y@satz28b:=t27".428":is(ts(x,suc),pl(ts(x,y),x)) +*428 @t28:=t11(1,times(1),id,t26(1),t13):is"e"([y:nat]nat,times(1),id) -428 x@satz28c:=fise(nat,nat,times(1),id".428",t28".428",x):is(ts(1,x),x) +*428 x@t29:=t11(suc,times(suc),[y:nat]pl(times,y),t26(suc),t23(b1,times,t26)):is"e"([y:nat]nat,times(suc),[y:nat]pl(times,y)) -428 y@satz28d:=fise(nat,nat,times(suc),[z:nat]pl(times,z),t29".428",y):is(ts(suc,y),pl(ts(x,y),y)) x@satz28e:=symis(nat,ts(x,1),x,satz28a):is(x,ts(x,1)) y@satz28f:=symis(nat,ts(x,suc),pl(ts(x,y),x),satz28b):is(pl(ts(x,y),x),ts(x,suc)) x@satz28g:=symis(nat,ts(1,x),x,satz28c):is(x,ts(1,x)) y@satz28h:=symis(nat,ts(suc,y),pl(ts(x,y),y),satz28d):is(pl(ts(x,y),y),ts(suc,y)) z@[i:is(x,y)] ists1:=isf(nat,nat,[u:nat]ts(u,z),x,y,i):is(ts(x,z),ts(y,z)) ists2:=isf(nat,nat,[u:nat]ts(z,u),x,y,i):is(ts(z,x),ts(z,y)) z@[u:nat][i:is(x,y)][j:is(z,u)] 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)) +429 y@prop1:=is(ts(x,y),ts(y,x)):'prop' t1:=satz28a(y):is(ts(y,1),y) t2:=satz28c(y):is(ts(1,y),y) t3:=tris2(nat,ts(1,y),ts(y,1),y,t2,t1):prop1(1,y) [p:prop1(x,y)] t4:=tris(nat,pl(ts(x,y),y),pl(ts(y,x),y),ts(y,suc),ispl1(ts(x,y),ts(y,x),y,p),satz28f(y,x)):is(pl(ts(x,y),y),ts(y,suc)) t5:=satz28d:is(ts(suc,y),pl(ts(x,y),y)) t6:=tris(nat,ts(suc,y),pl(ts(x,y),y),ts(y,suc),t5,t4):prop1(suc,y) -429 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)) comts:=satz29:is(ts(x,y),ts(y,x)) +*429 x@t7:=tris(nat,ts(x,1),x,ts(1,x),satz28a,satz28g):prop1(1) y@[p:prop1(y)] t8:=tr3is(nat,ts(x,suc),pl(ts(x,y),x),pl(ts(y,x),x),ts(suc,x),satz28b(x,y),ispl1(ts(x,y),ts(y,x),x,p),satz28h(y,x)):prop1(suc) y@anders:=induction([z:nat]prop1(z),t7,[z:nat][u:prop1(z)]t8(z,u),y):is(ts(x,y),ts(y,x)) -429 +430 z@prop1:=is(ts(x,pl(y,z)),pl(ts(x,y),ts(x,z))):'prop' y@t1:=tr3is(nat,ts(x,pl(y,1)),ts(x,suc),pl(ts(x,y),x),pl(ts(x,y),ts(x,1)),ists2(pl(y,1),suc,x,satz4a(y)),satz28b,ispl2(x,ts(x,1),ts(x,y),satz28e(x))):prop1(1) z@[p:prop1(z)] t2:=tr3is(nat,ts(x,pl(y,suc)),ts(x,suc),pl(ts(x,pl(y,z)),x),pl(pl(ts(x,y),ts(x,z)),x),ists2(pl(y,suc),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,suc)),pl(pl(ts(x,y),ts(x,z)),x)) t3:=tr3is(nat,ts(x,pl(y,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,suc)),t2,asspl1(ts(x,y),ts(x,z),x),ispl2(pl(ts(x,z),x),ts(x,suc),ts(x,y),satz28f(x,z))):prop1(suc) -430 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))) 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))) disttp2:=satz30:is(ts(x,pl(y,z)),pl(ts(x,y),ts(x,z))) 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)) 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))) +431 prop1:=is(ts(ts(x,y),z),ts(x,ts(y,z))):'prop' 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) z@[p:prop1(z)] t2:=tr4is(nat,ts(ts(x,y),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,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,suc),x,satz28f(y,z))):prop1(suc) -431 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))) assts1:=satz31:is(ts(ts(x,y),z),ts(x,ts(y,z))) 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)) [m:more(x,y)] +432 [u:nat][du:diffprop(u)] 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))) t2:=somei(nat,[v:nat]diffprop(ts(x,z),ts(y,z),v),ts(u,z),t1):more(ts(x,z),ts(y,z)) -432 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)) z@[i:is(x,y)] satz32b:=ists1(x,y,z,i):is(ts(x,z),ts(y,z)) z@[l:less(x,y)] satz32c:=satz11(ts(y,z),ts(x,z),satz32a(y,x,z,satz12(x,y,l))):less(ts(x,z),ts(y,z)) +*432 l@anders1:=satz32a(y,x,z,l):less(ts(x,z),ts(y,z)) -432 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)) i@satz32e:=ists2(x,y,z,i):is(ts(z,x),ts(z,y)) 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)) +*432 l@anders2:=satz32d(y,x,z,l):less(ts(z,x),ts(z,y)) -432 z@[u:nat][i:is(x,y)][m:more(z,u)] 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)) 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)) i@[l:less(z,u)] 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)) 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)) z@[m:moreis(x,y)] +*432 m@[n:more(x,y)] t3:=moreisi1(ts(x,z),ts(y,z),satz32a(n)):moreis(ts(x,z),ts(y,z)) m@[i:is(x,y)] t4:=moreisi2(ts(x,z),ts(y,z),ists1(x,y,z,i)):moreis(ts(x,z),ts(y,z)) -432 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)) 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)) z@[l:lessis(x,y)] satz32n:=satz13(ts(y,z),ts(x,z),satz32l(y,x,z,satz14(l))):lessis(ts(x,z),ts(y,z)) satz32o:=satz13(ts(z,y),ts(z,x),satz32m(y,x,z,satz14(l))):lessis(ts(z,x),ts(z,y)) +433 z@t1:=satz10a(x,y):or3(is(x,y),more(x,y),less(x,y)) 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))) -433 z@[m:more(ts(x,z),ts(y,z))] 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) z@[i:is(ts(x,z),ts(y,z))] 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) z@[l:less(ts(x,z),ts(y,z))] 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) +*433 l@anders:=satz33a(y,x,z,l):less(x,y) -433 u@[m:more(x,y)][n:more(z,u)] +434 t1:=satz32a(x,y,z,m):more(ts(x,z),ts(y,z)) 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)) -434 satz34:=trmore(ts(x,z),ts(y,z),ts(y,u),t1".434",t2".434"):more(ts(x,z),ts(y,u)) +*434 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)) -434 u@[l:less(x,y)][k:less(z,u)] satz34a:=satz34(y,x,u,z,l,k):less(ts(x,z),ts(y,u)) +*434 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)) -434 u@[m:moreis(x,y)][n:more(z,u)] 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)) u@[m:more(x,y)][n:moreis(z,u)] 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)) u@[l:lessis(x,y)][k:less(z,u)] satz35c:=satz35a(y,x,u,z,satz14(x,y,l),k):less(ts(x,z),ts(y,u)) u@[l:less(x,y)][k:lessis(z,u)] satz35d:=satz35b(y,x,u,z,l,satz14(z,u,k)):less(ts(x,z),ts(y,u)) u@[m:moreis(x,y)][n:moreis(z,u)] +436 [i:is(x,y)][j:is(z,u)] 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)) i@[o:more(z,u)] t2:=moreisi1(ts(x,z),ts(y,u),satz35a(m,o)):moreis(ts(x,z),ts(y,u)) 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)) n@[o:more(x,y)] t4:=moreisi1(ts(x,z),ts(y,u),satz35b(o,n)):moreis(ts(x,z),ts(y,u)) -436 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)) +*436 n@[o:more(x,y)] t5:=moreisi1(ts(x,z),ts(y,u),satz35b(o,n)):moreis(ts(x,z),ts(y,u)) n@[i:is(x,y)] 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)) 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)) -436 u@[l:lessis(x,y)][k:lessis(z,u)] 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)) y@[m:more(x,y)] +mn t1:=onei(nat,[z:nat]diffprop(x,y,z),satz8b(x,y),m):one([z:nat]diffprop(x,y,z)) -mn mn:=ind(nat,[z:nat]diffprop(x,y,z),t1".mn"):nat +*mn m@th1a:=oneax(nat,[z:nat]diffprop(x,y,z),t1):is(x,pl(y,mn(x,y,m))) th1b:=symis(nat,x,pl(y,mn(x,y,m)),th1a):is(pl(y,mn(x,y,m)),x) 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)) th1d:=symis(nat,x,pl(mn(x,y,m),y),th1c):is(pl(mn(x,y,m),y),x) y@[z:nat][m:more(x,y)][i:is(pl(y,z),x)] th1e:=satz8b(x,y):is(z,mn(x,y,m)) -mn z@[u:nat][m:more(x,z)][n:more(y,u)][i:is(x,y)][j:is(z,u)] +*mn 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) -mn j@ismn12:=th1e".mn"(y,u,mn(x,z,m),n,t2".mn"):is(mn(x,z,m),mn(y,u,n)) @[n:nat] 1to:=ot(nat,[x:nat]lessis(x,n)):'type' [x:nat][l:lessis(x,n)] outn:=out(nat,[y:nat]lessis(y,n),x,l):1to(n) n@[xn:1to(n)] inn:=in"e"(nat,[y:nat]lessis(y,n),xn):nat 1top:=inp(nat,[y:nat]lessis(y,n),xn):lessis(inn,n) l@[y:nat][k:lessis(y,n)][i:is(x,y)] isoutni:=isouti(nat,[z:nat]lessis(z,n),x,l,y,k,i):is"e"(1to(n),outn(x,l),outn(y,k)) k@[i:is"e"(1to(n),outn(x,l),outn(y,k))] isoutne:=isoute(nat,[z:nat]lessis(z,n),x,l,y,k,i):is(x,y) xn@[yn:1to(n)][i:is"e"(1to(n),xn,yn)] isinni:=isini(nat,[z:nat]lessis(z,n),xn,yn,i):is(inn(xn),inn(yn)) yn@[i:is(inn(xn),inn(yn))] isinne:=isine(nat,[z:nat]lessis(z,n),xn,yn,i):is"e"(1to(n),xn,yn) xn@isoutinn:=isoutin(nat,[y:nat]lessis(y,n),xn):is"e"(1to(n),xn,outn(inn(xn),1top(xn))) l@isinoutn:=isinout(nat,[y:nat]lessis(y,n),x,l):is(x,inn(outn(x,l))) @1o:=outn(1,1,lessisi2(1,1,refis(nat,1))):1to(1) [u:1to(1)] +singlet u0:=inn(1,u):nat t1:=1top(1,u):lessis(u0,1) t2:=ore2(more(u0,1),is(u0,1),satz24(u0),satz10d(u0,1,t1)):is(u0,1) 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) -singlet @2:=pl(1,1):nat [x:nat] +pair [l:lessis(x,2)][n:nis(x,2)] t1:=satz26(1,x,ore1(less(x,2),is(x,2),l,n)):lessis(x,1) t2:=ore2(more(x,1),is(x,1),satz24(x),satz10d(x,1,t1)):is(x,1) l@th1:=th2"l.or"(is(x,1),is(x,2),[t:nis(x,2)]t2(t)):or(is(x,1),is(x,2)) @th2:=th1"e.notis"(nat,<1>suc,1,2,<1>ax3,satz4e(1)):nis(2,1) -pair @1t:=outn(2,1,satz24a(2)):1to(2) 2t:=outn(2,2,lessisi2(2,2,refis(nat,2))):1to(2) +*pair @[u:1to(2)] u0:=inn(2,u):nat t3:=1top(2,u):lessis(u0,2) [i:is(u0,1)] t4:=isoutni(2,u0,t3,1,satz24a(2),i):is"e"(1to(2),outn(2,u0,t3),1t) t5:=tris(1to(2),u,outn(2,u0,t3),1t,isoutinn(2,u),t4):is"e"(1to(2),u,1t) u@[i:is(u0,2)] t6:=isoutni(2,u0,t3,2,lessisi2(2,2,refis(nat,2)),i):is"e"(1to(2),outn(2,u0,t3),2t) t7:=tris(1to(2),u,outn(2,u0,t3),2t,isoutinn(2,u),t6):is"e"(1to(2),u,2t) 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)) @[i:is"e"(1to(2),2t,1t)] t9:=isini(nat,[x:nat]lessis(x,2),2t,1t,i):is(u0(2t),u0(1t)) 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) @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)) -pair @[alpha:'type'] pair1type:=[x:1to(2)]alpha:'type' [a:alpha][b:alpha] pair1:=[x:1to(2)]ite(is"e"(1to(2),x,1t),alpha,a,b):pair1type alpha@[p:pair1type] first1:=<1t>p:alpha second1:=<2t>p:alpha b@first1is1:=itet(is"e"(1to(2),1t,1t),alpha,a,b,refis(1to(2),1t)):is"e"(alpha,first1(pair1),a) first1is2:=symis(alpha,first1(pair1),a,first1is1):is"e"(alpha,a,first1(pair1)) second1is1:=itef(is"e"(1to(2),2t,1t),alpha,a,b,th4".pair"):is"e"(alpha,second1(pair1),b) second1is2:=symis(alpha,second1(pair1),b,second1is1):is"e"(alpha,b,second1(pair1)) +*pair 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)] t11:=isf(1to(2),alpha,p,u,1t,u1):is"e"(alpha,p,first1(p)) t12:=symis(alpha,q,first1(q),isf(1to(2),alpha,q,u,1t,u1)):is"e"(alpha,first1(q),q) t13:=tr3is(alpha,p,first1(p),first1(q),q,t11,i,t12):is"e"(alpha,p,q) u@[u2:is"e"(1to(2),u,2t)] t14:=isf(1to(2),alpha,p,u,2t,u2):is"e"(alpha,p,second1(p)) t15:=symis(alpha,q,second1(q),isf(1to(2),alpha,q,u,2t,u2)):is"e"(alpha,second1(q),q) t16:=tr3is(alpha,p,second1(p),second1(q),q,t14,j,t15):is"e"(alpha,p,q) u@t17:=orapp(is"e"(1to(2),u,1t),is"e"(1to(2),u,2t),is"e"(alpha,p,q),th3(u),[t:is"e"(1to(2),u,1t)]t13(t),[t:is"e"(1to(2),u,2t)]t16(t)):is"e"(alpha,p,q) j@th5:=fisi(1to(2),alpha,p,q,[t:1to(2)]t17(t)):is"e"(pair1type,p,q) p@q0:=pair1(first1,second1):pair1type t18:=first1is1(first1(p),second1):is"e"(alpha,first1(q0),first1(p)) t19:=second1is1(first1,second1):is"e"(alpha,second1(q0),second1(p)) -pair p@pair1is1:=th5".pair"(q0".pair",p,t18".pair",t19".pair"):is"e"(pair1type,pair1(first1,second1),p) pair1is2:=symis(pair1type,pair1(first1,second1),p,pair1is1):is"e"(pair1type,p,pair1(first1,second1)) @[x:nat] lessisi3:=lessisi2(x,x,refis(nat,x)):lessis(x,x) 1out:=outn(x,1,satz24a(x)):1to(x) xout:=outn(x,x,lessisi3(x)):1to(x) [y:nat][l:lessis(y,x)][u:1to(y)] +left ui:=inn(y,u):nat t1:=1top(y,u):lessis(ui,y) t2:=trlessis(ui,y,x,t1,l):lessis(ui,x) -left left1to:=outn(x,ui".left",t2".left"):1to(x) [v:1to(y)][i:is"e"(1to(x),left1to(u),left1to(v))] +*left i@t3:=isoutne(x,ui,t2,ui(v),t2(v),i):is(ui,ui(v)) -left i@thleft1:=isinne(y,u,v,t3".left"):is"e"(1to(y),u,v) 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)) y@[u:1to(y)] +right ui:=inn(y,u):nat t4:=1top(y,u):lessis(ui,y) t5:=satz19o(ui,y,x,t4):lessis(pl(x,ui),pl(x,y)) -right right1to:=outn(pl(x,y),pl(x,ui".right"),t5".right"):1to(pl(x,y)) [v:1to(y)][i:is"e"(1to(pl(x,y)),right1to(u),right1to(v))] +*right 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))) t7:=satz20e(ui(u),ui(v),x,t6):is(ui(u),ui(v)) -right i@thright1:=isinne(y,u,v,t7".right"):is"e"(1to(y),u,v) @[alpha:'type'][x:nat][y:nat][l:lessis(y,x)][f:[t:1to(x)]alpha] left:=[t:1to(y)]f:[t:1to(y)]alpha y@[f:[t:1to(pl(x,y))]alpha] right:=[t:1to(y)]f:[t:1to(y)]alpha y@[i:is(y,x)][f:[t:1to(y)]alpha] +*left f@t4:=lessisi2(y,x,i):lessis(y,x) t5:=lessisi2(x,y,symis(nat,y,x,i)):lessis(x,y) f1:=left(y,x,t5,f):[t:1to(x)]alpha f2:=left(t4,f1):[t:1to(y)]alpha [u:1to(y)] 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))) 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))) t8:=isf(1to(y),alpha,f,u,left1to(y,x,t5,left1to(x,y,t4,u)),t7):is"e"(alpha,f,f2) -left 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))) @frac:=pair1type(nat):'type' [x1:nat][x2:nat] fr:=pair1(nat,x1,x2):frac @[x:frac] num:=first1(nat,x):nat den:=second1(nat,x):nat x2@numis:=first1is1(nat,x1,x2):is(num(fr(x1,x2)),x1) isnum:=first1is2(nat,x1,x2):is(x1,num(fr(x1,x2))) denis:=second1is1(nat,x1,x2):is(den(fr(x1,x2)),x2) isden:=second1is2(nat,x1,x2):is(x2,den(fr(x1,x2))) x@1x:=num(x):nat 2x:=den(x):nat fris:=pair1is1(nat,x):is"e"(frac,fr(1x,2x),x) isfr:=pair1is2(nat,x):is"e"(frac,x,fr(1x,2x)) x2@[y1:nat][y2:nat] 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)))) 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)) x@[n1:nat][n2:nat] 1disnd:=ists1(n1,num(fr(n1,n2)),2x,isnum(n1,n2)):is(ts(n1,2x),ts(num(fr(n1,n2)),2x)) ndis1d:=symis(nat,ts(n1,2x),ts(num(fr(n1,n2)),2x),1disnd):is(ts(num(fr(n1,n2)),2x),ts(n1,2x)) n2isnd:=ists2(n2,den(fr(n1,n2)),1x,isden(n1,n2)):is(ts(1x,n2),ts(1x,den(fr(n1,n2)))) ndisn2:=symis(nat,ts(1x,n2),ts(1x,den(fr(n1,n2))),n2isnd):is(ts(1x,den(fr(n1,n2))),ts(1x,n2)) x2@[n:nat][i:is(x1,n)] isn:=isf(nat,frac,[t:nat]fr(t,x2),x1,n,i):is"e"(frac,fr(x1,x2),fr(n,x2)) n@[i:is(x2,n)] isd:=isf(nat,frac,[t:nat]fr(x1,t),x2,n,i):is"e"(frac,fr(x1,x2),fr(x1,n)) y2@[i:is(x1,y1)][j:is(x2,y2)] 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)) x@[y:frac] 1y:=num(y):nat 2y:=den(y):nat eq:=is(ts(1x,2y),ts(1y,2x)):'prop' y2@[i:is(ts(x1,y2),ts(y1,x2))] 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)) n2@[i:is(ts(1x,n2),ts(n1,2x))] 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)) n2@[i:is(ts(n1,2x),ts(1x,n2))] 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) x@satz37:=refis(nat,ts(1x,2x)):eq(x,x) refeq:=satz37:eq(x,x) y@[i:is"e"(frac,x,y)] refeq1:=isp(frac,[t:frac]eq(x,t),x,y,refeq,i):eq(x,y) refeq2:=isp(frac,[t:frac]eq(t,x),x,y,refeq,i):eq(y,x) y2@[i:is(x1,y1)][j:is(x2,y2)] eqnd:=refeq1(fr(x1,x2),fr(y1,y2),isnd(i,j)):eq(fr(x1,x2),fr(y1,y2)) x2@[n:nat][i:is(x1,n)] eqn:=refeq1(fr(x1,x2),fr(n,x2),isn(n,i)):eq(fr(x1,x2),fr(n,x2)) n@[i:is(x2,n)] eqd:=refeq1(fr(x1,x2),fr(x1,n),isd(n,i)):eq(fr(x1,x2),fr(x1,n)) y@[e:eq(x,y)] satz38:=symis(nat,ts(1x,2y),ts(1y,2x),e):eq(y,x) symeq:=satz38:eq(y,x) @[a:nat][b:nat][c:nat][d:nat] +ii1 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))) -ii1 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))) +*ii1 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))) 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))) -ii1 y@[z:frac] 1z:=num(z):nat 2z:=den(z):nat [e:eq(x,y)][f:eq(y,z)] +139 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))) t2:=stets(1x,2y,1y,2z):is(ts(ts(1x,2y),ts(1y,2z)),ts(ts(1x,2z),ts(1y,2y))) 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))) 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))) -139 satz39:=satz33b(ts(1x,2z),ts(1z,2x),ts(1y,2y),t4".139"):eq(x,z) +*139 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))) -139 f@treq:=satz39:eq(x,z) z@[e:eq(z,x)][f:eq(z,y)] treq1:=treq(x,z,y,symeq(z,x,e),f):eq(x,y) z@[e:eq(x,z)][f:eq(y,z)] treq2:=treq(x,z,y,e,symeq(y,z,f)):eq(x,y) z@[u:frac][e:eq(x,y)][f:eq(y,z)][g:eq(z,u)] tr3eq:=treq(x,y,u,e,treq(y,z,u,f,g)):eq(x,u) u@[v:frac][e:eq(x,y)][f:eq(y,z)][g:eq(z,u)][h:eq(u,v)] tr4eq:=tr3eq(x,y,z,v,e,f,treq(z,u,v,g,h)):eq(x,v) x@[n:nat] 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))) satz40a:=symeq(x,fr(ts(1x,n),ts(2x,n)),satz40):eq(fr(ts(1x,n),ts(2x,n)),x) x2@[n:nat] 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))) satz40c:=symeq(fr(x1,x2),fr(ts(x1,n),ts(x2,n)),satz40b):eq(fr(ts(x1,n),ts(x2,n)),fr(x1,x2)) y@moref:=more(ts(1x,2y),ts(1y,2x)):'prop' lessf:=less(ts(1x,2y),ts(1y,2x)):'prop' y2@[m:more(ts(x1,y2),ts(y1,x2))] 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)) y2@[l:less(ts(x1,y2),ts(y1,x2))] 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)) n2@[m:more(ts(1x,n2),ts(n1,2x))] 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)) n2@[m:more(ts(n1,2x),ts(1x,n2))] 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) n2@[l:less(ts(1x,n2),ts(n1,2x))] 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)) n2@[l:less(ts(n1,2x),ts(1x,n2))] 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) y@satz41:=satz10(ts(1x,2y),ts(1y,2x)):orec3(eq(x,y),moref(x,y),lessf(x,y)) satz41a:=satz10a(ts(1x,2y),ts(1y,2x)):or3(eq(x,y),moref(x,y),lessf(x,y)) satz41b:=satz10b(ts(1x,2y),ts(1y,2x)):ec3(eq(x,y),moref(x,y),lessf(x,y)) [m:moref(x,y)] satz42:=satz11(ts(1x,2y),ts(1y,2x),m):lessf(y,x) y@[l:lessf(x,y)] satz43:=satz12(ts(1x,2y),ts(1y,2x),l):moref(y,x) u@1u:=num(u):nat 2u:=den(u):nat [m:moref(x,y)][e:eq(x,z)][f:eq(y,u)] +244 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))) 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))) 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))) -244 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) u@[e:eq(x,y)][f:eq(z,u)][m:moref(x,z)] eqmoref12:=satz44(x,z,y,u,m,e,f):moref(y,u) z@[e:eq(x,y)][m:moref(x,z)] eqmoref1:=satz44(x,z,y,z,m,e,refeq(z)):moref(y,z) e@[m:moref(z,x)] eqmoref2:=satz44(z,x,z,y,m,refeq(z),e):moref(z,y) u@[l:lessf(x,y)][e:eq(x,z)][f:eq(y,u)] satz45:=satz42(u,z,satz44(y,x,u,z,satz43(x,y,l),f,e)):lessf(z,u) u@[e:eq(x,y)][f:eq(z,u)][l:lessf(x,z)] eqlessf12:=satz45(x,z,y,u,l,e,f):lessf(y,u) z@[e:eq(x,y)][l:lessf(x,z)] eqlessf1:=satz45(x,z,y,z,l,e,refeq(z)):lessf(y,z) e@[l:lessf(z,x)] eqlessf2:=satz45(z,x,z,y,l,refeq(z),e):lessf(z,y) y@moreq:=or(moref(x,y),eq(x,y)):'prop' lesseq:=or(lessf(x,y),eq(x,y)):'prop' [e:eq(x,y)] moreqi2:=ori2(moref(x,y),eq(x,y),e):moreq(x,y) lesseqi2:=ori2(lessf(x,y),eq(x,y),e):lesseq(x,y) y@[m:moref(x,y)] moreqi1:=ori1(moref(x,y),eq(x,y),m):moreq(x,y) y@[l:lessf(x,y)] lesseqi1:=ori1(lessf(x,y),eq(x,y),l):lesseq(x,y) y@[m:moreq(x,y)] 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)) y@[l:lesseq(x,y)] satz41d:=th9"l.ec3"(eq(x,y),moref(x,y),lessf(x,y),satz41b,l):not(moref(x,y)) y@[n:not(moref(x,y))] satz41e:=th2"l.or3"(eq(x,y),moref(x,y),lessf(x,y),satz41a,n):lesseq(x,y) y@[n:not(lessf(x,y))] 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) y@[m:moref(x,y)] 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)) y@[l:lessf(x,y)] 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)) y@[n:not(moreq(x,y))] 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) y@[n:not(lesseq(x,y))] 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) u@[m:moreq(x,y)][e:eq(x,z)][f:eq(y,u)] +246 [n:moref(x,y)] t1:=ori1(moref(z,u),eq(z,u),satz44(n,e,f)):moreq(z,u) f@[g:eq(x,y)] t2:=ori2(moref(z,u),eq(z,u),tr3eq(z,x,y,u,symeq(x,z,e),g,f)):moreq(z,u) -246 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) u@[e:eq(x,y)][f:eq(z,u)][m:moreq(x,z)] eqmoreq12:=satz46(x,z,y,u,m,e,f):moreq(y,u) z@[e:eq(x,y)][m:moreq(x,z)] eqmoreq1:=satz46(x,z,y,z,m,e,refeq(z)):moreq(y,z) e@[m:moreq(z,x)] eqmoreq2:=satz46(z,x,z,y,m,refeq(z),e):moreq(z,y) u@[l:lesseq(x,y)][e:eq(x,z)][f:eq(y,u)] +247 [k:lessf(x,y)] t1:=ori1(lessf(z,u),eq(z,u),satz45(k,e,f)):lesseq(z,u) f@[g:eq(x,y)] t2:=ori2(lessf(z,u),eq(z,u),tr3eq(z,x,y,u,symeq(x,z,e),g,f)):lesseq(z,u) -247 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) u@[e:eq(x,y)][f:eq(z,u)][l:lesseq(x,z)] eqlesseq12:=satz47(x,z,y,u,l,e,f):lesseq(y,u) z@[e:eq(x,y)][l:lesseq(x,z)] eqlesseq1:=satz47(x,z,y,z,l,e,refeq(z)):lesseq(y,z) e@[l:lesseq(z,x)] eqlesseq2:=satz47(z,x,z,y,l,refeq(z),e):lesseq(z,y) y@[m:moreq(x,y)] 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) y@[l:lesseq(x,y)] 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) z@[l:lessf(x,y)][k:lessf(y,z)] +250 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))) 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))) -250 satz50:=satz33c(ts(1x,2z),ts(1z,2x),ts(1y,2y),t2".250"):lessf(x,z) trlessf:=satz50:lessf(x,z) z@[m:moref(x,y)][n:moref(y,z)] trmoref:=satz43(z,x,satz50(z,y,x,satz42(y,z,n),satz42(x,y,m))):moref(x,z) z@[l:lesseq(x,y)][k:lessf(y,z)] 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) z@[l:lessf(x,y)][k:lesseq(y,z)] 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) z@[m:moreq(x,y)][n:moref(y,z)] satz51c:=satz43(z,x,satz51b(z,y,x,satz42(y,z,n),satz48(x,y,m))):moref(x,z) z@[m:moref(x,y)][n:moreq(y,z)] satz51d:=satz43(z,x,satz51a(z,y,x,satz48(y,z,n),satz42(x,y,m))):moref(x,z) z@[l:lesseq(x,y)][k:lesseq(y,z)] +252 [e:eq(x,y)][f:eq(y,z)] t1:=ori2(lessf(x,z),eq(x,z),treq(x,y,z,e,f)):lesseq(x,z) e@[j:lessf(y,z)] t2:=ori1(lessf(x,z),eq(x,z),satz51a(l,j)):lesseq(x,z) 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) k@[j:lessf(x,y)] t4:=ori1(lessf(x,z),eq(x,z),satz51b(j,k)):lesseq(x,z) -252 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) trlesseq:=satz52:lesseq(x,z) +*252 k@[j:lessf(x,y)] t5:=ori1(lessf(x,z),eq(x,z),satz51b(j,k)):lesseq(x,z) k@[e:eq(x,y)] t6:=eqlesseq1(y,x,z,symeq(x,y,e),k):lesseq(x,z) 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) -252 z@[m:moreq(x,y)][n:moreq(y,z)] trmoreq:=satz49(z,x,satz52(z,y,x,satz48(y,z,n),satz48(x,y,m))):moreq(x,z) +253 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)) t2:=morefi2(pl(1x,1x),2x,t1):moref(fr(pl(1x,1x),2x),x) -253 x@satz53:=somei(frac,[t:frac]moref(t,x),fr(pl(1x,1x),2x),t2".253"):some"l"(frac,[t:frac]moref(t,x)) +254 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))) t2:=lessfi2(1x,pl(2x,2x),t1):lessf(fr(1x,pl(2x,2x)),x) -254 satz54:=somei(frac,[t:frac]lessf(t,x),fr(1x,pl(2x,2x)),t2".254"):some"l"(frac,[t:frac]lessf(t,x)) y@[l:lessf(x,y)] +255 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))) 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))) 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)) t4:=lessfi1(pl(1x,1y),pl(2x,2y),t3):lessf(x,fr(pl(1x,1y),pl(2x,2y))) 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))) t6:=lessfi2(y,pl(1x,1y),pl(2x,2y),t5):lessf(fr(pl(1x,1y),pl(2x,2y)),y) 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)) -255 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))) y@pf:=fr(pl(ts(1x,2y),ts(1y,2x)),ts(2x,2y)):frac +ii3 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))) 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)) -ii3 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))) +*ii3 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))) t4:=ists2(den(fr(n1,n2)),n2,2x,denis(n1,n2)):is(ts(2x,den(fr(n1,n2))),ts(2x,n2)) -ii3 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))) +*ii3 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))) t6:=ists1(den(fr(n1,n2)),n2,2x,denis(n1,n2)):is(ts(den(fr(n1,n2)),2x),ts(n2,2x)) -ii3 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))) 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))) 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))) 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))) 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))) 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))) 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)) u@[e:eq(x,y)][f:eq(z,u)] +356 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))) t2:=t1(z,u,x,y,f,e):is(ts(ts(1z,2u),ts(2x,2y)),ts(ts(1u,2z),ts(2x,2y))) 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))) 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))) 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))) 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)))) 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))) -356 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)) eqpf12:=satz56:eq(pf(x,z),pf(y,u)) z@[e:eq(x,y)] eqpf1:=eqpf12(x,y,z,z,e,refeq(z)):eq(pf(x,z),pf(y,z)) eqpf2:=eqpf12(z,z,x,y,refeq(z),e):eq(pf(z,x),pf(z,y)) x2@[n:nat] 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)) 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))) 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)) compf:=satz58:eq(pf(x,y),pf(y,x)) +359 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)) 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))) 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))) 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)) 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))) 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))) 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))) -359 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))) asspf1:=satz59:eq(pf(pf(x,y),z),pf(x,pf(y,z))) asspf2:=symeq(pf(pf(x,y),z),pf(x,pf(y,z)),asspf1):eq(pf(x,pf(y,z)),pf(pf(x,y),z)) 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)) +*359 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))) 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)) 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))) -359 +360 y@t1:=satz18(ts(1x,2y),ts(1y,2x)):more(pl(ts(1x,2y),ts(1y,2x)),ts(1x,2y)) 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)) 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))) 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))) -360 y@satz60:=morefi2(pl(ts(1x,2y),ts(1y,2x)),ts(2x,2y),t4".360"):moref(pf(x,y),x) satz60a:=satz42(pf(x,y),x,satz60):lessf(x,pf(x,y)) z@[m:moref(x,y)] +361 t1:=satz32a(ts(1x,2y),ts(1y,2x),2z,m):more(ts(ts(1x,2y),2z),ts(ts(1y,2x),2z)) 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)) t3:=stets1(1z,2x,2y):is(ts(ts(1z,2x),2y),ts(ts(1z,2y),2x)) 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))) 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)) 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)) 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))) -361 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)) z@[m:moref(x,y)] satz62a:=satz61(m):moref(pf(x,z),pf(y,z)) z@[e:eq(x,y)] satz62b:=eqpf1(x,y,z,e):eq(pf(x,z),pf(y,z)) z@[l:lessf(x,y)] satz62c:=satz42(pf(y,z),pf(x,z),satz61(y,x,z,satz43(l))):lessf(pf(x,z),pf(y,z)) 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)) e@satz62e:=eqpf2(x,y,z,e):eq(pf(z,x),pf(z,y)) 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)) u@[e:eq(x,y)][m:moref(z,u)] 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)) 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)) e@[l:lessf(z,u)] 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)) 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)) +363 z@t1:=satz41a(x,y):or3(eq(x,y),moref(x,y),lessf(x,y)) 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))) -363 z@[m:moref(pf(x,z),pf(y,z))] 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) z@[e:eq(pf(x,z),pf(y,z))] 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) z@[l:lessf(pf(x,z),pf(y,z))] 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) z@[m:moref(pf(z,x),pf(z,y))] 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) z@[e:eq(pf(z,x),pf(z,y))] 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) z@[f:lessf(pf(z,x),pf(z,y))] 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) u@[m:moref(x,y)][n:moref(z,u)] +364 t1:=satz61(x,y,z,m):moref(pf(x,z),pf(y,z)) 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)) -364 satz64:=trmoref(pf(x,z),pf(y,z),pf(y,u),t1".364",t2".364"):moref(pf(x,z),pf(y,u)) u@[l:lessf(x,y)][k:lessf(z,u)] 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)) u@[m:moreq(x,y)][n:moref(z,u)] 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)) u@[m:moref(x,y)][n:moreq(z,u)] 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)) u@[l:lesseq(x,y)][k:lessf(z,u)] 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)) u@[l:lessf(x,y)][k:lesseq(z,u)] 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)) u@[m:moreq(x,y)][n:moreq(z,u)] +366 [e:eq(x,y)][f:eq(z,u)] t1:=moreqi2(pf(x,z),pf(y,u),satz56(e,f)):moreq(pf(x,z),pf(y,u)) e@[o:moref(z,u)] t2:=moreqi1(pf(x,z),pf(y,u),satz65a(m,o)):moreq(pf(x,z),pf(y,u)) 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)) n@[o:moref(x,y)] t4:=moreqi1(pf(x,z),pf(y,u),satz65b(o,n)):moreq(pf(x,z),pf(y,u)) -366 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)) u@[l:lesseq(x,y)][k:lesseq(z,u)] 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)) y@[l:lesseq(x,y)] +367 [v:frac][e:eq(pf(y,v),x)] t1:=eqmoref1(pf(y,v),x,y,e,satz60(y,v)):moref(x,y) 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)) -367 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))) y@[v:frac][w:frac][e:eq(pf(y,v),x)][f:eq(pf(y,w),x)] satz67b:=satz63e(v,w,y,treq2(pf(y,v),pf(y,w),x,e,f)):eq(v,w) y@[m:moref(x,y)] +*367 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)) vo:=ind(nat,[t:nat]diffprop(ts(1x,2y),ts(1y,2x),t),t3):nat t4:=oneax(nat,[t:nat]diffprop(ts(1x,2y),ts(1y,2x),t),t3):is(ts(1x,2y),pl(ts(1y,2x),vo)) w:=fr(vo,ts(2x,2y)):frac 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))) 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) -367 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)) mf:=w".367":frac satz67c:=t6".367":eq(pf(y,mf(x,y,m)),x) satz67d:=symeq(pf(y,mf(x,y,m)),x,satz67c):eq(x,pf(y,mf(x,y,m))) y@[v:frac][m:moref(x,y)][e:eq(pf(y,v),x)] satz67e:=satz67b(v,mf(x,y,m),e,satz67c(m)):eq(v,mf(x,y,m)) y@tf:=fr(ts(1x,1y),ts(2x,2y)):frac +ii4 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)) 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)) -ii4 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))) +*ii4 n2@t3:=ists2(num(fr(n1,n2)),n1,1x,numis(n1,n2)):is(ts(1x,num(fr(n1,n2))),ts(1x,n1)) t4:=ists2(den(fr(n1,n2)),n2,2x,denis(n1,n2)):is(ts(2x,den(fr(n1,n2))),ts(2x,n2)) -ii4 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))) +*ii4 n2@t5:=ists1(num(fr(n1,n2)),n1,1x,numis(n1,n2)):is(ts(num(fr(n1,n2)),1x),ts(n1,1x)) t6:=ists1(den(fr(n1,n2)),n2,2x,denis(n1,n2)):is(ts(den(fr(n1,n2)),2x),ts(n2,2x)) -ii4 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))) 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))) 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))) 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))) 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))) 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))) 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)) u@[e:eq(x,y)][f:eq(z,u)] +468 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))) -468 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))) +*468 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))) -468 f@satz68:=eqi12(ts(1x,1z),ts(2x,2z),ts(1y,1u),ts(2y,2u),t2".468"):eq(tf(x,z),tf(y,u)) eqtf12:=satz68:eq(tf(x,z),tf(y,u)) z@[e:eq(x,y)] eqtf1:=eqtf12(x,y,z,z,e,refeq(z)):eq(tf(x,z),tf(y,z)) eqtf2:=eqtf12(z,z,x,y,refeq(z),e):eq(tf(z,x),tf(z,y)) 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)) comtf:=satz69:eq(tf(x,y),tf(y,x)) 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))) asstf1:=satz70:eq(tf(tf(x,y),z),tf(x,tf(y,z))) asstf2:=symeq(tf(tf(x,y),z),tf(x,tf(y,z)),asstf1):eq(tf(x,tf(y,z)),tf(tf(x,y),z)) +471 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))))) 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)) 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)) -471 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))) 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))) disttpf2:=satz71:eq(tf(x,pf(y,z)),pf(tf(x,y),tf(x,z))) 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)) 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))) [m:moref(x,y)] +472 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))) 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))) -472 satz72a:=morefi12(ts(1x,1z),ts(2x,2z),ts(1y,1z),ts(2y,2z),t2".472"):moref(tf(x,z),tf(y,z)) z@[e:eq(x,y)] satz72b:=satz68(x,y,z,z,e,refeq(z)):eq(tf(x,z),tf(y,z)) z@[l:lessf(x,y)] satz72c:=satz42(tf(y,z),tf(x,z),satz72a(y,x,z,satz43(x,y,l))):lessf(tf(x,z),tf(y,z)) 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)) e@satz72e:=eqtf2(x,y,z,e):eq(tf(z,x),tf(z,y)) 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)) u@[e:eq(x,y)][m:moref(z,u)] 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)) 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)) e@[l:lessf(z,u)] 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)) 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)) +473 z@t1:=satz41a(x,y):or3(eq(x,y),moref(x,y),lessf(x,y)) 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))) -473 z@[m:moref(tf(x,z),tf(y,z))] 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) z@[e:eq(tf(x,z),tf(y,z))] 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) z@[l:lessf(tf(x,z),tf(y,z))] 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) z@[m:moref(tf(z,x),tf(z,y))] 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) z@[e:eq(tf(z,x),tf(z,y))] 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) z@[l:lessf(tf(z,x),tf(z,y))] 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) u@[m:moref(x,y)][n:moref(z,u)] +474 t1:=satz72a(x,y,z,m):moref(tf(x,z),tf(y,z)) 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)) -474 satz74:=trmoref(tf(x,z),tf(y,z),tf(y,u),t1".474",t2".474"):moref(tf(x,z),tf(y,u)) u@[l:lessf(x,y)][k:lessf(z,u)] 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)) u@[m:moreq(x,y)][n:moref(z,u)] 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)) u@[m:moref(x,y)][n:moreq(z,u)] 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)) u@[l:lesseq(x,y)][k:lessf(z,u)] 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)) u@[l:lessf(x,y)][k:lesseq(z,u)] 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)) u@[m:moreq(x,y)][n:moreq(z,u)] +476 [e:eq(x,y)][f:eq(z,u)] t1:=moreqi2(tf(x,z),tf(y,u),satz68(e,f)):moreq(tf(x,z),tf(y,u)) e@[o:moref(z,u)] t2:=moreqi1(tf(x,z),tf(y,u),satz75a(m,o)):moreq(tf(x,z),tf(y,u)) 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)) n@[o:moref(x,y)] t4:=moreqi1(tf(x,z),tf(y,u),satz75b(o,n)):moreq(tf(x,z),tf(y,u)) -476 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)) u@[l:lesseq(x,y)][k:lesseq(z,u)] 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)) y@[v:frac][w:frac][e:eq(tf(y,v),x)][f:eq(tf(y,w),x)] satz77b:=satz73e(v,w,y,treq2(tf(y,v),tf(y,w),x,e,f)):eq(v,w) +477 y@v:=fr(ts(1x,2y),ts(2x,1y)):frac 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) -477 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)) +rt @eq:=[x:frac][y:frac]eq"n"(x,y):[x:frac][y:frac]'prop' refeq:=[x:frac]refeq"n"(x):[x:frac]eq symeq:=[x:frac][y:frac][t:eq]symeq"n"(x,y,t):[x:frac][y:frac][t:eq]eq treq:=[x:frac][y:frac][z:frac][t:eq][u:eq]treq"n"(x,y,z,t,u):[x:frac][y:frac][z:frac][t:eq][u:eq]eq [x:frac][s:set(frac)] inf:=esti(frac,x,s):'prop' @rat:=ect"eq"(frac,eq,refeq,symeq,treq):'type' [x0:rat][y0:rat] is:=is"e"(rat,x0,y0):'prop' nis:=not(is(x0,y0)):'prop' @[p:[x:rat]'prop'] some:=some"l"(rat,p):'prop' all:=all"l"(rat,p):'prop' one:=one"e"(rat,p):'prop' x0@[s:set(rat)] in:=esti(rat,x0,s):'prop' x@ratof:=ectelt"eq"(frac,eq,refeq,symeq,treq,x):rat x0@class:=ecect"eq"(frac,eq,refeq,symeq,treq,x0):set(frac) x@inclass:=th5"eq.4"(frac,eq,refeq,symeq,treq,x):inf(x,class(ratof(x))) x0@[x:frac][y:frac][xix0:inf(x,class(x0))][e:eq"n"(x,y)] lemmaeq1:=th8"eq.4"(frac,eq,refeq,symeq,treq,x0,x,xix0,y,e):inf(y,class(x0)) x0@[a:'prop'][a1:[x:frac][xi:inf(x,class(x0))]a] ratapp1:=th3"eq.4"(frac,eq,refeq,symeq,treq,x0,a,a1):a 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))] +ii5 t1:=ratapp1(y0,a,[y:frac][yi:inf(y,class(y0))]a1):a -ii5 a1@ratapp2:=ratapp1(x0,a,[x:frac][xi:inf(x,class(x0))]t1".ii5"(x,xi)):a 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))] +*ii5 xix0@t2:=ratapp2(y0,z0,a,[y:frac][z:frac][yi:inf(y,class(y0))][zi:inf(z,class(z0))]a1):a -ii5 a1@ratapp3:=ratapp1(x0,a,[x:frac][xi:inf(x,class(x0))]t2".ii5"(x,xi)):a 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))] +*ii5 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))]a1):a -ii5 a1@ratapp4:=ratapp1(x0,a,[x:frac][xi:inf(x,class(x0))]t3".ii5"(x,xi)):a y0@[x1:frac][y1:frac][x1ix0:inf(x1,class(x0))][y1iy0:inf(y1,class(y0))][e:eq"n"(x1,y1)] isi:=th3"eq.5"(frac,eq,refeq,symeq,treq,x0,y0,x1,x1ix0,y1,y1iy0,e):is(x0,y0) y1iy0@[i:is(x0,y0)] ise:=th5"eq.5"(frac,eq,refeq,symeq,treq,x0,y0,x1,x1ix0,y1,y1iy0,i):eq"n"(x1,y1) y1iy0@[n:not(eq"n"(x1,y1))] nisi:=th3"l.imp"(is(x0,y0),eq"n"(x1,y1),n,[t:is(x0,y0)]ise(t)):nis(x0,y0) y1iy0@[n:nis(x0,y0)] nise:=th3"l.imp"(eq"n"(x1,y1),is(x0,y0),n,[t:eq"n"(x1,y1)]isi(t)):not(eq"n"(x1,y1)) @[alpha:'type'][f:[x:frac][y:frac]alpha] fixf:=fixfu2"eq"(frac,eq,refeq,symeq,treq,alpha,f):'prop' y0@[alpha:'type'][f:[x:frac][y:frac]alpha][ff:fixf(alpha,f)] indrat:=indeq2"eq"(frac,eq,refeq,symeq,treq,alpha,f,ff,x0,y0):alpha [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))] isindrat:=th1"eq.11"(frac,eq,refeq,symeq,treq,alpha,f,ff,x0,y0,x,xix0,y,yiy0):is"e"(alpha,f,indrat) x0@satz78:=refis(rat,x0):is(x0,x0) y0@[i:is(x0,y0)] satz79:=symis(rat,x0,y0,i):is(y0,x0) z0@[i:is(x0,y0)][j:is(y0,z0)] satz80:=tris(rat,x0,y0,z0,i,j):is(x0,z0) 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' +*ii5 y1@propm:=and3(inf(x1,class(x0)),inf(y1,class(y0)),moref(x1,y1)):'prop' -ii5 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)] +*ii5 p@t4:=and3e1(inf(t,class(x0)),inf(u,class(y0)),moref(t,u),p):inf(t,class(x0)) t5:=and3e2(inf(t,class(x0)),inf(u,class(y0)),moref(t,u),p):inf(u,class(y0)) t6:=and3e3(inf(t,class(x0)),inf(u,class(y0)),moref(t,u),p):moref(t,u) 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) 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) -ii5 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) y1iy0@[m:moref(x1,y1)] +*ii5 m@t9:=and3i(inf(x1,class(x0)),inf(y1,class(y0)),moref(x1,y1),x1ix0,y1iy0,m):propm(x1,y1) t10:=somei(frac,[t:frac]propm(x1,t),y1,t9):some"l"(frac,[t:frac]propm(x1,t)) -ii5 m@morei:=somei(frac,[u:frac]some"l"(frac,[t:frac]propm".ii5"(u,t)),x1,t10".ii5"):more(x0,y0) y1iy0@[m:more(x0,y0)] moree:=also18(m,x1,y1,x1ix0,y1iy0):moref(x1,y1) z0@[i:is(x0,y0)][m:more(x0,z0)] ismore1:=isp(rat,[t:rat]more(t,z0),x0,y0,m,i):more(y0,z0) i@[m:more(z0,x0)] ismore2:=isp(rat,[t:rat]more(z0,t),x0,y0,m,i):more(z0,y0) u0@[i:is(x0,y0)][j:is(z0,u0)][m:more(x0,z0)] ismore12:=ismore2(z0,u0,y0,j,ismore1(x0,y0,z0,i,m)):more(y0,u0) 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' +*ii5 y1@propl:=and3(inf(x1,class(x0)),inf(y1,class(y0)),lessf(x1,y1)):'prop' -ii5 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)] +*ii5 p@t11:=and3e1(inf(t,class(x0)),inf(u,class(y0)),lessf(t,u),p):inf(t,class(x0)) t12:=and3e2(inf(t,class(x0)),inf(u,class(y0)),lessf(t,u),p):inf(u,class(y0)) t13:=and3e3(inf(t,class(x0)),inf(u,class(y0)),lessf(t,u),p):lessf(t,u) 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) 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) -ii5 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) y1iy0@[l:lessf(x1,y1)] +*ii5 l@t16:=and3i(inf(x1,class(x0)),inf(y1,class(y0)),lessf(x1,y1),x1ix0,y1iy0,l):propl(x1,y1) t17:=somei(frac,[t:frac]propl(x1,t),y1,t16):some"l"(frac,[t:frac]propl(x1,t)) -ii5 l@lessi:=somei(frac,[u:frac]some"l"(frac,[t:frac]propl".ii5"(u,t)),x1,t17".ii5"):less(x0,y0) y1iy0@[l:less(x0,y0)] lesse:=also19(l,x1,y1,x1ix0,y1iy0):lessf(x1,y1) z0@[i:is(x0,y0)][l:less(x0,z0)] isless1:=isp(rat,[t:rat]less(t,z0),x0,y0,l,i):less(y0,z0) i@[l:less(z0,x0)] isless2:=isp(rat,[t:rat]less(z0,t),x0,y0,l,i):less(z0,y0) u0@[i:is(x0,y0)][j:is(z0,u0)][l:less(x0,z0)] isless12:=isless2(z0,u0,y0,j,isless1(x0,y0,z0,i,l)):less(y0,u0) +581 y1iy0@t1:=satz41a(x1,y1):or3(eq"n"(x1,y1),moref(x1,y1),lessf(x1,y1)) [e:eq"n"(x1,y1)] t2:=or3i1(is(x0,y0),more(x0,y0),less(x0,y0),isi(e)):or3(is(x0,y0),more(x0,y0),less(x0,y0)) y1iy0@[m:moref(x1,y1)] t3:=or3i2(is(x0,y0),more(x0,y0),less(x0,y0),morei(m)):or3(is(x0,y0),more(x0,y0),less(x0,y0)) y1iy0@[l:lessf(x1,y1)] t4:=or3i3(is(x0,y0),more(x0,y0),less(x0,y0),lessi(l)):or3(is(x0,y0),more(x0,y0),less(x0,y0)) 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)) t6:=satz41b(x1,y1):ec3(eq"n"(x1,y1),moref(x1,y1),lessf(x1,y1)) [i:is(x0,y0)] 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)) y1iy0@[m:more(x0,y0)] 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)) y1iy0@[l:less(x0,y0)] 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) 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)) t11:=orec3i(is(x0,y0),more(x0,y0),less(x0,y0),t5,t10):orec3(is(x0,y0),more(x0,y0),less(x0,y0)) -581 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)) satz81a:=orec3e1(is(x0,y0),more(x0,y0),less(x0,y0),satz81):or3(is(x0,y0),more(x0,y0),less(x0,y0)) satz81b:=orec3e2(is(x0,y0),more(x0,y0),less(x0,y0),satz81):ec3(is(x0,y0),more(x0,y0),less(x0,y0)) [m:more(x0,y0)] +582 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))] t1:=lessi(y0,x0,y,x,yiy0,xix0,satz42(x,y,moree(x0,y0,x,y,xix0,yiy0,m))):less(y0,x0) -582 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) y0@[l:less(x0,y0)] +583 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))] t1:=morei(y0,x0,y,x,yiy0,xix0,satz43(x,y,lesse(x0,y0,x,y,xix0,yiy0,l))):more(y0,x0) -583 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) y0@moreis:=or(more(x0,y0),is(x0,y0)):'prop' [m:more(x0,y0)] moreisi1:=ori1(more,is,m):moreis(x0,y0) y0@[i:is(x0,y0)] moreisi2:=ori2(more,is,i):moreis(x0,y0) y1iy0@[m:moreq(x1,y1)] 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) y1iy0@[m:moreis(x0,y0)] 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) z0@[i:is(x0,y0)][m:moreis(x0,z0)] ismoreis1:=isp(rat,[t:rat]moreis(t,z0),x0,y0,m,i):moreis(y0,z0) i@[m:moreis(z0,x0)] ismoreis2:=isp(rat,[t:rat]moreis(z0,t),x0,y0,m,i):moreis(z0,y0) u0@[i:is(x0,y0)][j:is(z0,u0)][m:moreis(x0,z0)] ismoreis12:=ismoreis2(z0,u0,y0,j,ismoreis1(x0,y0,z0,i,m)):moreis(y0,u0) y0@lessis:=or(less(x0,y0),is(x0,y0)):'prop' [l:less(x0,y0)] lessisi1:=ori1(less,is,l):lessis(x0,y0) y0@[i:is(x0,y0)] lessisi2:=ori2(less,is,i):lessis(x0,y0) y1iy0@[l:lesseq(x1,y1)] 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) y1iy0@[l:lessis(x0,y0)] 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) z0@[i:is(x0,y0)][l:lessis(x0,z0)] islessis1:=isp(rat,[t:rat]lessis(t,z0),x0,y0,l,i):lessis(y0,z0) i@[l:lessis(z0,x0)] islessis2:=isp(rat,[t:rat]lessis(z0,t),x0,y0,l,i):lessis(z0,y0) u0@[i:is(x0,y0)][j:is(z0,u0)][l:lessis(x0,z0)] islessis12:=islessis2(z0,u0,y0,j,islessis1(x0,y0,z0,i,l)):lessis(y0,u0) y0@[m:moreis(x0,y0)] 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)) y0@[l:lessis(x0,y0)] satz81d:=th9"l.ec3"(is(x0,y0),more(x0,y0),less(x0,y0),satz81b,l):not(more(x0,y0)) y0@[n:not(more(x0,y0))] satz81e:=th2"l.or3"(is(x0,y0),more(x0,y0),less(x0,y0),satz81a,n):lessis(x0,y0) y0@[n:not(less(x0,y0))] 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) y0@[m:more(x0,y0)] 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)) y0@[l:less(x0,y0)] 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)) y0@[n:not(moreis(x0,y0))] 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) y0@[n:not(lessis(x0,y0))] 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) y0@[m:moreis(x0,y0)] +584 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))] t1:=lessisi(y0,x0,y,x,yiy0,xix0,satz48(x,y,moreise(x0,y0,x,y,xix0,yiy0,m))):lessis(y0,x0) -584 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) y0@[l:lessis(x0,y0)] +585 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))] t1:=moreisi(y0,x0,y,x,yiy0,xix0,satz49(x,y,lessise(x0,y0,x,y,xix0,yiy0,l))):moreis(y0,x0) -585 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) z0@[l:less(x0,y0)][k:less(y0,z0)] +586 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -586 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) trless:=satz86:less(x0,z0) z0@[m:more(x0,y0)][n:more(y0,z0)] trmore:=satz83(z0,x0,satz86(z0,y0,x0,satz82(y0,z0,n),satz82(m))):more(x0,z0) z0@[l:lessis(x0,y0)][k:less(y0,z0)] +587 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -587 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) z0@[l:less(x0,y0)][k:lessis(y0,z0)] +*587 k@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -587 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) z0@[m:moreis(x0,y0)][n:more(y0,z0)] satz87c:=satz83(z0,x0,satz87b(z0,y0,x0,satz82(y0,z0,n),satz84(m))):more(x0,z0) z0@[m:more(x0,y0)][n:moreis(y0,z0)] satz87d:=satz83(z0,x0,satz87a(z0,y0,x0,satz84(y0,z0,n),satz82(m))):more(x0,z0) z0@[l:lessis(x0,y0)][k:lessis(y0,z0)] +588 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -588 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) trlessis:=satz88:lessis(x0,z0) z0@[m:moreis(x0,y0)][n:moreis(y0,z0)] trmoreis:=satz85(z0,x0,satz88(z0,y0,x0,satz84(y0,z0,n),satz84(m))):moreis(x0,z0) +589 x0@[x:frac][xix0:inf(x,class(x0))][z:frac][m:moref(z,x)] 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)) 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)) -589 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)) +590 z"rt.589"@[l:lessf(z,x)] 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)) 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)) -590 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)) y0@[l:less(x0,y0)] +591 [x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][z:frac][a:and(lessf(x,z),lessf(z,y))] t1:=lessi(x0,ratof(z),x,z,xix0,inclass(z),ande1(lessf(x,z),lessf(z,y),a)):less(x0,ratof(z)) t2:=lessi(ratof(z),y0,z,y,inclass(z),yiy0,ande2(lessf(x,z),lessf(z,y),a)):less(ratof(z),y0) t3:=andi(less(x0,ratof(z)),less(ratof(z),y0),t1,t2):and(less(x0,ratof(z)),less(ratof(z),y0)) 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))) 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))) -591 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))) @plusfrt:=[x:frac][y:frac]ratof(pf(x,y)):[x:frac][y:frac]rat [x:frac][y:frac][z:frac][u:frac][e:eq"n"(x,y)][f:eq"n"(z,u)] +*ii5 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(plusfrt,plusfrt) -ii5 @fplusfrt:=[x:frac][y:frac][z:frac][u:frac][v:eq][w:eq]t18".ii5"(x,y,z,u,v,w):fixf(rat,plusfrt) y0@pl:=indrat(rat,plusfrt,fplusfrt):rat +*ii5 y1iy0@t19:=isindrat(rat,plusfrt,fplusfrt,x1,y1,x1ix0,y1iy0):is(ratof(pf(x1,y1)),pl(x0,y0)) -ii5 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))) z0@[i:is(x0,y0)] ispl1:=isf(rat,rat,[t:rat]pl(t,z0),x0,y0,i):is(pl(x0,z0),pl(y0,z0)) ispl2:=isf(rat,rat,[t:rat]pl(z0,t),x0,y0,i):is(pl(z0,x0),pl(z0,y0)) u0@[i:is(x0,y0)][j:is(z0,u0)] 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)) +592 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)) -592 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)) compl:=satz92:is(pl(x0,y0),pl(y0,x0)) +593 z0@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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))) 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)))) 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))) -593 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))) asspl1:=satz93:is(pl(pl(x0,y0),z0),pl(x0,pl(y0,z0))) 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)) +594 y1iy0@t1:=morei(pl(x0,y0),x0,pf(x1,y1),x1,picp,x1ix0,satz60(x1,y1)):more(pl(x0,y0),x0) -594 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) satz94a:=satz82(pl(x0,y0),x0,satz94):less(x0,pl(x0,y0)) z0@[m:more(x0,y0)] +595 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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)) -595 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)) z0@[m:more(x0,y0)] +596 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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)) -596 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)) z0@[i:is(x0,y0)] +*596 i@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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)) -596 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)) z0@[l:less(x0,y0)] +*596 l@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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)) -596 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)) +*596 m@andersa:=satz95(m):more(pl(x0,z0),pl(y0,z0)) i@andersb:=ispl1(x0,y0,z0,i):is(pl(x0,z0),pl(y0,z0)) l@andersc:=satz82(pl(y0,z0),pl(x0,z0),satz95(y0,x0,z0,satz83(l))):less(pl(x0,z0),pl(y0,z0)) -596 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)) i@satz96e:=ispl2(x0,y0,z0,i):is(pl(z0,x0),pl(z0,y0)) 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)) z0@[m:more(pl(x0,z0),pl(y0,z0))] +597 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -597 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) z0@[i:is(pl(x0,z0),pl(y0,z0))] +*597 i@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -597 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) z0@[l:less(pl(x0,z0),pl(y0,z0))] +*597 l@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -597 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) +*597 l@anders:=satz82(y0,x0,satz97a(y0,x0,z0,satz83(pl(x0,z0),pl(y0,z0),l))):less(x0,y0) -597 u0@[m:more(x0,y0)][n:more(z0,u0)] +598 [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))] 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)) -598 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)) u0@[l:less(x0,y0)][k:less(z0,u0)] 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)) u0@[m:moreis(x0,y0)][n:more(z0,u0)] +599 [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))] 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)) -599 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)) u0@[m:more(x0,y0)][n:moreis(z0,u0)] +*599 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))] 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)) -599 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)) u0@[l:lessis(x0,y0)][k:less(z0,u0)] 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)) u0@[l:less(x0,y0)][k:lessis(z0,u0)] 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)) u0@[m:moreis(x0,y0)][n:moreis(z0,u0)] +5100 [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))] 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)) -5100 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)) u0@[l:lessis(x0,y0)][k:lessis(z0,u0)] 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)) y0@[l:lessis(x0,y0)] +5101 [v0:rat][i:is(pl(y0,v0),x0)] t1:=ismore1(pl(y0,v0),x0,y0,i,satz94(y0,v0)):more(x0,y0) 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) -5101 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))) y0@[m:more(x0,y0)] +*5101 m@[x:frac][y:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][v:frac][e:eq"n"(pf(y,v),x)] 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) t4:=somei(rat,[t:rat]is(pl(y0,t),x0),ratof(v),t3):some([t:rat]is(pl(y0,t),x0)) 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)) -5101 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)) y0@[v0:rat][w0:rat][i:is(pl(y0,v0),x0)][j:is(pl(y0,w0),x0)] +*5101 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))] 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) -5101 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) +*5101 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)) -5101 m@satz101:=onei(rat,[t:rat]is(pl(y0,t),x0),t7".5101",satz101a):one([t:rat]is(pl(y0,t),x0)) mn:=ind(rat,[t:rat]is(pl(y0,t),x0),satz101):rat satz101c:=oneax(rat,[t:rat]is(pl(y0,t),x0),satz101):is(pl(y0,mn(x0,y0,m)),x0) satz101d:=symis(rat,pl(y0,mn(x0,y0,m)),x0,satz101c):is(x0,pl(y0,mn(x0,y0,m))) 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) satz101f:=symis(rat,pl(mn(x0,y0,m),y0),x0,satz101e):is(x0,pl(mn(x0,y0,m),y0)) y0@[v0:rat][m:more(x0,y0)][i:is(pl(y0,v0),x0)] satz101g:=satz101b(v0,mn(x0,y0,m),i,satz101c(m)):is(v0,mn(x0,y0,m)) @timesfrt:=[x:frac][y:frac]ratof(tf(x,y)):[x:frac][y:frac]rat +*ii5 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(timesfrt,timesfrt) -ii5 @ftimesfrt:=[x:frac][y:frac][z:frac][u:frac][v:eq][w:eq]t20".ii5"(x,y,z,u,v,w):fixf(rat,timesfrt) y0@ts:=indrat(rat,timesfrt,ftimesfrt):rat +*ii5 y1iy0@t21:=isindrat(rat,timesfrt,ftimesfrt,x1,y1,x1ix0,y1iy0):is(ratof(tf(x1,y1)),ts(x0,y0)) -ii5 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))) z0@[i:is(x0,y0)] ists1:=isf(rat,rat,[t:rat]ts(t,z0),x0,y0,i):is(ts(x0,z0),ts(y0,z0)) ists2:=isf(rat,rat,[t:rat]ts(z0,t),x0,y0,i):is(ts(z0,x0),ts(z0,y0)) u0@[i:is(x0,y0)][j:is(z0,u0)] 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)) +5102 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)) -5102 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)) comts:=satz102:is(ts(x0,y0),ts(y0,x0)) +5103 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))) 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)))) 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))) -5103 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))) assts1:=satz103:is(ts(ts(x0,y0),z0),ts(x0,ts(y0,z0))) 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)) +5104 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)))) 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)))) 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))) -5104 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))) 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))) disttp2:=satz104:is(ts(x0,pl(y0,z0)),pl(ts(x0,y0),ts(x0,z0))) 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)) 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))) [m:more(x0,y0)] +5105 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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)) -5105 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)) z0@[i:is(x0,y0)] +*5105 i@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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)) -5105 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)) z0@[l:less(x0,y0)] +*5105 l@[x:frac][y:frac][z:frac][xi:inf(x,class(x0))][yi:inf(y,class(y0))][zi:inf(z,class(z0))] 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)) -5105 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)) +*5105 i@andersb:=ists1(x0,y0,z0,i):is(ts(x0,z0),ts(y0,z0)) l@andersc:=satz82(ts(y0,z0),ts(x0,z0),satz105a(y0,x0,z0,satz83(l))):less(ts(x0,z0),ts(y0,z0)) -5105 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)) i@satz105e:=ists2(x0,y0,z0,i):is(ts(z0,x0),ts(z0,y0)) 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)) z0@[m:more(ts(x0,z0),ts(y0,z0))] +5106 [x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -5106 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) z0@[i:is(ts(x0,z0),ts(y0,z0))] +*5106 i@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -5106 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) z0@[l:less(ts(x0,z0),ts(y0,z0))] +*5106 l@[x:frac][y:frac][z:frac][xix0:inf(x,class(x0))][yiy0:inf(y,class(y0))][ziz0:inf(z,class(z0))] 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) -5106 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) +*5106 l@anders:=satz82(y0,x0,satz106a(y0,x0,z0,satz83(ts(x0,z0),ts(y0,z0),l))):less(x0,y0) -5106 u0@[m:more(x0,y0)][n:more(z0,u0)] +5107 [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))] 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)) -5107 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)) u0@[l:less(x0,y0)][k:less(z0,u0)] 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)) u0@[m:moreis(x0,y0)][n:more(z0,u0)] +5108 [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))] 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)) -5108 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)) u0@[m:more(x0,y0)][n:moreis(z0,u0)] +*5108 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))] 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)) -5108 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)) u0@[l:lessis(x0,y0)][k:less(z0,u0)] 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)) u0@[l:less(x0,y0)][k:lessis(z0,u0)] 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)) u0@[m:moreis(x0,y0)][n:moreis(z0,u0)] +5109 [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))] 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)) -5109 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)) u0@[l:lessis(x0,y0)][k:lessis(z0,u0)] 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)) +5110 y1iy0@[v:frac][e:eq"n"(tf(y1,v),x1)] 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) t2:=somei(rat,[t:rat]is(ts(y0,t),x0),ratof(v),t1):some([t:rat]is(ts(y0,t),x0)) 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)) -5110 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)) [v0:rat][w0:rat][i:is(ts(y0,v0),x0)][j:is(ts(y0,w0),x0)] +*5110 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))] 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) -5110 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) +*5110 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)) -5110 y0@satz110:=onei(rat,[t:rat]is(ts(y0,t),x0),t5".5110",satz110a):one([t:rat]is(ts(y0,t),x0)) -rt @[x:nat][y:nat] +5111 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) 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)))) -5111 [m:moref(fr(x,1),fr(y,1))] 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) y@[e:eq(fr(x,1),fr(y,1))] 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) y@[l:lessf(fr(x,1),fr(y,1))] 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) y@[m:more(x,y)] 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)) y@[i:is(x,y)] 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)) y@[l:less(x,y)] 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)) +*rt @[x0:rat][x:nat] natprop:=inf(fr(x,1),class(x0)):'prop' x0@natrt:=some"n"([t:nat]natprop(t)):'prop' +*ii5 y0@[x:nat][y:nat][npx:natprop(x0,x)][npy:natprop(y0,y)][i:is(x0,y0)] t22:=satz111b(x,y,ise(x0,y0,fr(x,1),fr(y,1),npx,npy,i)):is"n"(x,y) 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)) -ii5 x0@[nx0:natrt(x0)] satz111g:=onei(nat,[t:nat]natprop(t),t23".ii5",nx0):one"n"([t:nat]natprop(x0,t)) nofrt:=ind(nat,[t:nat]natprop(t),satz111g):nat inclassn:=oneax(nat,[t:nat]natprop(t),satz111g):inf(fr(nofrt,1),class(x0)) [y0:rat][ny0:natrt(y0)][i:is(x0,y0)] 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)) ny0@[i:is"n"(nofrt(x0,nx0),nofrt(y0,ny0))] 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) @[x:nat] rtofn:=ratof(fr(x,1)):rat natrti:=somei(nat,[t:nat]natprop(rtofn,t),x,inclass(fr(x,1))):natrt(rtofn(x)) [y:nat][i:is"n"(x,y)] isnert:=isf(nat,rat,[t:nat]rtofn(t),x,y,i):is(rtofn(x),rtofn(y)) y@[i:is(rtofn(x),rtofn(y))] isnirt:=t22".ii5"(rtofn(x),rtofn(y),x,y,inclass(fr(x,1)),inclass(fr(y,1)),i):is"n"(x,y) 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))) 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))) -rt @[x:nat][y:nat] satz112a:=satz57(x,y,1):eq(pf(fr(x,1),fr(y,1)),fr(pl(x,y),1)) 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)) +*rt 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))) satz112d:=somei(nat,[t:nat]natprop(pl(x0,y0),t),pl"n"(nofrt(x0,nx0),nofrt(y0,ny0)),satz112c):natrt(pl(x0,y0)) 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))) satz112f:=somei(nat,[t:nat]natprop(ts(x0,y0),t),ts"n"(nofrt(x0,nx0),nofrt(y0,ny0)),satz112e):natrt(ts(x0,y0)) [m:more(x0,y0)] +5112 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)) [z:nat][d:diffprop(nofrt(x0,nx0),nofrt(y0,ny0),z)] 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)))) 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))) t4:=satz101g(x0,y0,rtofn(z),m,symis(rat,x0,pl(y0,rtofn(z)),t3)):is(rtofn(z),mn(x0,y0,m)) t5:=isp(rat,[t:rat]natrt(t),rtofn(z),mn(x0,y0,m),natrti(z),t4):natrt(mn(x0,y0,m)) -5112 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)) @[x:nat][y:nat] 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))) 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))) +nt @natt:=ot(rat,[t:rat]natrt(t)):'type' nx0@ntofrt:=out(rat,[t:rat]natrt(t),x0,nx0):natt @[xt:natt][yt:natt] is:=is"e"(natt,xt,yt):'prop' nis:=not(is(xt,yt)):'prop' @[p:[x:natt]'prop'] all:=all"l"(natt,p):'prop' some:=some"l"(natt,p):'prop' one:=one"e"(natt,p):'prop' xt@[st:set(natt)] in:=esti(natt,xt,st):'prop' xt@rtofnt:=in"e"(rat,[t:rat]natrt(t),xt):rat natrti:=inp(rat,[t:rat]natrt(t),xt):natrt(rtofnt(xt)) ny0@[i:is"rt"(x0,y0)] isrtent:=isouti(rat,[t:rat]natrt(t),x0,nx0,y0,ny0,i):is(ntofrt(x0,nx0),ntofrt(y0,ny0)) ny0@[i:is(ntofrt(x0,nx0),ntofrt(y0,ny0))] isrtint:=isoute(rat,[t:rat]natrt(t),x0,nx0,y0,ny0,i):is"rt"(x0,y0) yt@[i:is(xt,yt)] isntert:=isini(rat,[t:rat]natrt(t),xt,yt,i):is"rt"(rtofnt(xt),rtofnt(yt)) yt@[i:is"rt"(rtofnt(xt),rtofnt(yt))] isntirt:=isine(rat,[t:rat]natrt(t),xt,yt,i):is(xt,yt) nx0@isrtnt1:=isinout(rat,[t:rat]natrt(t),x0,nx0):is"rt"(x0,rtofnt(ntofrt(x0,nx0))) xt@isntrt1:=isoutin(rat,[t:rat]natrt(t),xt):is(xt,ntofrt(rtofnt(xt),natrti(xt))) x@ntofn:=ntofrt(rtofn(x),natrti"rt"(x)):natt y@[i:is"n"(x,y)] isnent:=isrtent(rtofn(x),natrti"rt"(x),rtofn(y),natrti"rt"(y),isnert(x,y,i)):is(ntofn(x),ntofn(y)) y@[i:is(ntofn(x),ntofn(y))] isnint:=isnirt(x,y,isrtint(rtofn"rt"(x),natrti"rt"(x),rtofn(y),natrti"rt"(y),i)):is"n"(x,y) xt@nofnt:=nofrt(rtofnt(xt),natrti(xt)):nat yt@[i:is(xt,yt)] isnten:=isrten(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt),isntert(xt,yt,i)):is"n"(nofnt(xt),nofnt(yt)) yt@[i:is"n"(nofnt(xt),nofnt(yt))] isntin:=isntirt(xt,yt,isrtin(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt),i)):is(xt,yt) +ii5 x@t24:=isrtnt1(rtofn(x),natrti"rt"(x)):is"rt"(rtofn(x),rtofnt(ntofn(x))) 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))) -ii5 x@isnnt1:=tris(nat,x,nofrt(rtofn(x),natrti"rt"(x)),nofnt(ntofn(x)),isnrt1(x),t25".ii5"):is"n"(x,nofnt(ntofn(x))) +*ii5 xt@t26:=isrtn1(rtofnt(xt),natrti(xt)):is"rt"(rtofnt(xt),rtofn(nofnt(xt))) t27:=isrtent(rtofnt(xt),natrti(xt),rtofn(nofnt(xt)),natrti"rt"(nofnt(xt)),t26):is(ntofrt(rtofnt(xt),natrti(xt)),ntofn(nofnt(xt))) -ii5 xt@isntn1:=tris(natt,xt,ntofrt(rtofnt(xt),natrti(xt)),ntofn(nofnt(xt)),isntrt1(xt),t27".ii5"):is(xt,ntofn(nofnt(xt))) x@isnnt2:=symis(nat,x,nofnt(ntofn(x)),isnnt1):is"n"(nofnt(ntofn(x)),x) xt@isntn2:=symis(natt,xt,ntofn(nofnt(xt)),isntn1):is(ntofn(nofnt(xt)),xt) @1t:=ntofn(1):natt suct:=[x:natt]ntofn(suc):[x:natt]natt +5113 xt@[i:is(suct,1t)] t1:=isnint(suc,1,i):is"n"(suc,1) -5113 xt@satz113a:=th3"l.imp"(is(suct,1t),is"n"(suc,1),ax3,[t:is(suct,1t)]t1".5113"(t)):nis(suct,1t) yt@[i:is(suct,suct)] +*5113 i"nt"@t2:=isnint(suc,suc,i):is"n"(suc,suc) -5113 i@satz113b:=isntin(xt,yt,ax4):is(xt,yt) @[st:set(natt)] cond1:=in(1t,st):'prop' cond2:=all([x:natt]imp(in(x,st),in(suct,st))):'prop' [c1:cond1][c2:cond2] +*5113 c2@[x:nat] prop1:=in(ntofn(x),st):'prop' [p:prop1(x)] t3:=c2:imp(in(ntofn(x),st),in(suct,st)) t4:=mp(in(ntofn(x),st),in(suct,st),p,t3):in(suct,st) t5:=isp(nat,[t:nat]in(ntofn(suc),st),nofnt(ntofn(x)),x,t4,isnnt2(x)):prop1(suc) -5113 c2@[xt:natt] +*5113 xt@t6:=induction([t:nat]prop1(t),c1,[t:nat][u:prop1(t)]t5(t,u),nofnt(xt)):in(ntofn(nofnt(xt)),st) -5113 xt@satz113c:=isp(natt,[t:natt]in(t,st),ntofn(nofnt(xt)),xt,t6".5113",isntn2(xt)):in(xt,st) @ax3t:=[x:natt]satz113a(x):[x:natt]nis(suct,1t) ax4t:=[x:natt][y:natt][u:is(suct,suct)]satz113b(x,y,u):[x:natt][y:natt][u:is(suct,suct)]is(x,y) 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) yt@[n:nis(xt,yt)] +51 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)) t2:=satz1(nofnt(xt),nofnt(yt),t1):nis"n"(suc,suc) -51 satz1:=th3"l.imp"(is(suct,suct),is"n"(suc,suc),t2".51",[t:is(suct,suct)]isnint(suc,suc,t)):nis(suct,suct) +54 xt@x:=nofnt(xt):nat [ft:[x:natt]natt] prop1t:=all([t:natt]is(<suct>ft,<ft>suct)):'prop' prop2t:=and(is(<1t>ft,suct),prop1t):'prop' xt@[f:[x:nat]nat] prop1:=all"n"([t:nat]is"n"(<suc>f,<f>suc)):'prop' prop2:=and(is"n"(<1>f,suc),prop1):'prop' ft@g:=[t:nat]nofnt(ft):[x:nat]nat [p:prop2t] t1:=ande1(is(<1t>ft,suct),prop1t,p):is(<1t>ft,suct) t2:=tris(nat,<1>g,nofnt(suct),suc,isnten(<1t>ft,suct,t1),isnnt2(suc)):is"n"(<1>g,suc) t3:=ande2(is(<1t>ft,suct),prop1t,p):prop1t [u:nat] ut:=ntofn(u):natt t4:=isf(nat,nat,suc,u,nofnt(ut),isnnt1(u)):is"n"(suc,suc) t5:=isf(nat,nat,g,suc,suc,t4):is"n"(<suc>g,nofnt(<suct>ft)) t6:=t3:is(<suct>ft,<ft>suct) t7:=isnten(<suct>ft,<ft>suct,t6):is"n"(nofnt(<suct>ft),nofnt(<ft>suct)) t8:=isnnt2(<g>suc):is"n"(nofnt(<ft>suct),<g>suc) t9:=tr3is(nat,<suc>g,nofnt(<suct>ft),nofnt(<ft>suct),<g>suc,t5,t7,t8):is"n"(<suc>g,<g>suc) p@t10:=[u:nat]t9(u):prop1(g) t11:=andi(is"n"(<1>g,suc),prop1(g),t2,t10):prop2(g) xt@[a:[t:natt]natt][b:[t:natt]natt][pa:prop2t(a)][pb:prop2t(b)] t12:=onee1([t:nat]nat,[u:[t:nat]nat]prop2(u),satz4(x)):amone([t:nat]nat,[u:[t:nat]nat]prop2(u)) t13:=t12:is"e"([t:nat]nat,g(a),g(b)) [yt:natt] y:=nofnt(yt):nat t14:=fise(nat,nat,g(a),g(b),t13,y):is"n"(nofnt(a),nofnt(b)) t15:=isntin(a,b,t14):is(a,b) t16:=tr3is(natt,a,a,b,b,isf(natt,natt,a,yt,ntofn(y),isntn1(yt)),t15,isf(natt,natt,b,ntofn(y),yt,isntn2(yt))):is(a,b) pb@t17:=fisi(natt,natt,a,b,[t:natt]t16(t)):is"e"([t:natt]natt,a,b) 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)) t19:=onee2([t:nat]nat,[u:[t:nat]nat]prop2(u),satz4(x)):some"l"([t:nat]nat,[u:[t:nat]nat]prop2(u)) f@gt:=[t:natt]ntofn(f):[x:natt]natt [p:prop2] t20:=ande1(is"n"(<1>f,suc),prop1,p):is"n"(<1>f,suc) t21:=isf(nat,nat,f,nofnt(1t),1,isnnt2(1)):is"n"(f,<1>f) t22:=isnent(f,suc,tris(nat,f,<1>f,suc,t21,t20)):is(<1t>gt,suct) t23:=ande2(is"n"(<1>f,suc),prop1,p):prop1 [zt:natt] z:=nofnt(zt):nat t24:=isf(nat,nat,f,nofnt(suct),suc,isnnt2(suc)):is"n"(suct)>f,<suc>f) t25:=t23:is"n"(<suc>f,<f>suc) t26:=isf(nat,nat,suc,f,nofnt(gt),isnnt1(f)):is"n"(<f>suc,gt)>suc) t27:=isnent(suct)>f,gt)>suc,tr3is(nat,suct)>f,<suc>f,<f>suc,gt)>suc,t24,t25,t26)):is(<suct>gt,<gt>suct) p@t28:=[u:natt]t27(u):prop1t(gt) t29:=andi(is(<1t>gt,suct),prop1t(gt),t22,t28):prop2t(gt) t30:=somei([t:natt]natt,[u:[t:natt]natt]prop2t(u),gt,t29):some"l"([t:natt]natt,[u:[t:natt]natt]prop2t(u)) 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)) -54 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,suct),all([v:natt]is(<suct>u,<u>suct)))) yt@pl:=ntofrt(pl"rt"(rtofnt(xt),rtofnt(yt)),satz112d(rtofnt(xt),natrti(xt),rtofnt(yt),natrti(yt))):natt +*ii5 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)))) 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)))) -ii5 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)))) 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)) 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))) 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))) [zt:natt] +55 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))) 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)))) 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)))) -55 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))) diffprop:=is(xt,pl(yt,zt)):'prop' [d:diffprop] 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)) zt@[d:diffprop"n"(nofnt(xt),nofnt(yt),nofnt(zt))] +*ii5 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))) -ii5 d@diffpropi:=isntin(xt,pl(yt,zt),t30".ii5"):diffprop +59 yt@it:=is(xt,yt):'prop' iit:=some([u:natt]diffprop(xt,yt,u)):'prop' iiit:=some([u:natt]diffprop(yt,xt,u)):'prop' i:=is"n"(nofnt(xt),nofnt(yt)):'prop' ii:=some"n"([u:nat]diffprop"n"(nofnt(xt),nofnt(yt),u)):'prop' iii:=some"n"([u:nat]diffprop"n"(nofnt(yt),nofnt(xt),u)):'prop' [one:i] t1:=or3i1(it,iit,iiit,isntin(xt,yt,one)):or3(it,iit,iiit) yt@[two:ii][v:nat][d:diffprop"n"(nofnt(xt),nofnt(yt),v)] 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))) t3:=somei(natt,[u:natt]diffprop(xt,yt,u),ntofn(v),diffpropi(xt,yt,ntofn(v),t2)):iit 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 t5:=or3i2(it,iit,iiit,t4):or3(it,iit,iiit) yt@[three:iii] t6:=or3i3(it,iit,iiit,t4(yt,xt,three)):or3(it,iit,iiit) 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) [onet:it] t8:=isnten(xt,yt,onet):i yt@[twot:iit][vt:natt][d:diffprop(xt,yt,vt)] t9:=somei(nat,[u:nat]diffprop"n"(nofnt(xt),nofnt(yt),u),nofnt(vt),diffprope(xt,yt,vt,d)):ii twot@t10:=someapp(natt,[u:natt]diffprop(xt,yt,u),twot,ii,[u:natt][v:diffprop(xt,yt,u)]t9(u,v)):ii yt@[threet:iiit] t11:=t10(yt,xt,threet):iii yt@t12:=satz9b(nofnt(xt),nofnt(yt)):ec3(i,ii,iii) onet@t13:=ec3e12(i,ii,iii,t12,t8):not(ii) t14:=th3"l.imp"(iit,ii,t13,[x:iit]t10(x)):not(iit) yt@t15:=th1"l.ec"(it,iit,[x:it]t14(x)):ec(it,iit) twot@t16:=ec3e23(i,ii,iii,t12,t10):not(iii) t17:=th3"l.imp"(iiit,iii,t16,[x:iiit]t11(x)):not(iiit) yt@t18:=th1"l.ec"(iit,iiit,[x:iit]t17(x)):ec(iit,iiit) threet@t19:=ec3e31(i,ii,iii,t12,t11):not(i) t20:=th3"l.imp"(it,i,t19,[x:it]t8(x)):not(it) yt@t21:=th1"l.ec"(iiit,it,[x:iiit]t20(x)):ec(iiit,it) t22:=th6"l.ec3"(it,iit,iiit,t15,t18,t21):ec3(it,iit,iiit) -59 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)))) more:=more"rt"(rtofnt(xt),rtofnt(yt)):'prop' [m:more(xt,yt)] +*ii5 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)) -ii5 m@moree:=satz111a(nofnt(xt),nofnt(yt),t31".ii5"):more"n"(nofnt(xt),nofnt(yt)) yt@[m:more"n"(nofnt(xt),nofnt(yt))] +*ii5 m@t32:=satz111d(nofnt(xt),nofnt(yt),m):moref(fr(nofnt(xt),1),fr(nofnt(yt),1)) -ii5 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) yt@less:=less"rt"(rtofnt(xt),rtofnt(yt)):'prop' [l:less(xt,yt)] +*ii5 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)) -ii5 l@lesse:=satz111c(nofnt(xt),nofnt(yt),t33".ii5"):less"n"(nofnt(xt),nofnt(yt)) yt@[l:less"n"(nofnt(xt),nofnt(yt))] +*ii5 l@t34:=satz111f(nofnt(xt),nofnt(yt),l):lessf(fr(nofnt(xt),1),fr(nofnt(yt),1)) -ii5 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) yt@moreis:=moreis"rt"(rtofnt(xt),rtofnt(yt)):'prop' [m:moreis(xt,yt)] 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)) yt@[m:moreis"n"(nofnt(xt),nofnt(yt))] 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) yt@lessis:=lessis"rt"(rtofnt(xt),rtofnt(yt)):'prop' [l:lessis(xt,yt)] 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)) yt@[l:lessis"n"(nofnt(xt),nofnt(yt))] 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) zt@[l:less(xt,yt)][k:less(yt,zt)] +515 t1:=satz15(nofnt(xt),nofnt(yt),nofnt(zt),lesse(xt,yt,l),lesse(yt,zt,k)):less"n"(nofnt(xt),nofnt(zt)) -515 satz15:=lessi(xt,zt,t1".515"):less(xt,zt) zt@[ut:natt][m:more(xt,yt)][n:more(zt,ut)] +521 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))) 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))) -521 satz21:=morei(pl(xt,zt),pl(yt,ut),t2".521"):more(pl(xt,zt),pl(yt,ut)) @[p:[x:natt]'prop'][n:natt] lb:=all([x:natt]imp(p,lessis(n,x))):'prop' min:=and(lb,p):'prop' p@[s:some(p)] +527 q:=[x:nat]p:[x:nat]'prop' [n:natt][np:p] t1:=isp(natt,p,n,ntofn(nofnt(n)),np,isntn1(n)):q t2:=somei(nat,q,nofnt(n),t1):some"n"(q) s@t3:=someapp(natt,p,s,some"n"(q),[u:natt][v:p]t2(u,v)):some"n"(q) t4:=satz27(q,t3):some"n"([x:nat]min"n"(q,x)) [m:nat][mqm:min"n"(q,m)] t5:=ande1(lb"n"(q,m),q,mqm):lb"n"(q,m) [n:nat][nq:q] t6:=mp(q,lessis"n"(m,n),nq,t5):lessis"n"(m,n) mqm@[n:natt][np:p] t7:=t6(nofnt(n),t1(n,np)):lessis"n"(m,nofnt(n)) t8:=islessis1"n"(m,nofnt(ntofn(m)),nofnt(n),isnnt1(m),t7):lessis"n"(nofnt(ntofn(m)),nofnt(n)) t9:=lessisi(ntofn(m),n,t8):lessis(ntofn(m),n) n@t10:=[u:p]t9(u):imp(p,lessis(ntofn(m),n)) mqm@t11:=[x:natt]t10(x):lb(ntofn(m)) t12:=ande2(lb"n"(q,m),q,mqm):p t13:=andi(lb(ntofn(m)),p,t11,t12):min(ntofn(m)) t14:=somei(natt,[x:natt]min(x),ntofn(m),t13):some([x:natt]min(x)) -527 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)) -nt @1rt:=rtofn(1):rat x0@[x:frac][xix0:inf(x,class(x0))] +*ii5 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) 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) -ii5 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) example1b:=symis(rat,ts(x0,1rt),x0,example1a):is(x0,ts(x0,1rt)) example1c:=tris(rat,ts(1rt,x0),ts(x0,1rt),x0,comts(1rt,x0),example1a):is(ts(1rt,x0),x0) example1d:=symis(rat,ts(1rt,x0),x0,example1c):is(x0,ts(1rt,x0)) @[x:frac] +5114 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)) -5114 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))) @[x1:nat][x2:nat] 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)) x0@[y0:rat] ov:=ind(rat,[t:rat]is(ts(y0,t),x0),satz110(x0,y0)):rat satz110c:=oneax(rat,[t:rat]is(ts(y0,t),x0),satz110(x0,y0)):is(ts(y0,ov(x0,y0)),x0) satz110d:=symis(rat,ts(y0,ov(x0,y0)),x0,satz110c):is(x0,ts(y0,ov(x0,y0))) 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) satz110f:=symis(rat,ts(ov(x0,y0),y0),x0,satz110e):is(x0,ts(ov(x0,y0),y0)) [v0:rat][i:is(ts(y0,v0),x0)] satz110g:=satz110b(x0,y0,v0,ov(x0,y0),i,satz110c):is(v0,ov(x0,y0)) 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))) satz114c:=symis(rat,ratof(fr(x1,x2)),ov(rtofn(x1),rtofn(x2)),satz114b):is(ov(rtofn(x1),rtofn(x2)),ratof(fr(x1,x2))) +5115 y0@t1:=satz89(ov(y0,x0)):some([t:rat]more(t,ov(y0,x0))) [u0:rat][m:more(u0,ov(y0,x0))][u:frac][uiu0:inf(u,class(u0))] z:=num(u):nat v:=den(u):nat 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)) 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) 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))) [n:more(rtofn(v),1rt)] 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)) 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)) uiu0@[i:is(rtofn(v),1rt)] 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)) 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)) 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) t10:=satz87c(ts(rtofn(z),x0),ts(ts(x0,ov(rtofn(z),rtofn(v))),1rt),y0,t8,t9):more(ts(rtofn(z),x0),y0) t11:=somei(nat,[t:nat]more(ts(rtofn(t),x0),y0),z,t10):some"n"([t:nat]more(ts(rtofn(t),x0),y0)) 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)) -5115 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)) +*5115 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)) 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))) 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))) -5115 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))) @[s:set(rat)] cutprop1a:=nonempty(rat,s):'prop' cutprop1b:=not(all([x:rat]in(x,s))):'prop' cutprop1:=and(cutprop1a,cutprop1b):'prop' [x0:rat] cutprop2a:=all([x:rat]imp(not(in(x,s)),less(x0,x))):'prop' s@cutprop2:=all([x:rat]imp(in(x,s),cutprop2a(x))):'prop' x0@[y0:rat] +iii1 ubprop:=imp(in(y0,s),moreis(x0,y0)):'prop' -iii1 x0@ub:=all([x:rat]ubprop".iii1"(x0,x)):'prop' max:=and(ub(x0),in(x0,s)):'prop' s@cutprop3:=not(some([x:rat]max(s,x))):'prop' cutprop:=and3(cutprop1,cutprop2,cutprop3):'prop' +*iii1 y0@lbprop:=imp(in(y0,s),lessis(x0,y0)):'prop' -iii1 x0@lb:=all([x:rat]lbprop".iii1"(x0,x)):'prop' min:=and(lb(x0),in(x0,s)):'prop' @cut:=ot(set(rat),[x:set(rat)]cutprop(x)):'type' [ksi:cut] lcl:=in"e"(set(rat),[x:set(rat)]cutprop(x),ksi):set(rat) [x0:rat] lrt:=in(x0,lcl(ksi)):'prop' urt:=not(in(x0,lcl(ksi))):'prop' ksi@clcl:=inp(set(rat),[x:set(rat)]cutprop(x),ksi):cutprop(lcl(ksi)) clcl1:=and3e1(cutprop1(lcl),cutprop2(lcl),cutprop3(lcl),clcl):cutprop1(lcl(ksi)) clcl2:=and3e2(cutprop1(lcl),cutprop2(lcl),cutprop3(lcl),clcl):cutprop2(lcl(ksi)) clcl3:=and3e3(cutprop1(lcl),cutprop2(lcl),cutprop3(lcl),clcl):cutprop3(lcl(ksi)) clcl1a:=ande1(cutprop1a(lcl),cutprop1b(lcl),clcl1):cutprop1a(lcl(ksi)) clcl1b:=ande2(cutprop1a(lcl),cutprop1b(lcl),clcl1):cutprop1b(lcl(ksi)) [p:'prop'][p1:[x:rat][t:lrt(ksi,x)]p] cutapp1a:=nonemptyapp(rat,lcl,clcl1a,p,p1):p +*iii1 ksi@t1:=th1"l.some"(rat,[x:rat]lrt(ksi,x),clcl1b):some([x:rat]urt(ksi,x)) -iii1 p@[p1:[x:rat][t:urt(ksi,x)]p] cutapp1b:=someapp(rat,[x:rat]urt(ksi,x),t1".iii1",p,p1):p ksi@[x0:rat][lx:lrt(ksi,x0)] +*iii1 lx@t2:=mp(lrt(ksi,x0),cutprop2a(lcl,x0),lx,clcl2):cutprop2a(lcl,x0) -iii1 lx@[y0:rat][uy:urt(ksi,y0)] cutapp2a:=mp(urt(ksi,y0),less(x0,y0),uy,t2".iii1"):less(x0,y0) cutapp2b:=satz83(x0,y0,cutapp2a):more(y0,x0) +*iii1 lx@t3:=th4"l.some"(rat,[x:rat]max(lcl,x),clcl3,x0):not(max(lcl,x0)) t4:=th4"l.and"(ub(lcl,x0),lrt(ksi,x0),t3,lx):not(ub(lcl,x0)) t5:=th1"l.some"(rat,[x:rat]ubprop(lcl,x0,x),t4):some([x:rat]not(ubprop(lcl,x0,x))) -iii1 lx@[p:'prop'][p1:[y:rat][t:lrt(ksi,y)][u:less(x0,y)]p][y0:rat][n:not(ubprop".iii1"(lcl,x0,y0))] +*iii1 n@t6:=th5"l.imp"(lrt(ksi,y0),moreis(x0,y0),n):lrt(ksi,y0) t7:=th6"l.imp"(lrt(ksi,y0),moreis(x0,y0),n):not(moreis(x0,y0)) t8:=satz81j(x0,y0,t7):less(x0,y0) t9:=p1:p -iii1 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 s@[x0:rat][i:in(x0,s)][y0:rat][n:not(in(y0,s))] +*iii1 n@t10:=andi(cutprop1a,cutprop1b,nonemptyi(rat,s,x0,i),th1"l.all"(rat,[x:rat]in(x,s),y0,n)):cutprop1 -iii1 s@[n:[x:rat]not(max(s,x))] +*iii1 n@t11:=th5"l.some"(rat,[x:rat]max(s,x),n):cutprop3 -iii1 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))] cut1:=and3i(cutprop1,cutprop2,cutprop3,t10".iii1"(x0,i,y0,n),l,t11".iii1"(m)):cutprop(s) +rp ksi@[eta:cut] is:=is"e"(cut,ksi,eta):'prop' nis:=not(is(ksi,eta)):'prop' [i:is(ksi,eta)] ise:=isini(set(rat),[x:set(rat)]cutprop(x),ksi,eta,i):is"e"(set(rat),lcl(ksi),lcl(eta)) [x0:rat][lx:lrt(ksi,x0)] ise1:=issete1(rat,lcl(ksi),lcl(eta),ise,x0,lx):lrt(eta,x0) eta@[i:is"e"(set(rat),lcl(ksi),lcl(eta))] isi:=isine(set(rat),[x:set(rat)]cutprop(x),ksi,eta,i):is(ksi,eta) eta@[l:[x:rat][t:lrt(ksi,x)]lrt(eta,x)][k:[x:rat][t:lrt(eta,x)]lrt(ksi,x)] isi1:=isi(isseti(rat,lcl(ksi),lcl(eta),l,k)):is(ksi,eta) @[s:set(rat)][cs:cutprop(s)] cutof:=out(set(rat),[x:set(rat)]cutprop(x),s,cs):cut [x0:rat][i:in(x0,s)] 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) x0@[lx:lrt(cutof(s,cs),x0)] ini:=issete2(rat,s,lcl(cutof(s,cs)),isinout(set(rat),[x:set(rat)]cutprop(x),s,cs),x0,lx):in(x0,s) 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)] 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)) @[p:[x:cut]'prop'] all:=all"l"(cut,p):'prop' some:=some"l"(cut,p):'prop' one:=one"e"(cut,p):'prop' ksi@satz116:=refis(cut,ksi):is(ksi,ksi) eta@[i:is(ksi,eta)] satz117:=symis(cut,ksi,eta,i):is(eta,ksi) eta@[zeta:cut][i:is(ksi,eta)][j:is(eta,zeta)] satz118:=tris(cut,ksi,eta,zeta,i,j):is(ksi,zeta) +1119 @[x0:rat][y0:rat][m:more(x0,y0)] t1:=ec3e23(is"rt"(x0,y0),more(x0,y0),less(x0,y0),satz81b(x0,y0),m):not(less(x0,y0)) -1119 ksi@[x0:rat][ux:urt(ksi,x0)][y0:rat][m:more(y0,x0)] 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) y0@[l:less(x0,y0)] satz119a:=satz119(satz83(x0,y0,l)):urt(ksi,y0) +1120 @[x0:rat][y0:rat][l:less(x0,y0)] t1:=ec3e32(is"rt"(x0,y0),more(x0,y0),less(x0,y0),satz81b(x0,y0),l):not(more(x0,y0)) -1120 ksi@[x0:rat][lx:lrt(ksi,x0)][y0:rat][l:less(y0,x0)] 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) y0@[m:more(x0,y0)] satz120a:=satz120(satz82(x0,y0,m)):lrt(ksi,y0) -rp 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))] +*iii1 n@t12:=i:[u:less(y0,x0)]in(y0,s) t13:=th3"l.imp"(less(y0,x0),in(y0,s),n,t12):not(less(y0,x0)) t14:=satz81f(y0,x0,t13):moreis(y0,x0) -iii1 n@[k:is(y0,x0)] +*iii1 k@t15:=isp1(rat,[x:rat]in(x,s),x0,y0,j,k):in(y0,s) n@t16:=th3"l.imp"(is(y0,x0),in(y0,s),n,[t:is(y0,x0)]t15(t)):nis(y0,x0) t17:=ore1(more(y0,x0),is(y0,x0),t14,t16):more(y0,x0) t18:=satz82(y0,x0,t17):less(x0,y0) i@t19:=[x:rat][t:in(x,s)][y:rat][u:not(in(y,s))]t18(x,t,y,u):cutprop2 -iii1 s@[s1:[x:rat][t:in(x,s)]some([y:rat]and(in(y,s),more(y,x)))][x0:rat][i:in(x0,s)] +*iii1 i@t20:=s1:some([y:rat]and(in(y,s),more(y,x0))) -iii1 i@[y0:rat][a:and(in(y0,s),more(y0,x0))] +*iii1 a@t21:=ande1(in(y0,s),more(y0,x0),a):in(y0,s) t22:=ande2(in(y0,s),more(y0,x0),a):more(y0,x0) t23:=satz81g(y0,x0,t22):not(lessis(y0,x0)) t24:=th3"l.imp"(moreis(x0,y0),lessis(y0,x0),t23,[t:moreis(x0,y0)]satz84(x0,y0,t)):not(moreis(x0,y0)) t25:=th4"l.imp"(in(y0,s),moreis(x0,y0),t21,t24):not(ubprop(x0,y0)) t26:=th1"l.all"(rat,[y:rat]ubprop(x0,y),y0,t25):not(ub(s,x0)) t27:=th1"l.and"(ub(s,x0),in(x0,s),t26):not(max(s,x0)) 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)) -iii1 x0@[n:not(in(x0,s))] +*iii1 n@t29:=th2"l.and"(ub(x0),in(x0,s),n):not(max(s,x0)) 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)) s1@t31:=t11([x:rat]t30(x)):cutprop3 -iii1 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)))] cut2:=and3i(cutprop1,cutprop2,cutprop3,t10".iii1"(x0,i,y0,n),t19".iii1"(j),t31".iii1"(s1)):cutprop(s) +*rp eta@more:=some"rt"([x:rat]and(lrt(ksi,x),urt(eta,x))):'prop' [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))] +iii2 t1:=ande1(lrt(ksi,x0),urt(eta,x0),a):lrt(ksi,x0) t2:=ande2(lrt(ksi,x0),urt(eta,x0),a):urt(eta,x0) t3:=p1:p -iii2 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 eta@less:=some"rt"([x:rat]and(urt(ksi,x),lrt(eta,x))):'prop' [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))] +*iii2 a@t4:=ande1(urt(ksi,x0),lrt(eta,x0),a):urt(ksi,x0) t5:=ande2(urt(ksi,x0),lrt(eta,x0),a):lrt(eta,x0) t6:=p1:p -iii2 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 eta@[m:more(ksi,eta)] +2121 [x0:rat][lx:lrt(ksi,x0)][ux:urt(eta,x0)] t1:=andi(urt(eta,x0),lrt(ksi,x0),ux,lx):and(urt(eta,x0),lrt(ksi,x0)) t2:=somei(rat,[x:rat]and(urt(eta,x),lrt(ksi,x)),x0,t1):less(eta,ksi) -2121 satz121:=moreapp(m,less(eta,ksi),[x:rat][t:lrt(ksi,x)][u:urt(eta,x)]t2".2121"(x,t,u)):less(eta,ksi) eta@[l:less(ksi,eta)] +2122 [x0:rat][ux:urt(ksi,x0)][lx:lrt(eta,x0)] t1:=andi(lrt(eta,x0),urt(ksi,x0),lx,ux):and(lrt(eta,x0),urt(ksi,x0)) t2:=somei(rat,[x:rat]and(lrt(eta,x),urt(ksi,x)),x0,t1):more(eta,ksi) -2122 satz122:=lessapp(l,more(eta,ksi),[x:rat][t:urt(ksi,x)][u:lrt(eta,x)]t2".2122"(x,t,u)):more(eta,ksi) +2123 eta@[m:more(ksi,eta)][x0:rat][lx:lrt(ksi,x0)][ux:urt(eta,x0)] t1:=th3"st.isset"(rat,lcl(ksi),lcl(eta),x0,lx,ux):not(is"e"(set(rat),lcl(ksi),lcl(eta))) 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)) 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)) eta@t4:=th2"l.ec"(is(ksi,eta),more(ksi,eta),[t:more(ksi,eta)]t3(t)):ec(is(ksi,eta),more(ksi,eta)) [l:less(ksi,eta)][x0:rat][ux:urt(ksi,x0)][lx:lrt(eta,x0)] t5:=th4"st.isset"(rat,lcl(eta),lcl(ksi),x0,lx,ux):not(is"e"(set(rat),lcl(ksi),lcl(eta))) 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)) 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)) eta@t8:=th1"l.ec"(less(ksi,eta),is(ksi,eta),[t:less(ksi,eta)]t7(t)):ec(less(ksi,eta),is(ksi,eta)) 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)] t9:=cutapp2a(ksi,x0,lx,y0,uy):less"rt"(x0,y0) t10:=cutapp2b(eta,y0,ly,x0,ux):more"rt"(x0,y0) 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 ux@t12:=lessapp(ksi,eta,l,con,[x:rat][t:urt(ksi,x)][u:lrt(eta,x)]t11(x,t,u)):con l@t13:=moreapp(ksi,eta,m,con,[x:rat][t:lrt(ksi,x)][u:urt(eta,x)]t12(x,t,u)):con m@t14:=[t:less(ksi,eta)]t13(t):not(less(ksi,eta)) eta@t15:=th1"l.ec"(more(ksi,eta),less(ksi,eta),[t:more(ksi,eta)]t14(t)):ec(more(ksi,eta),less(ksi,eta)) 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)) [i:is(ksi,eta)] t16:=or3i1(is(ksi,eta),more(ksi,eta),less(ksi,eta),i):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta)) eta@[n:nis(ksi,eta)] 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))) t18:=th5"st.isset"(rat,lcl(ksi),lcl(eta),t17):or(more(ksi,eta),more(eta,ksi)) 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)) t20:=th7"l.or3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),t19):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta)) 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)) -2123 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)) satz123a:=orec3e1(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta)) satz123b:=orec3e2(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123):ec3(is(ksi,eta),more(ksi,eta),less(ksi,eta)) moreis:=or(more(ksi,eta),is(ksi,eta)):'prop' lessis:=or(less(ksi,eta),is(ksi,eta)):'prop' [m:moreis(ksi,eta)] 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) eta@[l:lessis(ksi,eta)] 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) zeta@[i:is(ksi,eta)][m:more(ksi,zeta)] ismore1:=isp(cut,[u:cut]more(u,zeta),ksi,eta,m,i):more(eta,zeta) i@[m:more(zeta,ksi)] ismore2:=isp(cut,[u:cut]more(zeta,u),ksi,eta,m,i):more(zeta,eta) i@[l:less(ksi,zeta)] isless1:=isp(cut,[u:cut]less(u,zeta),ksi,eta,l,i):less(eta,zeta) i@[l:less(zeta,ksi)] isless2:=isp(cut,[u:cut]less(zeta,u),ksi,eta,l,i):less(zeta,eta) i@[m:moreis(ksi,zeta)] ismoreis1:=isp(cut,[u:cut]moreis(u,zeta),ksi,eta,m,i):moreis(eta,zeta) i@[m:moreis(zeta,ksi)] ismoreis2:=isp(cut,[u:cut]moreis(zeta,u),ksi,eta,m,i):moreis(zeta,eta) i@[l:lessis(ksi,zeta)] islessis1:=isp(cut,[u:cut]lessis(u,zeta),ksi,eta,l,i):lessis(eta,zeta) i@[l:lessis(zeta,ksi)] islessis2:=isp(cut,[u:cut]lessis(zeta,u),ksi,eta,l,i):lessis(zeta,eta) eta@[i:is(ksi,eta)] moreisi2:=ori2(more(ksi,eta),is(ksi,eta),i):moreis(ksi,eta) lessisi2:=ori2(less(ksi,eta),is(ksi,eta),i):lessis(ksi,eta) eta@[m:more(ksi,eta)] moreisi1:=ori1(more(ksi,eta),is(ksi,eta),m):moreis(ksi,eta) eta@[l:less(ksi,eta)] lessisi1:=ori1(less(ksi,eta),is(ksi,eta),l):lessis(ksi,eta) zeta@[upsilon:cut][i:is(ksi,eta)][j:is(zeta,upsilon)][m:more(ksi,zeta)] ismore12:=ismore2(zeta,upsilon,eta,j,ismore1(ksi,eta,zeta,i,m)):more(eta,upsilon) j@[l:less(ksi,zeta)] isless12:=isless2(zeta,upsilon,eta,j,isless1(ksi,eta,zeta,i,l)):less(eta,upsilon) j@[m:moreis(ksi,zeta)] ismoreis12:=ismoreis2(zeta,upsilon,eta,j,ismoreis1(ksi,eta,zeta,i,m)):moreis(eta,upsilon) j@[l:lessis(ksi,zeta)] islessis12:=islessis2(zeta,upsilon,eta,j,islessis1(ksi,eta,zeta,i,l)):lessis(eta,upsilon) eta@[m:moreis(ksi,eta)] 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)) eta@[l:lessis(ksi,eta)] satz123d:=th9"l.ec3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123b,l):not(more(ksi,eta)) eta@[n:not(more(ksi,eta))] satz123e:=th2"l.or3"(is(ksi,eta),more(ksi,eta),less(ksi,eta),satz123a,n):lessis(ksi,eta) eta@[n:not(less(ksi,eta))] 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) eta@[m:more(ksi,eta)] 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)) eta@[l:less(ksi,eta)] 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)) eta@[n:not(moreis(ksi,eta))] 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) eta@[n:not(lessis(ksi,eta))] 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) zeta@[l:less(ksi,eta)][k:less(eta,zeta)] +2126 [x0:rat][ux:urt(ksi,x0)][lx:lrt(eta,x0)][y0:rat][uy:urt(eta,y0)][ly:lrt(zeta,y0)] t1:=cutapp2a(eta,x0,lx,y0,uy):less"rt"(x0,y0) t2:=satz119a(ksi,x0,ux,y0,t1):urt(ksi,y0) t3:=andi(urt(ksi,y0),lrt(zeta,y0),t2,ly):and(urt(ksi,y0),lrt(zeta,y0)) t4:=somei(rat,[x:rat]and(urt(ksi,x),lrt(zeta,x)),y0,t3):less(ksi,zeta) 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) -2126 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) trless:=satz126:less(ksi,zeta) zeta@[m:more(ksi,eta)][n:more(eta,zeta)] trmore:=satz122(zeta,ksi,satz126(zeta,eta,ksi,satz121(eta,zeta,n),satz121(ksi,eta,m))):more(ksi,zeta) zeta@[l:lessis(ksi,eta)][k:less(eta,zeta)] 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) zeta@[l:less(ksi,eta)][k:lessis(eta,zeta)] 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) zeta@[m:moreis(ksi,eta)][n:more(eta,zeta)] satz127c:=satz122(zeta,ksi,satz127b(zeta,eta,ksi,satz121(eta,zeta,n),satz124(ksi,eta,m))):more(ksi,zeta) zeta@[m:more(ksi,eta)][n:moreis(eta,zeta)] satz127d:=satz122(zeta,ksi,satz127a(zeta,eta,ksi,satz124(eta,zeta,n),satz121(ksi,eta,m))):more(ksi,zeta) zeta@[l:lessis(ksi,eta)][k:lessis(eta,zeta)] +2128 [i:is(ksi,eta)][j:is(eta,zeta)] t1:=lessisi2(ksi,zeta,tris(cut,ksi,eta,zeta,i,j)):lessis(ksi,zeta) i@[j:less(eta,zeta)] t2:=lessisi1(ksi,zeta,satz127a(l,j)):lessis(ksi,zeta) 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) k@[j:less(ksi,eta)] t4:=lessisi1(ksi,zeta,satz127b(j,k)):lessis(ksi,zeta) -2128 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) trlessis:=satz128:lessis(ksi,zeta) zeta@[m:moreis(ksi,eta)][n:moreis(eta,zeta)] trmoreis:=satz125(zeta,ksi,satz128(zeta,eta,ksi,satz124(eta,zeta,n),satz124(ksi,eta,m))):moreis(ksi,zeta) eta@[z0:rat][x0:rat][y0:rat] sumprop1:=and3(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0))):'prop' z0@sumprop:=some"rt"([x:rat]some"rt"([y:rat]sumprop1(z0,x,y))):'prop' eta@sum:=setof(rat,[z:rat]sumprop(z)):set(rat) x0@[lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,pl(x0,y0))] +iii3 t1:=and3i(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0)),lx,ly,i):sumprop1(z0,x0,y0) t2:=somei(rat,[y:rat]sumprop1(z0,x0,y),y0,t1):some"rt"([y:rat]sumprop1(z0,x0,y)) t3:=somei(rat,[x:rat]some"rt"([y:rat]sumprop1(z0,x,y)),x0,t2):sumprop(z0) -iii3 sum1:=estii(rat,[z:rat]sumprop(z),z0,t3".iii3"):in(z0,sum) 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] +*iii3 p1@t4:=estie(rat,[z:rat]sumprop(z),z0,i):sumprop(z0) -iii3 p1@[x0:rat][px:some"rt"([y:rat]sumprop1(z0,x0,y))][y0:rat][py:sumprop1(z0,x0,y0)] +*iii3 py@t5:=and3e1(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0)),py):lrt(ksi,x0) t6:=and3e2(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0)),py):lrt(eta,y0) t7:=and3e3(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,pl(x0,y0)),py):is"rt"(z0,pl(x0,y0)) t8:=p1:p px@t9:=someapp(rat,[y:rat]sumprop1(z0,x0,y),px,p,[y:rat][t:sumprop1(z0,x0,y)]t8(y,t)):p -iii3 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 eta@[x1:rat][ux:urt(ksi,x1)][y1:rat][uy:urt(eta,y1)] +3129 [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))] t1:=cutapp2a(ksi,x0,lx,x1,ux):less"rt"(x0,x1) t2:=cutapp2a(eta,y0,ly,y1,uy):less"rt"(y0,y1) 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)) 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)) 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)) -3129 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)) +*3129 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))] t6:=isless2"rt"(u0,pl(x0,y0),z0,j,l):less"rt"(z0,pl(x0,y0)) z1:=ov(z0,pl(x0,y0)):rat 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))) t8:=satz106c(z1,1rt,pl(x0,y0),t7):less"rt"(z1,1rt) t9:=isless2"rt"(ts(x0,1rt),x0,ts(x0,z1),example1a(x0),satz105f(z1,1rt,x0,t8)):less"rt"(ts(x0,z1),x0) t10:=isless2"rt"(ts(y0,1rt),y0,ts(y0,z1),example1a(y0),satz105f(z1,1rt,y0,t8)):less"rt"(ts(y0,z1),y0) t11:=satz120(ksi,x0,lx,ts(x0,z1),t9):lrt(ksi,ts(x0,z1)) t12:=satz120(eta,y0,ly,ts(y0,z1),t10):lrt(eta,ts(y0,z1)) 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) t14:=symis(rat,pl(ts(x0,z1),ts(y0,z1)),z0,t13):is"rt"(z0,pl(ts(x0,z1),ts(y0,z1))) t15:=sum1(ksi,eta,z0,ts(x0,z1),t11,ts(y0,z1),t12,t14):in(z0,sum) 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) 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)] t17:=sum1(ksi,eta,pl(x1,y0),x1,lx1,y0,ly,refis(rat,pl(x1,y0))):in(pl(x1,y0),sum) t18:=satz96a(x1,x0,y0,satz83(x0,x1,l)):more"rt"(pl(x1,y0),pl(x0,y0)) t19:=ismore2"rt"(pl(x0,y0),z0,pl(x1,y0),symis(rat,z0,pl(x0,y0),j),t18):more"rt"(pl(x1,y0),z0) 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)) 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))) 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))) 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))) 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)] 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) ux@t25:=cutapp1b(eta,cutprop(sum),[y:rat][t:urt(eta,y)]t24(y,t)):cutprop(sum) ly@t26:=cutapp1b(ksi,cutprop(sum),[x:rat][t:urt(ksi,x)]t25(x,t)):cutprop(sum) lx@t27:=cutapp1a(eta,cutprop(sum),[y:rat][t:lrt(eta,y)]t26(y,t)):cutprop(sum) -3129 eta@satz129:=cutapp1a(ksi,cutprop(sum),[x:rat][t:lrt(ksi,x)]t27".3129"(x,t)):cutprop(sum) pl:=cutof(sum,satz129):cut [z0:rat][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,pl"rt"(x0,y0))] lrtpl:=ine(sum,satz129,z0,sum1(z0,x0,lx,y0,ly,i)):lrt(pl(ksi,eta),z0) eta@[z0:rat][x0:rat][ux:urt(ksi,x0)][y0:rat][uy:urt(eta,y0)][i:is"rt"(z0,pl"rt"(x0,y0))] +*iii3 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)) -iii3 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) 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] +*iii3 p1@t11:=ini(sum,satz129,z0,lz):in(z0,sum) -iii3 p1@plapp:=sumapp(z0,t11".iii3",p,p1):p zeta@[i:is(ksi,eta)] ispl1:=isf(cut,cut,[u:cut]pl(u,zeta),ksi,eta,i):is(pl(ksi,zeta),pl(eta,zeta)) ispl2:=isf(cut,cut,[u:cut]pl(zeta,u),ksi,eta,i):is(pl(zeta,ksi),pl(zeta,eta)) upsilon@[i:is(ksi,eta)][j:is(zeta,upsilon)] 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)) +3130 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))] t1:=tris(rat,z0,pl"rt"(x0,y0),pl"rt"(y0,x0),i,compl(x0,y0)):is"rt"(z0,pl"rt"(y0,x0)) t2:=lrtpl(eta,ksi,z0,y0,ly,x0,lx,t1):lrt(pl(eta,ksi),z0) 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) -3130 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)) compl:=satz130:is(pl(ksi,eta),pl(eta,ksi)) +3131 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))] 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))) 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)) t3:=lrtpl(ksi,pl(eta,zeta),u0,x0,lx,pl"rt"(y0,z0),t2,t1):lrt(pl(ksi,pl(eta,zeta)),u0) 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) 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) 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))] 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)) 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)) t8:=lrtpl(pl(ksi,eta),zeta,u0,pl"rt"(x0,y0),t7,z0,lz,t6):lrt(pl(pl(ksi,eta),zeta),u0) 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) 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) -3131 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))) asspl1:=satz131:is(pl(pl(ksi,eta),zeta),pl(ksi,pl(eta,zeta))) 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)) ksi@[a0:rat] +3132 [x0:rat][y0:rat] prop1:=and(lrt(ksi,x0),urt(ksi,y0)):'prop' [p:prop1] t1:=cutapp2b(x0,ande1(lrt(ksi,x0),urt(ksi,y0),p),y0,ande2(lrt(ksi,x0),urt(ksi,y0),p)):more"rt"(y0,x0) prop2:=is"rt"(mn(y0,x0,t1),a0):'prop' y0@prop3:=and(prop1,[t:prop1]prop2(t)):'prop' a0@prop4:=some"rt"([x:rat]some"rt"([y:rat]prop3(x,y))):'prop' x0@[lx:lrt(ksi,x0)][y0:rat][uy:urt(ksi,y0)] t2:=cutapp2b(x0,lx,y0,uy):more"rt"(y0,x0) [n:nat][m:more"rt"(ts(rtofn(n),a0),mn(y0,x0,t2))] 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))) 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) t5:=satz119(y0,uy,pl"rt"(x0,ts(rtofn(n),a0)),t4):urt(pl"rt"(x0,ts(rtofn(n),a0))) 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)))) 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)))) [u:nat][m:min"n"([x:nat]urt(pl"rt"(x0,ts(rtofn(x),a0))),u)] 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) 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))) [i:is"n"(u,1)] u0:=pl"rt"(x0,a0):rat 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) t11:=isp(rat,[x:rat]urt(pl"rt"(x0,x)),ts(rtofn(u),a0),a0,t9,t10):urt(ksi,u0) t12:=andi(lrt(ksi,x0),urt(ksi,u0),lx,t11):prop1(x0,u0) [p:prop1(x0,u0)] 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) i@t14:=andi(prop1(x0,u0),[t:prop1(x0,u0)]prop2(x0,u0,t),t12,[t:prop1(x0,u0)]t13(t)):prop3(x0,u0) t15:=somei(rat,[y:rat]prop3(x0,y),u0,t14):some"rt"([y:rat]prop3(x0,y)) t16:=somei(rat,[x:rat]some"rt"([y:rat]prop3(x,y)),x0,t15):prop4 m@[o:more"n"(u,1)] 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) um10:=mn(rtofn(u),1rt,t17):rat t18:=satz112g(rtofn(u),natrti(u),1rt,natrti(1),t17):natrt(um10) um1:=nofrt(um10,t18):nat v0:=pl"rt"(x0,ts(um10,a0)):rat w0:=pl"rt"(x0,ts(rtofn(u),a0)):rat t19:=isless2"rt"(pl"rt"(um10,1rt),rtofn(u),um10,satz101e(rtofn(u),1rt,t17),satz94a(um10,1rt)):less"rt"(um10,rtofn(u)) 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)) t21:=satz111c(um1,u,t20):less"n"(um1,u) 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)) 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,t8)):lrt(pl"rt"(x0,ts(rtofn(um1),a0))) t24:=isp1(rat,[x:rat]lrt(pl"rt"(x0,ts(x,a0))),rtofn(um1),um10,t23,isrtn1(um10,t18)):lrt(ksi,v0) t25:=andi(lrt(ksi,v0),urt(ksi,w0),t24,t9):prop1(v0,w0) 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)) 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) [p:prop1(v0,w0)] t28:=symis(rat,a0,mn(w0,v0,t1(v0,w0,p)),satz101g(w0,v0,a0,t1(v0,w0,p),t27)):prop2(v0,w0,p) o@t29:=andi(prop1(v0,w0),[t:prop1(v0,w0)]prop2(v0,w0,t),t25,[t:prop1(v0,w0)]t28(t)):prop3(v0,w0) t30:=somei(rat,[y:rat]prop3(v0,y),w0,t29):some"rt"([y:rat]prop3(v0,y)) t31:=somei(rat,[x:rat]some"rt"([y:rat]prop3(x,y)),v0,t30):prop4 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 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 lx@t35:=cutapp1b(prop4,[y:rat][t:urt(ksi,y)]t34(y,t)):prop4 -3132 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)))) 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] +*3132 p1@[x0:rat][s:some"rt"([y:rat]prop3(a0,x0,y))][y0:rat][p3:prop3(a0,x0,y0)] t36:=ande1(prop1(a0,x0,y0),[t:prop1(a0,x0,y0)]prop2(a0,x0,y0,t),p3):prop1(a0,x0,y0) t37:=ande2"l.r"(prop1(a0,x0,y0),[t:prop1(a0,x0,y0)]prop2(a0,x0,y0,t),p3):prop2(a0,x0,y0,t36) t38:=ande1(lrt(ksi,x0),urt(ksi,y0),t36):lrt(ksi,x0) t39:=ande2(lrt(ksi,x0),urt(ksi,y0),t36):urt(ksi,y0) 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))) 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) t42:=p1:p s@t43:=someapp(rat,[y:rat]prop3(a0,x0,y),s,p,[y:rat][t:prop3(a0,x0,y)]t42(y,t)):p -3132 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 +3133 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)] 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)) t2:=lrtpl(ksi,eta,u0,x0,lx,y0,ly,t1):lrt(pl(ksi,eta),u0) t3:=andi(lrt(pl(ksi,eta),u0),urt(ksi,u0),t2,uu):and(lrt(pl(ksi,eta),u0),urt(ksi,u0)) t4:=somei(rat,[x:rat]and(lrt(pl(ksi,eta),x),urt(ksi,x)),u0,t3):more(pl(ksi,eta),ksi) 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) -3133 eta@satz133:=cutapp1a(eta,more(pl(ksi,eta),ksi),[x:rat][t:lrt(eta,x)]t5".3133"(x,t)):more(pl(ksi,eta),ksi) satz133a:=satz121(pl(ksi,eta),ksi,satz133):less(ksi,pl(ksi,eta)) zeta@[m:more(ksi,eta)] +3134 [y0:rat][ly:lrt(ksi,y0)][uy:urt(eta,y0)][x0:rat][lx:lrt(ksi,x0)][l:less"rt"(y0,x0)] t1:=satz119a(eta,y0,uy,x0,l):urt(eta,x0) t2:=satz83(y0,x0,l):more"rt"(x0,y0) [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))] 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)) 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)) t5:=lrtpl(ksi,zeta,pl"rt"(y0,z0),x0,lx,u0,lu,t4):lrt(pl(ksi,zeta),pl"rt"(y0,z0)) 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)) 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))) 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)) 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)) 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)) -3134 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)) zeta@[m:more(ksi,eta)] satz135a:=satz134(m):more(pl(ksi,zeta),pl(eta,zeta)) zeta@[i:is(ksi,eta)] satz135b:=ispl1(ksi,eta,zeta,i):is(pl(ksi,zeta),pl(eta,zeta)) zeta@[l:less(ksi,eta)] satz135c:=satz121(pl(eta,zeta),pl(ksi,zeta),satz134(eta,ksi,zeta,satz122(ksi,eta,l))):less(pl(ksi,zeta),pl(eta,zeta)) 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)) i@satz135e:=ispl2(ksi,eta,zeta,i):is(pl(zeta,ksi),pl(zeta,eta)) 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)) upsilon@[i:is(ksi,eta)][m:more(zeta,upsilon)] 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)) 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)) i@[l:less(zeta,upsilon)] 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)) 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)) +3136 zeta@t1:=satz123a(ksi,eta):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta)) 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))) -3136 zeta@[m:more(pl(ksi,zeta),pl(eta,zeta))] 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) zeta@[i:is(pl(ksi,zeta),pl(eta,zeta))] 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) zeta@[l:less(pl(ksi,zeta),pl(eta,zeta))] 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) zeta@[m:more(pl(zeta,ksi),pl(zeta,eta))] 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) zeta@[i:is(pl(zeta,ksi),pl(zeta,eta))] 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) zeta@[l:less(pl(zeta,ksi),pl(zeta,eta))] 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) upsilon@[m:more(ksi,eta)][n:more(zeta,upsilon)] +3137 t1:=satz134(ksi,eta,zeta,m):more(pl(ksi,zeta),pl(eta,zeta)) 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)) -3137 satz137:=trmore(pl(ksi,zeta),pl(eta,zeta),pl(eta,upsilon),t1".3137",t2".3137"):more(pl(ksi,zeta),pl(eta,upsilon)) upsilon@[l:less(ksi,eta)][k:less(zeta,upsilon)] 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)) upsilon@[m:moreis(ksi,eta)][n:more(zeta,upsilon)] 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)) upsilon@[m:more(ksi,eta)][n:moreis(zeta,upsilon)] 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)) upsilon@[l:lessis(ksi,eta)][k:less(zeta,upsilon)] 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)) upsilon@[l:less(ksi,eta)][k:lessis(zeta,upsilon)] 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)) upsilon@[m:moreis(ksi,eta)][n:moreis(zeta,upsilon)] +3139 [i:is(ksi,eta)][j:is(zeta,upsilon)] t1:=moreisi2(pl(ksi,zeta),pl(eta,upsilon),ispl12(ksi,eta,zeta,upsilon,i,j)):moreis(pl(ksi,zeta),pl(eta,upsilon)) i@[o:more(zeta,upsilon)] t2:=moreisi1(pl(ksi,zeta),pl(eta,upsilon),satz138a(m,o)):moreis(pl(ksi,zeta),pl(eta,upsilon)) 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)) n@[o:more(ksi,eta)] t4:=moreisi1(pl(ksi,zeta),pl(eta,upsilon),satz138b(o,n)):moreis(pl(ksi,zeta),pl(eta,upsilon)) -3139 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)) upsilon@[l:lessis(ksi,eta)][k:lessis(zeta,upsilon)] 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)) eta@[l:lessis(ksi,eta)] +3140 [phi:cut][i:is(pl(eta,phi),ksi)] t1:=ismore1(pl(eta,phi),ksi,eta,i,satz133(eta,phi)):more(ksi,eta) 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) -3140 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))) eta@[phi:cut][psi:cut] +*3140 psi@[m:more(phi,psi)] t3:=satz135d(phi,psi,eta,m):more(pl(eta,phi),pl(eta,psi)) 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)) psi@[l:less(phi,psi)] t5:=satz135f(phi,psi,eta,l):less(pl(eta,phi),pl(eta,psi)) 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)) psi@[n:nis(phi,psi)] t7:=th1"l.or3"(is(phi,psi),more(phi,psi),less(phi,psi),satz123a(phi,psi),n):or(more(phi,psi),less(phi,psi)) 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)) -3140 psi@[i:is(pl(eta,phi),ksi)][j:is(pl(eta,psi),ksi)] 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) eta@[z0:rat][x0:rat][y0:rat] diffprop1:=and(more"rt"(x0,y0),[t:more"rt"(x0,y0)]is"rt"(z0,mn(x0,y0,t))):'prop' diffprop2:=and3(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0)):'prop' z0@diffprop:=some"rt"([x:rat]some"rt"([y:rat]diffprop2(z0,x,y))):'prop' eta@diff:=setof(rat,[z:rat]diffprop(z)):set(rat) x0@[lx:lrt(ksi,x0)][y0:rat][uy:urt(eta,y0)][m:more"rt"(x0,y0)][i:is"rt"(z0,mn(x0,y0,m))] +*iii3 i@[m1:more"rt"(x0,y0)] 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)) 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) t13:=and3i(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0),lx,uy,t12):diffprop2(z0,x0,y0) t14:=somei(rat,[y:rat]diffprop2(z0,x0,y),y0,t13):some"rt"([y:rat]diffprop2(z0,x0,y)) t15:=somei(rat,[x:rat]some"rt"([y:rat]diffprop2(z0,x,y)),x0,t14):diffprop(z0) -iii3 i@diff1:=estii(rat,[z:rat]diffprop(z),z0,t15".iii3"):in(z0,diff) 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] +*iii3 p1@t16:=estie(rat,[z:rat]diffprop(z),z0,i):diffprop(z0) -iii3 p1@[x0:rat][px:some"rt"([y:rat]diffprop2(z0,x0,y))][y0:rat][py:diffprop2(z0,x0,y0)] +*iii3 py@t17:=and3e1(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0),py):lrt(ksi,x0) t18:=and3e2(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0),py):urt(eta,y0) t19:=and3e3(lrt(ksi,x0),urt(eta,y0),diffprop1(z0,x0,y0),py):diffprop1(z0,x0,y0) t20:=ande1(more"rt"(x0,y0),[t:more"rt"(x0,y0)]is"rt"(z0,mn(x0,y0,t)),t19):more"rt"(x0,y0) 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)) t22:=p1:p px@t23:=someapp(rat,[y:rat]diffprop2(z0,x0,y),px,p,[y:rat][t:diffprop2(z0,x0,y)]t22(y,t)):p -iii3 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 eta@[m:more(ksi,eta)] +*3140 m"rp"@[y0:rat][ly:lrt(ksi,y0)][uy:urt(eta,y0)][x0:rat][lx:lrt(ksi,x0)][l:less"rt"(y0,x0)] t9:=t2"rp.3134"(eta,m,y0,ly,uy,x0,lx,l):more"rt"(x0,y0) t10:=diff1(mn(x0,y0,t9),x0,lx,y0,uy,t9,refis(rat,mn(x0,y0,t9))):in(mn(x0,y0,t9),diff) 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))] 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) t12:=trless"rt"(z0,x0,x1,t11,cutapp2a(ksi,x0,lx,x1,ux)):less"rt"(z0,x1) t13:=ec3e31(is"rt"(z0,x1),more"rt"(z0,x1),less"rt"(z0,x1),satz81b(z0,x1),t12):nis"rt"(z0,x1) 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) 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)) 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))] 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) x2:=pl"rt"(u0,y0):rat t17:=isless2"rt"(pl"rt"(z0,y0),x0,x2,t16,satz96c(u0,z0,y0,l)):less"rt"(x2,x0) t18:=satz120(ksi,x0,lx,x2,t17):lrt(ksi,x2) t19:=ismore1"rt"(pl"rt"(y0,u0),pl"rt"(u0,y0),y0,compl"rt"(y0,u0),satz94(y0,u0)):more"rt"(x2,y0) t20:=satz101g(x2,y0,u0,t19,compl"rt"(y0,u0)):is"rt"(u0,mn(x2,y0,t19)) t21:=diff1(u0,x2,t18,y0,uy,t19,t20):in(u0,diff) 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) 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)] t23:=satz83(x0,x3,l):more"rt"(x3,x0) t24:=trmore"rt"(x3,x0,y0,t23,n):more"rt"(x3,y0) 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)) t26:=satz97a(mn(x3,y0,t24),mn(x0,y0,n),y0,t25):more"rt"(mn(x3,y0,t24),mn(x0,y0,n)) 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) t28:=diff1(mn(x3,y0,t24),x3,lx3,y0,uy,t24,refis(rat,mn(x3,y0,t24))):in(mn(x3,y0,t24),diff) 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)) 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))) 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))) 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))) 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)] 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) l@t34:=cutapp1b(ksi,cutprop(diff),[x:rat][t:urt(ksi,x)]t33(x,t)):cutprop(diff) 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) -3140 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) +*3140 m"rp"@chi:=cutof(diff,satz140h):cut [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))] t36:=cutapp2b(eta,y1,ly,y0,uy):more"rt"(y0,y1) 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) 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) 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)) 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) t41:=satz120(ksi,x0,lx,z0,t40):lrt(ksi,z0) 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) 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) m"rp"@[y0:rat][ly:lrt(ksi,y0)][uy:urt(eta,y0)][x0:rat][lx:lrt(ksi,x0)][l:less"rt"(y0,x0)] t43:=satz83(y0,x0,l):more"rt"(x0,y0) [y1:rat][ly1:lrt(eta,y1)][y2:rat][uy2:urt(eta,y2)] t44:=cutapp2b(eta,y1,ly1,y2,uy2):more"rt"(y2,y1) [i:is"rt"(mn(y2,y1,t44),mn(x0,y0,t43))] t45:=cutapp2b(eta,y1,ly1,y0,uy):more"rt"(y0,y1) 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)) 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) t48:=ismore1"rt"(pl"rt"(y2,mn(y0,y1,t45)),x0,y2,t47,satz94(y2,mn(y0,y1,t45))):more"rt"(x0,y2) t49:=satz101g(x0,y2,mn(y0,y1,t45),t48,t47):is"rt"(mn(y0,y1,t45),mn(x0,y2,t48)) 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)) 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)) 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) 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) 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) ly@[ly0:lrt(eta,y0)][y1:rat][ly1:lrt(ksi,y1)][uy1:urt(eta,y1)] t55:=t54(y1,ly1,uy1):lrt(pl(eta,chi),y1) t56:=satz120(pl(eta,chi),y1,t55,y0,cutapp2a(eta,y0,ly0,y1,uy1)):lrt(pl(eta,chi),y0) 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) 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) 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) -3140 m@satz140a:=somei(cut,[a:cut]is(pl(eta,a),ksi),chi".3140",t58".3140"):some([a:cut]is(pl(eta,a),ksi)) +*3140 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)) -3140 m@satz140:=onei(cut,[a:cut]is(pl(eta,a),ksi),t59".3140",satz140a):one([a:cut]is(pl(eta,a),ksi)) mn:=ind(cut,[a:cut]is(pl(eta,a),ksi),satz140):cut satz140c:=oneax(cut,[a:cut]is(pl(eta,a),ksi),satz140):is(pl(eta,mn(ksi,eta,m)),ksi) satz140d:=symis(cut,pl(eta,mn(ksi,eta,m)),ksi,satz140c):is(ksi,pl(eta,mn(ksi,eta,m))) 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) satz140f:=symis(cut,pl(mn(ksi,eta,m),eta),ksi,satz140e):is(ksi,pl(mn(ksi,eta,m),eta)) eta@[phi:cut][m:more(ksi,eta)][i:is(pl(eta,phi),ksi)] satz140g:=satz140b(phi,mn(ksi,eta,m),i,satz140c(m)):is(phi,mn(ksi,eta,m)) upsilon@[m:more(ksi,zeta)][n:more(eta,upsilon)][i:is(ksi,eta)][j:is(zeta,upsilon)] +*3140 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) -3140 j@ismn12:=satz140g(eta,upsilon,mn(ksi,zeta,m),n,t60".3140"):is(mn(ksi,zeta,m),mn(eta,upsilon,n)) zeta@[m:more(ksi,zeta)][n:more(eta,zeta)][i:is(ksi,eta)] ismn1:=ismn12(zeta,m,n,i,refis(cut,zeta)):is(mn(ksi,zeta,m),mn(eta,zeta,n)) zeta@[m:more(zeta,ksi)][n:more(zeta,eta)][i:is(ksi,eta)] ismn2:=ismn12(zeta,zeta,ksi,eta,m,n,refis(cut,zeta),i):is(mn(zeta,ksi,m),mn(zeta,eta,n)) eta@[z0:rat][x0:rat][y0:rat] prodprop1:=and3(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0))):'prop' z0@prodprop:=some"rt"([x:rat]some"rt"([y:rat]prodprop1(z0,x,y))):'prop' eta@prod:=setof(rat,[z:rat]prodprop(z)):set(rat) x0@[lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,ts(x0,y0))] +iii4 t1:=and3i(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0)),lx,ly,i):prodprop1(z0,x0,y0) t2:=somei(rat,[y:rat]prodprop1(z0,x0,y),y0,t1):some"rt"([y:rat]prodprop1(z0,x0,y)) t3:=somei(rat,[x:rat]some"rt"([y:rat]prodprop1(z0,x,y)),x0,t2):prodprop(z0) -iii4 prod1:=estii(rat,[z:rat]prodprop(z),z0,t3".iii4"):in(z0,prod) 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] +*iii4 p1@t4:=estie(rat,[z:rat]prodprop(z),z0,i):prodprop(z0) -iii4 p1@[x0:rat][px:some"rt"([y:rat]prodprop1(z0,x0,y))][y0:rat][py:prodprop1(z0,x0,y0)] +*iii4 py@t5:=and3e1(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0)),py):lrt(ksi,x0) t6:=and3e2(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0)),py):lrt(eta,y0) t7:=and3e3(lrt(ksi,x0),lrt(eta,y0),is"rt"(z0,ts(x0,y0)),py):is"rt"(z0,ts(x0,y0)) t8:=p1:p px@t9:=someapp(rat,[y:rat]prodprop1(z0,x0,y),px,p,[y:rat][t:prodprop1(z0,x0,y)]t8(y,t)):p -iii4 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 eta@[x1:rat][ux:urt(ksi,x1)][y1:rat][uy:urt(eta,y1)] +4141 [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))] t1:=cutapp2a(ksi,x0,lx,x1,ux):less"rt"(x0,x1) t2:=cutapp2a(eta,y0,ly,y1,uy):less"rt"(y0,y1) 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)) 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)) 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)) -4141 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)) -rp @[x0:rat][y0:rat] +4141 v0:=ts(ov(1rt,y0),x0):rat 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) -4141 satz141b:=satz110g(x0,y0,v0".4141",t6".4141"):is(ts(ov(1rt,y0),x0),ov(x0,y0)) satz141c:=symis(rat,ts(ov(1rt,y0),x0),ov(x0,y0),satz141b):is"rt"(ov(x0,y0),ts(ov(1rt,y0),x0)) +*rp +*4141 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))] t7:=isless2"rt"(u0,ts(x0,y0),z0,j,l):less"rt"(z0,ts(x0,y0)) 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) 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) t10:=satz120(eta,y0,ly,ov(z0,x0),t9):lrt(eta,ov(z0,x0)) t11:=prod1(z0,x0,lx,ov(z0,x0),t10,satz110d(z0,x0)):in(z0,prod) 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) 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)] t13:=prod1(ts(x1,y0),x1,lx1,y0,ly,refis(rat,ts(x1,y0))):in(ts(x1,y0),prod) t14:=satz105a(x1,x0,y0,satz83(x0,x1,l)):more"rt"(ts(x1,y0),ts(x0,y0)) t15:=ismore2"rt"(ts(x0,y0),z0,ts(x1,y0),symis(rat,z0,ts(x0,y0),j),t14):more"rt"(ts(x1,y0),z0) 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)) 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))) 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))) 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))) 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)] 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) ux@t21:=cutapp1b(eta,cutprop(prod),[y:rat][t:urt(eta,y)]t20(y,t)):cutprop(prod) ly@t22:=cutapp1b(ksi,cutprop(prod),[x:rat][t:urt(ksi,x)]t21(x,t)):cutprop(prod) lx@t23:=cutapp1a(eta,cutprop(prod),[y:rat][t:lrt(eta,y)]t22(y,t)):cutprop(prod) -4141 eta@satz141:=cutapp1a(ksi,cutprop(prod),[x:rat][t:lrt(ksi,x)]t23".4141"(x,t)):cutprop(prod) ts:=cutof(prod,satz141):cut [z0:rat][x0:rat][lx:lrt(ksi,x0)][y0:rat][ly:lrt(eta,y0)][i:is"rt"(z0,ts"rt"(x0,y0))] lrtts:=ine(prod,satz141,z0,prod1(z0,x0,lx,y0,ly,i)):lrt(ts(ksi,eta),z0) eta@[z0:rat][x0:rat][ux:urt(ksi,x0)][y0:rat][uy:urt(eta,y0)][i:is"rt"(z0,ts"rt"(x0,y0))] +*iii4 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)) -iii4 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) 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] +*iii4 p1@t11:=ini(prod,satz141,z0,lz):in(z0,prod) -iii4 p1@tsapp:=prodapp(z0,t11".iii4",p,p1):p zeta@[i:is(ksi,eta)] ists1:=isf(cut,cut,[u:cut]ts(u,zeta),ksi,eta,i):is(ts(ksi,zeta),ts(eta,zeta)) ists2:=isf(cut,cut,[u:cut]ts(zeta,u),ksi,eta,i):is(ts(zeta,ksi),ts(zeta,eta)) upsilon@[i:is(ksi,eta)][j:is(zeta,upsilon)] 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)) +4142 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))] t1:=tris(rat,z0,ts"rt"(x0,y0),ts"rt"(y0,x0),i,comts(x0,y0)):is"rt"(z0,ts"rt"(y0,x0)) t2:=lrtts(eta,ksi,z0,y0,ly,x0,lx,t1):lrt(ts(eta,ksi),z0) 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) -4142 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)) comts:=satz142:is(ts(ksi,eta),ts(eta,ksi)) +4143 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))] 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))) 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)) t3:=lrtts(ksi,ts(eta,zeta),u0,x0,lx,ts"rt"(y0,z0),t2,t1):lrt(ts(ksi,ts(eta,zeta)),u0) 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) 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) 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))] 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)) 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)) t8:=lrtts(ts(ksi,eta),zeta,u0,ts"rt"(x0,y0),t7,z0,lz,t6):lrt(ts(ts(ksi,eta),zeta),u0) 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) 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) -4143 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))) assts1:=satz143:is(ts(ts(ksi,eta),zeta),ts(ksi,ts(eta,zeta))) 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)) +4144 [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))] 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))) 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)) 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)) 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) 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) 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) 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))] 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))) x2:=ite(moreis"rt"(x0,x1),rat,x0,x1):rat [m:moreis"rt"(x0,x1)] t8:=itet(moreis"rt"(x0,x1),rat,x0,x1,m):is"rt"(x2,x0) t9:=isp1(rat,[t:rat]lrt(ksi,t),x0,x2,lx,t8):lrt(ksi,x2) t10:=lessisi2"rt"(x0,x2,symis(rat,x2,x0,t8)):lessis"rt"(x0,x2) t11:=satz88(x1,x0,x2,satz84(x0,x1,m),t10):lessis"rt"(x1,x2) k@[n:not(moreis"rt"(x0,x1))] t12:=itef(moreis"rt"(x0,x1),rat,x0,x1,n):is"rt"(x2,x1) t13:=isp1(rat,[t:rat]lrt(ksi,t),x1,x2,lx1,t12):lrt(ksi,x2) t14:=lessisi2"rt"(x1,x2,symis(rat,x2,x1,t12)):lessis"rt"(x1,x2) t15:=lessisi1"rt"(x0,x2,satz87b(x0,x1,x2,satz81j(x0,x1,n),t14)):lessis"rt"(x0,x2) 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) 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) 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) 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)) 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))) t21:=satz109a(x0,x2,y0,y0,t17,lessisi2"rt"(y0,y0,refis(rat,y0))):lessis"rt"(ts"rt"(x0,y0),ts"rt"(x2,y0)) t22:=satz109a(x1,x2,z0,z0,t18,lessisi2"rt"(z0,z0,refis(rat,z0))):lessis"rt"(ts"rt"(x1,z0),ts"rt"(x2,z0)) 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))) 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) 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) 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) 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) -4144 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))) 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))) disttp2:=satz144:is(ts(ksi,pl(eta,zeta)),pl(ts(ksi,eta),ts(ksi,zeta))) 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)) 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))) [m:more(ksi,eta)] +4145 phi:=mn(ksi,eta,m):cut t1:=satz140d(ksi,eta,m):is(ksi,pl(eta,phi)) 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))) -4145 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)) zeta@[i:is(ksi,eta)] satz145b:=ists1(ksi,eta,zeta,i):is(ts(ksi,zeta),ts(eta,zeta)) zeta@[l:less(ksi,eta)] satz145c:=satz121(ts(eta,zeta),ts(ksi,zeta),satz145a(eta,ksi,zeta,satz122(ksi,eta,l))):less(ts(ksi,zeta),ts(eta,zeta)) 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)) i@satz145e:=ists2(ksi,eta,zeta,i):is(ts(zeta,ksi),ts(zeta,eta)) 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)) upsilon@[i:is(ksi,eta)][m:more(zeta,upsilon)] 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)) 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)) i@[l:less(zeta,upsilon)] 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)) 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)) +4146 zeta@t1:=satz123a(ksi,eta):or3(is(ksi,eta),more(ksi,eta),less(ksi,eta)) 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))) -4146 zeta@[m:more(ts(ksi,zeta),ts(eta,zeta))] 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) zeta@[i:is(ts(ksi,zeta),ts(eta,zeta))] 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) zeta@[l:less(ts(ksi,zeta),ts(eta,zeta))] 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) zeta@[m:more(ts(zeta,ksi),ts(zeta,eta))] 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) zeta@[i:is(ts(zeta,ksi),ts(zeta,eta))] 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) zeta@[l:less(ts(zeta,ksi),ts(zeta,eta))] 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) upsilon@[m:more(ksi,eta)][n:more(zeta,upsilon)] +4147 t1:=satz145a(ksi,eta,zeta,m):more(ts(ksi,zeta),ts(eta,zeta)) 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)) -4147 satz147:=trmore(ts(ksi,zeta),ts(eta,zeta),ts(eta,upsilon),t1".4147",t2".4147"):more(ts(ksi,zeta),ts(eta,upsilon)) upsilon@[l:less(ksi,eta)][k:less(zeta,upsilon)] 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)) upsilon@[m:moreis(ksi,eta)][n:more(zeta,upsilon)] 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)) upsilon@[m:more(ksi,eta)][n:moreis(zeta,upsilon)] 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)) upsilon@[l:lessis(ksi,eta)][k:less(zeta,upsilon)] 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)) upsilon@[l:less(ksi,eta)][k:lessis(zeta,upsilon)] 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)) upsilon@[m:moreis(ksi,eta)][n:moreis(zeta,upsilon)] +4149 [i:is(ksi,eta)][j:is(zeta,upsilon)] t1:=moreisi2(ts(ksi,zeta),ts(eta,upsilon),ists12(ksi,eta,zeta,upsilon,i,j)):moreis(ts(ksi,zeta),ts(eta,upsilon)) i@[o:more(zeta,upsilon)] t2:=moreisi1(ts(ksi,zeta),ts(eta,upsilon),satz148a(m,o)):moreis(ts(ksi,zeta),ts(eta,upsilon)) 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)) n@[o:more(ksi,eta)] t4:=moreisi1(ts(ksi,zeta),ts(eta,upsilon),satz148b(o,n)):moreis(ts(ksi,zeta),ts(eta,upsilon)) -4149 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)) upsilon@[l:lessis(ksi,eta)][k:lessis(zeta,upsilon)] 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)) -rp @[r0:rat] ratset:=setof(rat,[x:rat]less(x,r0)):set(rat) +4150 t1:=satz90(r0):some([x:rat]less(x,r0)) [x0:rat][l:less(x0,r0)] t2:=estii(rat,[x:rat]less(x,r0),x0,l):in(x0,ratset) r0@t3:=ec3e13(is(r0,r0),more(r0,r0),less(r0,r0),satz81b(r0,r0),refis(rat,r0)):not(less(r0,r0)) 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)) x0@[i:in(x0,ratset)] t5:=estie(rat,[x:rat]less(x,r0),x0,i):less(x0,r0) [x1:rat][k:less(x1,x0)] t6:=t2(x1,trless(x1,x0,r0,k,t5)):in(x1,ratset) i@t7:=satz91(x0,r0,t5):some([x:rat]and(less(x0,x),less(x,r0))) [x1:rat][a:and(less(x0,x1),less(x1,r0))] t8:=ande1(less(x0,x1),less(x1,r0),a):less(x0,x1) t9:=ande2(less(x0,x1),less(x1,r0),a):less(x1,r0) t10:=andi(in(x1,ratset),more(x1,x0),t2(x1,t9),satz83(x0,x1,t8)):and(in(x1,ratset),more(x1,x0)) 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))) 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) -4150 satz150:=someapp(rat,[x:rat]less(x,r0),t1".4150",cutprop(ratset),[x:rat][t:less(x,r0)]t12".4150"(x,t)):cutprop(ratset) +*rp r0@rpofrt:=cutof(ratset,satz150):cut [x0:rat][l:less"rt"(x0,r0)] lrtrpofrt:=ine(ratset,satz150,x0,t2"rt.4150"(x0,l)):lrt(rpofrt,x0) r0@[x0:rat][lx:lrt(rpofrt,x0)] lrtrpofrte:=t5"rt.4150"(x0,ini(ratset,satz150,x0,lx)):less"rt"(x0,r0) r0@[x0:rat][m:moreis"rt"(x0,r0)] +*iii4 m@t12:=satz81c(x0,r0,m):not(less"rt"(x0,r0)) -iii4 m@urtrpofrt:=th3"l.imp"(lrt(rpofrt,x0),less"rt"(x0,r0),t12".iii4",[t:lrt(rpofrt,x0)]lrtrpofrte(x0,t)):urt(rpofrt,x0) @1rp:=rpofrt(1rt):cut +4151 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))] t1:=lrtrpofrte(1rt,y0,ly):less"rt"(y0,1rt) 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) t3:=satz120(ksi,x0,lx,z0,t2):lrt(ksi,z0) 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) ksi@[x0:rat][lx:lrt(ksi,x0)][x1:rat][lx1:lrt(ksi,x1)][l:less"rt"(x0,x1)] y1:=ts"rt"(ov(1rt,x1),x0):rat 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) t6:=lrtrpofrt(1rt,y1,t5):lrt(1rp,y1) 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) t8:=lrtts(ksi,1rp,x0,x1,lx1,y1,t6,symis(rat,ts"rt"(x1,y1),x0,t7)):lrt(ts(ksi,1rp),x0) 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) -4151 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) satz151a:=symis(cut,ts(ksi,1rp),ksi,satz151):is(ksi,ts(ksi,1rp)) satz151b:=tris(cut,ts(1rp,ksi),ts(ksi,1rp),ksi,comts(1rp,ksi),satz151):is(ts(1rp,ksi),ksi) satz151c:=symis(cut,ts(1rp,ksi),ksi,satz151b):is(ksi,ts(1rp,ksi)) +4152 [x0:rat][y0:rat] invprop1:=and(urt(ksi,y0),less"rt"(y0,x0)):'prop' ksi@[z0:rat][x0:rat] invprop2:=and3(urt(ksi,x0),some"rt"([x:rat]invprop1(x0,x)),is"rt"(z0,ov(1rt,x0))):'prop' z0@invprop:=some"rt"([x:rat]invprop2(z0,x)):'prop' ksi@inv:=setof(rat,[z:rat]invprop(z)):set(rat) x0@[ux:urt(ksi,x0)][y0:rat][uy:urt(ksi,y0)][l:less"rt"(y0,x0)][i:is"rt"(z0,ov(1rt,x0))] t1:=andi(urt(ksi,y0),less"rt"(y0,x0),uy,l):invprop1(x0,y0) t2:=somei(rat,[x:rat]invprop1(x0,x),y0,t1):some"rt"([x:rat]invprop1(x0,x)) t3:=and3i(urt(ksi,x0),some"rt"([x:rat]invprop1(x0,x)),is"rt"(z0,ov(1rt,x0)),ux,t2,i):invprop2(z0,x0) t4:=somei(rat,[x:rat]invprop2(z0,x),x0,t3):invprop(z0) inv1:=estii(rat,[z:rat]invprop(z),z0,t4):in(z0,inv) 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] t5:=estie(rat,[x:rat]invprop(x),z0,i):invprop(z0) [x0:rat][px:invprop2(z0,x0)] t6:=and3e1(urt(ksi,x0),some"rt"([x:rat]invprop1(x0,x)),is"rt"(z0,ov(1rt,x0)),px):urt(ksi,x0) 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)) 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)) [y0:rat][py:invprop1(x0,y0)] t9:=ande1(urt(ksi,y0),less"rt"(y0,x0),py):urt(ksi,y0) t10:=ande2(urt(ksi,y0),less"rt"(y0,x0),py):less"rt"(y0,x0) t11:=p1:p px@t12:=someapp(rat,[x:rat]invprop1(x0,x),t7,p,[x:rat][t:invprop1(x0,x)]t11(x,t)):p p1@invapp:=someapp(rat,[x:rat]invprop2(z0,x),t5,p,[x:rat][t:invprop2(z0,x)]t12(x,t)):p ksi@[x0:rat][ux:urt(ksi,x0)] 2x0:=pl"rt"(x0,x0):rat t13:=satz94a(x0,x0):less"rt"(x0,2x0) t14:=satz119a(ksi,x0,ux,2x0,t13):urt(ksi,2x0) t15:=inv1(ov(1rt,2x0),2x0,t14,x0,ux,t13,refis(rat,ov(1rt,2x0))):in(ov(1rt,2x0),inv) ksi@[x1:rat][lx:lrt(ksi,x1)][x0:rat][ux:urt(ksi,x0)] 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) t17:=satz110e(1rt,x0):is"rt"(ts"rt"(ov(1rt,x0),x0),1rt) t18:=satz110e(1rt,x1):is"rt"(ts"rt"(ov(1rt,x1),x1),1rt) [i:is"rt"(ov(1rt,x0),ov(1rt,x1))] 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) t20:=satz110b(1rt,ov(1rt,x0),x0,x1,t17,t19):is"rt"(x0,x1) 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)) lx@[i:in(ov(1rt,x1),inv)] 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))]t21(x,t)):con lx@t23:=[t:in(ov(1rt,x1),inv)]t22(t):not(in(ov(1rt,x1),inv)) 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))] t24:=isless2"rt"(z0,ov(1rt,x0),u0,j,l):less"rt"(u0,ov(1rt,x0)) 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))) 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))) 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)) t28:=satz106c(x0,ov(1rt,u0),u0,t27):less"rt"(x0,ov(1rt,u0)) t29:=satz119a(x0,ux,ov(1rt,u0),t28):urt(ksi,ov(1rt,u0)) t30:=satz110e(1rt,u0):is"rt"(ts"rt"(ov(1rt,u0),u0),1rt) t31:=satz110g(1rt,ov(1rt,u0),u0,t30):is"rt"(u0,ov(1rt,ov(1rt,u0))) t32:=inv1(u0,ov(1rt,u0),t29,x0,ux,t28,t31):in(u0,inv) 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) 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))] t34:=satz91(x1,x0,l):some"rt"([x:rat]and(less"rt"(x1,x),less"rt"(x,x0))) [x2:rat][a:and(less"rt"(x1,x2),less"rt"(x2,x0))] t35:=ande1(less"rt"(x1,x2),less"rt"(x2,x0),a):less"rt"(x1,x2) t36:=satz119a(ksi,x1,ux1,x2,t35):urt(ksi,x2) t37:=inv1(ov(1rt,x2),x2,t36,x1,ux1,t35,refis(rat,ov(1rt,x2))):in(ov(1rt,x2),inv) t38:=ande2(less"rt"(x1,x2),less"rt"(x2,x0),a):less"rt"(x2,x0) 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))) 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))) 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)) t42:=satz106c(ov(1rt,x0),ov(1rt,x2),x2,t41):less"rt"(ov(1rt,x0),ov(1rt,x2)) 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) 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)) 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))) 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))) 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))) ksi@[x0:rat][lx:lrt(ksi,x0)][y0:rat][uy:urt(ksi,y0)] 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) lx@t49:=cutapp1b(ksi,cutprop(inv),[x:rat][t:urt(ksi,x)]t48(x,t)):cutprop(inv) ksi@t50:=cutapp1a(ksi,cutprop(inv),[x:rat][t:lrt(ksi,x)]t49(x,t)):cutprop(inv) chi:=cutof(inv,t50):cut [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))] 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))) t52:=cutapp2a(ksi,x0,lx,x1,ux):less"rt"(x0,x1) 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) t54:=lrtrpofrt(1rt,z0,t53):lrt(1rp,z0) i@r1:=ini(inv,t50,u0,lu):in(u0,inv) 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) 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) ksi@[u0:rat][lu:lrt(1rp,u0)] t55:=lrtrpofrte(1rt,u0,lu):less"rt"(u0,1rt) t56:=satz83(u0,1rt,t55):more"rt"(1rt,u0) [x0:rat][lx:lrt(ksi,x0)][x1:rat][lx1:lrt(ksi,x1)][x2:rat][ux2:urt(ksi,x2)] t57:=cutapp2b(x1,lx1,x2,ux2):more"rt"(x2,x1) [i:is"rt"(mn"rt"(x2,x1,t57),ts"rt"(mn"rt"(1rt,u0,t56),x0))] t58:=cutapp2a(x0,lx,x2,ux2):less"rt"(x0,x2) 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)) 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)) 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)) 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))) 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)) 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))) t65:=satz97c(ts"rt"(u0,x2),x1,mn"rt"(x2,x1,t57),t64):less"rt"(ts"rt"(u0,x2),x1) 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) 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)) t68:=satz119a(x2,ux2,ov(x1,u0),t67):urt(ksi,ov(x1,u0)) t69:=satz110e(x1,u0):is"rt"(ts"rt"(ov(x1,u0),u0),x1) 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)))) 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) t72:=ine(inv,t50,ov(1rt,ov(x1,u0)),t71):lrt(chi,ov(1rt,ov(x1,u0))) t73:=lrtts(ksi,chi,u0,x1,lx1,ov(1rt,ov(x1,u0)),t72,t70):lrt(ts(ksi,chi),u0) 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) lu@t75:=cutapp1a(ksi,lrt(ts(ksi,chi),u0),[x:rat][t:lrt(ksi,x)]t74(x,t)):lrt(ts(ksi,chi),u0) 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) -4152 satz152:=somei(cut,[t:cut]is(ts(ksi,t),1rp),chi".4152",t76".4152"):some([c:cut]is(ts(ksi,c),1rp)) eta@[phi:cut][psi:cut] +4153 [m:more(phi,psi)] t1:=satz145d(phi,psi,eta,m):more(ts(eta,phi),ts(eta,psi)) 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)) psi@[l:less(phi,psi)] t3:=satz145f(phi,psi,eta,l):less(ts(eta,phi),ts(eta,psi)) 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)) psi@[n:nis(phi,psi)] t5:=th1"l.or3"(is(phi,psi),more(phi,psi),less(phi,psi),satz123a(phi,psi),n):or(more(phi,psi),less(phi,psi)) 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)) -4153 [i:is(ts(eta,phi),ksi)][j:is(ts(eta,psi),ksi)] 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) +*4153 eta@[tau:cut][i:is(ts(eta,tau),1rp)] chi:=ts(tau,ksi):cut 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) t8:=somei(cut,[c:cut]is(ts(eta,c),ksi),chi,t7):some([c:cut]is(ts(eta,c),ksi)) -4153 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)) +*4153 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)) -4153 eta@satz153:=onei(cut,[c:cut]is(ts(eta,c),ksi),t9".4153",satz153a):one([c:cut]is(ts(eta,c),ksi)) ov:=ind(cut,[a:cut]is(ts(eta,a),ksi),satz153):cut satz153c:=oneax(cut,[a:cut]is(ts(eta,a),ksi),satz153):is(ts(eta,ov(ksi,eta)),ksi) satz153d:=symis(cut,ts(eta,ov(ksi,eta)),ksi,satz153c):is(ksi,ts(eta,ov(ksi,eta))) 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) satz153f:=symis(cut,ts(ov(ksi,eta),eta),ksi,satz153e):is(ksi,ts(ov(ksi,eta),eta)) [phi:cut][i:is(ts(eta,phi),ksi)] satz153g:=satz153b(phi,ov(ksi,eta),i,satz153c):is(phi,ov(ksi,eta)) @[ksi:cut] ratrp:=image(rat,cut,[x:rat]rpofrt(x),ksi):'prop' @[x0:rat] ratrpi:=imagei(rat,cut,[x:rat]rpofrt(x),x0):ratrp(rpofrt(x0)) @[x:nat] rpofnt:=rpofrt(rtofn(x)):cut ksi@natrp:=image(nat,cut,[x:nat]rpofnt(x),ksi):'prop' x@natrpi:=imagei(nat,cut,[y:nat]rpofnt(y),x):natrp(rpofnt(x)) ksi@[n:natrp(ksi)] +iii5 [x:nat][i:is(ksi,rpofnt(x))] t1:=somei(rat,[y:rat]is(ksi,rpofrt(y)),rtofn(x),i):ratrp(ksi) -iii5 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) @[x0:rat][y0:rat][m:more"rt"(x0,y0)] +5154 t1:=lrtrpofrt(x0,y0,satz82(x0,y0,m)):lrt(rpofrt(x0),y0) t2:=urtrpofrt(y0,y0,moreisi2"rt"(y0,y0,refis(rat,y0))):urt(rpofrt(y0),y0) t3:=andi(lrt(rpofrt(x0),y0),urt(rpofrt(y0),y0),t1,t2):and(lrt(rpofrt(x0),y0),urt(rpofrt(y0),y0)) -5154 satz154a:=somei(rat,[x:rat]and(lrt(rpofrt(x0),x),urt(rpofrt(y0),x)),y0,t3".5154"):more(rpofrt(x0),rpofrt(y0)) y0@[i:is"rt"(x0,y0)] satz154b:=isf(rat,cut,[x:rat]rpofrt(x),x0,y0,i):is(rpofrt(x0),rpofrt(y0)) y0@[l:less"rt"(x0,y0)] satz154c:=satz121(rpofrt(y0),rpofrt(x0),satz154a(y0,x0,satz83(x0,y0,l))):less(rpofrt(x0),rpofrt(y0)) +*5154 y0@t4:=satz81a(x0,y0):or3(is"rt"(x0,y0),more"rt"(x0,y0),less"rt"(x0,y0)) t5:=satz123b(rpofrt(x0),rpofrt(y0)):ec3(is(rpofrt(x0),rpofrt(y0)),more(rpofrt(x0),rpofrt(y0)),less(rpofrt(x0),rpofrt(y0))) -5154 y0@[m:more(rpofrt(x0),rpofrt(y0))] 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) y0@[i:is(rpofrt(x0),rpofrt(y0))] 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) y0@[l:less(rpofrt(x0),rpofrt(y0))] 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) +*iii5 @t2:=[x:rat][y:rat][t:is(rpofrt(x),rpofrt(y))]satz154e(x,y,t):injective(rat,cut,[x:rat]rpofrt(x)) -iii5 y0@[i:is"rt"(x0,y0)] isrterp:=satz154b(x0,y0,i):is(rpofrt(x0),rpofrt(y0)) y0@[i:is(rpofrt(x0),rpofrt(y0))] isrtirp:=satz154e(x0,y0,i):is"rt"(x0,y0) ksi@[rtksi:ratrp(ksi)] rtofrp:=soft(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi):rat [eta:cut][rteta:ratrp(eta)][i:is(ksi,eta)] isrpert:=isinv(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi,eta,rteta,i):is"rt"(rtofrp(ksi,rtksi),rtofrp(eta,rteta)) rteta@[i:is"rt"(rtofrp(ksi,rtksi),rtofrp(eta,rteta))] isrpirt:=isinve(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi,eta,rteta,i):is(ksi,eta) x0@isrtrp1:=isst1(rat,cut,[x:rat]rpofrt(x),t2".iii5",x0):is"rt"(x0,rtofrp(rpofrt(x0),ratrpi(x0))) isrtrp2:=isst2(rat,cut,[x:rat]rpofrt(x),t2".iii5",x0):is"rt"(rtofrp(rpofrt(x0),ratrpi(x0)),x0) rtksi@isrprt1:=ists1"e"(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi):is(ksi,rpofrt(rtofrp(ksi,rtksi))) isrprt2:=ists2"e"(rat,cut,[x:rat]rpofrt(x),t2".iii5",ksi,rtksi):is(rpofrt(rtofrp(ksi,rtksi)),ksi) @[x:nat][y:nat][i:is"n"(x,y)] isnterp:=isf(nat,cut,[z:nat]rpofnt(z),x,y,i):is(rpofnt(x),rpofnt(y)) y@[i:is(rpofnt(x),rpofnt(y))] isntirp:=isnirt(x,y,isrtirp(rtofn(x),rtofn(y),i)):is"n"(x,y) +*iii5 @t3:=[x:nat][y:nat][t:is(rpofnt(x),rpofnt(y))]isntirp(x,y,t):injective(nat,cut,[x:nat]rpofnt(x)) -iii5 ksi@[ntksi:natrp(ksi)] ntofrp:=soft(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi):nat [eta:cut][nteta:natrp(eta)][i:is(ksi,eta)] isrpent:=isinv(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi,eta,nteta,i):is"n"(ntofrp(ksi,ntksi),ntofrp(eta,nteta)) nteta@[i:is"n"(ntofrp(ksi,ntksi),ntofrp(eta,nteta))] isrpint:=isinve(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi,eta,nteta,i):is(ksi,eta) x@isntrp1:=isst1(nat,cut,[y:nat]rpofnt(y),t3".iii5",x):is"n"(x,ntofrp(rpofnt(x),natrpi(x))) isntrp2:=isst2(nat,cut,[y:nat]rpofnt(y),t3".iii5",x):is"n"(ntofrp(rpofnt(x),natrpi(x)),x) ntksi@isrpnt1:=ists1"e"(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi):is(ksi,rpofnt(ntofrp(ksi,ntksi))) isrpnt2:=ists2"e"(nat,cut,[x:nat]rpofnt(x),t3".iii5",ksi,ntksi):is(rpofnt(ntofrp(ksi,ntksi)),ksi) @[x0:rat][y0:rat] +5155 [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))] t1:=lrtrpofrte(x0,u0,lu):less"rt"(u0,x0) t2:=lrtrpofrte(y0,v0,lv):less"rt"(v0,y0) t3:=satz98a(u0,x0,v0,y0,t1,t2):less"rt"(pl"rt"(u0,v0),pl"rt"(x0,y0)) 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)) t5:=lrtrpofrt(pl"rt"(x0,y0),z0,t4):lrt(rpofrt(pl"rt"(x0,y0)),z0) 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) y0@[u0:rat][lu:lrt(rpofrt(pl"rt"(x0,y0)),u0)] t7:=lrtrpofrte(pl"rt"(x0,y0),u0,lu):less"rt"(u0,pl"rt"(x0,y0)) u01:=ov"rt"(u0,pl"rt"(x0,y0)):rat 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))) t9:=satz106c(u01,1rt,pl"rt"(x0,y0),t8):less"rt"(u01,1rt) 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))) y0@[l:less"rt"(y0,1rt)] 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) t12:=lrtrpofrt(x0,ts"rt"(x0,y0),t11):lrt(rpofrt(x0),ts"rt"(x0,y0)) 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) -5155 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))) [m:more"rt"(x0,y0)] +*5155 m@t14:=satz101f(x0,y0,m):is"rt"(x0,pl"rt"(mn"rt"(x0,y0,m),y0)) 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))) 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)) -5155 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))) +*5155 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))] t17:=lrtrpofrte(x0,u0,lu):less"rt"(u0,x0) t18:=lrtrpofrte(y0,v0,lv):less"rt"(v0,y0) t19:=satz107a(u0,x0,v0,y0,t17,t18):less"rt"(ts"rt"(u0,v0),ts"rt"(x0,y0)) 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)) t21:=lrtrpofrt(ts"rt"(x0,y0),z0,t20):lrt(rpofrt(ts"rt"(x0,y0)),z0) 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) y0@[u0:rat][lu:lrt(rpofrt(ts"rt"(x0,y0)),u0)] t23:=lrtrpofrte(ts"rt"(x0,y0),u0,lu):less"rt"(u0,ts"rt"(x0,y0)) [u1:rat][a:and(less"rt"(u0,u1),less"rt"(u1,ts"rt"(x0,y0)))] t24:=ande1(less"rt"(u0,u1),less"rt"(u1,ts"rt"(x0,y0)),a):less"rt"(u0,u1) t25:=ande2(less"rt"(u0,u1),less"rt"(u1,ts"rt"(x0,y0)),a):less"rt"(u1,ts"rt"(x0,y0)) 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)) t27:=satz106c(ov"rt"(u0,u1),1rt,u1,t26):less"rt"(ov"rt"(u0,u1),1rt) 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)) t29:=satz106c(ov"rt"(u1,y0),x0,y0,t28):less"rt"(ov"rt"(u1,y0),x0) 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)))) 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) 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) -5155 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))) +*5155 y0@t33:=satz110f(x0,y0):is"rt"(x0,ts"rt"(ov"rt"(x0,y0),y0)) 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))) 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)) -5155 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))) @[x:nat][y:nat] 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))) 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))) +nt @natt:=ot(cut,[t:cut]natrp(t)):'type' [ksi:cut][nksi:natrp(ksi)] nttofrp:=out(cut,[t:cut]natrp(t),ksi,nksi):natt @[xt:natt][yt:natt] is:=is"e"(natt,xt,yt):'prop' nis:=not(is(xt,yt)):'prop' @[p:[x:natt]'prop'] all:=all"l"(natt,p):'prop' some:=some"l"(natt,p):'prop' one:=one"e"(natt,p):'prop' xt@[st:set(natt)] in:=esti(natt,xt,st):'prop' xt@rpofntt:=in"e"(cut,[t:cut]natrp(t),xt):cut natrpi:=inp(cut,[t:cut]natrp(t),xt):natrp(rpofntt(xt)) nksi@[eta:cut][neta:natrp(eta)][i:is"rp"(ksi,eta)] isrpentt:=isouti(cut,[t:cut]natrp(t),ksi,nksi,eta,neta,i):is(nttofrp(ksi,nksi),nttofrp(eta,neta)) neta@[i:is(nttofrp(ksi,nksi),nttofrp(eta,neta))] isrpintt:=isoute(cut,[t:cut]natrp(t),ksi,nksi,eta,neta,i):is"rp"(ksi,eta) yt@[i:is(xt,yt)] isntterp:=isini(cut,[t:cut]natrp(t),xt,yt,i):is"rp"(rpofntt(xt),rpofntt(yt)) yt@[i:is"rp"(rpofntt(xt),rpofntt(yt))] isnttirp:=isine(cut,[t:cut]natrp(t),xt,yt,i):is(xt,yt) nksi@isrpntt1:=isinout(cut,[t:cut]natrp(t),ksi,nksi):is"rp"(ksi,rpofntt(nttofrp(ksi,nksi))) xt@isnttrp1:=isoutin(cut,[t:cut]natrp(t),xt):is(xt,nttofrp(rpofntt(xt),natrpi(xt))) @[x:nat] nttofnt:=nttofrp(rpofnt(x),natrpi"rp"(x)):natt [y:nat][i:is"n"(x,y)] isntentt:=isrpentt(rpofnt(x),natrpi"rp"(x),rpofnt(y),natrpi"rp"(y),isnterp(x,y,i)):is(nttofnt(x),nttofnt(y)) y@[i:is(nttofnt(x),nttofnt(y))] isntintt:=isntirp(x,y,isrpintt(rpofnt(x),natrpi"rp"(x),rpofnt(y),natrpi"rp"(y),i)):is"n"(x,y) xt@ntofntt:=ntofrp(rpofntt(xt),natrpi(xt)):nat yt@[i:is(xt,yt)] isnttent:=isrpent(rpofntt(xt),natrpi(xt),rpofntt(yt),natrpi(yt),isntterp(xt,yt,i)):is"n"(ntofntt(xt),ntofntt(yt)) yt@[i:is"n"(ntofntt(xt),ntofntt(yt))] isnttint:=isnttirp(xt,yt,isrpint(rpofntt(xt),natrpi(xt),rpofntt(yt),natrpi(yt),i)):is(xt,yt) +iii5 x@t5:=isrpntt1(rpofnt(x),natrpi"rp"(x)):is"rp"(rpofnt(x),rpofntt(nttofnt(x))) 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))) -iii5 x@isntntt1:=tris(nat,x,ntofrp(rpofnt(x),natrpi"rp"(x)),ntofntt(nttofnt(x)),isntrp1(x),t6".iii5"):is"n"(x,ntofntt(nttofnt(x))) +*iii5 xt@t7:=isrpnt1(rpofntt(xt),natrpi(xt)):is"rp"(rpofntt(xt),rpofnt(ntofntt(xt))) t8:=isrpentt(rpofntt(xt),natrpi(xt),rpofnt(ntofntt(xt)),natrpi"rp"(ntofntt(xt)),t7):is(nttofrp(rpofntt(xt),natrpi(xt)),nttofnt(ntofntt(xt))) -iii5 xt@isnttnt1:=tris(natt,xt,nttofrp(rpofntt(xt),natrpi(xt)),nttofnt(ntofntt(xt)),isnttrp1(xt),t8".iii5"):is(xt,nttofnt(ntofntt(xt))) x@isntntt2:=symis(nat,x,ntofntt(nttofnt(x)),isntntt1):is"n"(ntofntt(nttofnt(x)),x) xt@isnttnt2:=symis(natt,xt,nttofnt(ntofntt(xt)),isnttnt1):is(nttofnt(ntofntt(xt)),xt) @1t:=nttofnt(1):natt suct:=[x:natt]nttofnt(suc):[x:natt]natt +5156 xt@[j:is(suct,1t)] t1:=isntintt(suc,1,j):is"n"(suc,1) -5156 xt@satz156a:=th3"l.imp"(is(suct,1t),is"n"(suc,1),ax3,[t:is(suct,1t)]t1".5156"(t)):nis(suct,1t) yt@[i:is(suct,suct)] +*5156 i@t2:=isntintt(suc,suc,i):is"n"(suc,suc) -5156 i@satz156b:=isnttint(xt,yt,ax4):is(xt,yt) @[st:set(natt)] cond1:=in(1t,st):'prop' cond2:=all([x:natt]imp(in(x,st),in(suct,st))):'prop' [c1:cond1][c2:cond2] +*5156 c2@[x:nat] prop1:=in(nttofnt(x),st):'prop' [p:prop1(x)] t3:=

c2:in(suct,st) t4:=isp(nat,[t:nat]in(nttofnt(suc),st),ntofntt(nttofnt(x)),x,t3,isntntt2(x)):prop1(suc) -5156 c2@[xt:natt] +*5156 xt@t5:=induction([t:nat]prop1(t),c1,[t:nat][u:prop1(t)]t4(t,u),ntofntt(xt)):in(nttofnt(ntofntt(xt)),st) -5156 xt@satz156c:=isp(natt,[t:natt]in(t,st),nttofnt(ntofntt(xt)),xt,t5".5156",isnttnt2(xt)):in(xt,st) @ax3t:=[x:natt]satz156a(x):[x:natt]nis(suct,1t) ax4t:=[x:natt][y:natt][u:is(suct,suct)]satz156b(x,y,u):[x:natt][y:natt][u:is(suct,suct)]is(x,y) 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) -nt +rtt @ratt:=ot(cut,[t:cut]ratrp(t)):'type' [ksi:cut][rtksi:ratrp(ksi)] rttofrp:=out(cut,[t:cut]ratrp(t),ksi,rtksi):ratt @[x0t:ratt][y0t:ratt] is:=is"e"(ratt,x0t,y0t):'prop' nis:=not(is(x0t,y0t)):'prop' @[p:[x:ratt]'prop'] all:=all"l"(ratt,p):'prop' some:=some"l"(ratt,p):'prop' one:=one"e"(ratt,p):'prop' x0t@rpofrtt:=in"e"(cut,[t:cut]ratrp(t),x0t):cut ratrpi:=inp(cut,[t:cut]ratrp(t),x0t):ratrp(rpofrtt(x0t)) rtksi@[eta:cut][rteta:ratrp(eta)][i:is"rp"(ksi,eta)] isrpertt:=isouti(cut,[t:cut]ratrp(t),ksi,rtksi,eta,rteta,i):is(rttofrp(ksi,rtksi),rttofrp(eta,rteta)) rteta@[i:is(rttofrp(ksi,rtksi),rttofrp(eta,rteta))] isrpirtt:=isoute(cut,[t:cut]ratrp(t),ksi,rtksi,eta,rteta,i):is"rp"(ksi,eta) y0t@[i:is(x0t,y0t)] isrtterp:=isini(cut,[t:cut]ratrp(t),x0t,y0t,i):is"rp"(rpofrtt(x0t),rpofrtt(y0t)) y0t@[i:is"rp"(rpofrtt(x0t),rpofrtt(y0t))] isrttirp:=isine(cut,[t:cut]ratrp(t),x0t,y0t,i):is(x0t,y0t) rtksi@isrprtt1:=isinout(cut,[t:cut]ratrp(t),ksi,rtksi):is"rp"(ksi,rpofrtt(rttofrp(ksi,rtksi))) x0t@isrttrp1:=isoutin(cut,[t:cut]ratrp(t),x0t):is(x0t,rttofrp(rpofrtt(x0t),ratrpi(x0t))) @[x0:rat] rttofrt:=rttofrp(rpofrt(x0),ratrpi"rp"(x0)):ratt [y0:rat][i:is"rt"(x0,y0)] isrtertt:=isrpertt(rpofrt(x0),ratrpi"rp"(x0),rpofrt(y0),ratrpi"rp"(y0),isrterp(x0,y0,i)):is(rttofrt(x0),rttofrt(y0)) y0@[i:is(rttofrt(x0),rttofrt(y0))] isrtirtt:=isrtirp(x0,y0,isrpirtt(rpofrt(x0),ratrpi"rp"(x0),rpofrt(y0),ratrpi"rp"(y0),i)):is"rt"(x0,y0) x0t@rtofrtt:=rtofrp(rpofrtt(x0t),ratrpi(x0t)):rat y0t@[i:is(x0t,y0t)] isrttert:=isrpert(rpofrtt(x0t),ratrpi(x0t),rpofrtt(y0t),ratrpi(y0t),isrtterp(x0t,y0t,i)):is"rt"(rtofrtt(x0t),rtofrtt(y0t)) y0t@[i:is"rt"(rtofrtt(x0t),rtofrtt(y0t))] isrttirt:=isrttirp(x0t,y0t,isrpirt(rpofrtt(x0t),ratrpi(x0t),rpofrtt(y0t),ratrpi(y0t),i)):is(x0t,y0t) +iii5 x0@t9:=isrprtt1(rpofrt(x0),ratrpi"rp"(x0)):is"rp"(rpofrt(x0),rpofrtt(rttofrt(x0))) 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))) -iii5 x0@isrtrtt1:=tris(rat,x0,rtofrp(rpofrt(x0),ratrpi"rp"(x0)),rtofrtt(rttofrt(x0)),isrtrp1(x0),t10".iii5"):is"rt"(x0,rtofrtt(rttofrt(x0))) +*iii5 x0t@t11:=isrprt1(rpofrtt(x0t),ratrpi(x0t)):is"rp"(rpofrtt(x0t),rpofrt(rtofrtt(x0t))) t12:=isrpertt(rpofrtt(x0t),ratrpi(x0t),rpofrt(rtofrtt(x0t)),ratrpi"rp"(rtofrtt(x0t)),t11):is(rttofrp(rpofrtt(x0t),ratrpi(x0t)),rttofrt(rtofrtt(x0t))) -iii5 x0t@isrttrt1:=tris(ratt,x0t,rttofrp(rpofrtt(x0t),ratrpi(x0t)),rttofrt(rtofrtt(x0t)),isrttrp1(x0t),t12".iii5"):is(x0t,rttofrt(rtofrtt(x0t))) -rtt @[ksi:cut] example2:=satz153c(1rp,ksi):is(ts(ksi,ov(1rp,ksi)),1rp) [rtksi:ratrp(ksi)] +5157 x01:=rtofrp(ksi,rtksi):rat ksi@s1:=setof(rat,[x:rat]urt(ksi,x)):set(rat) rtksi@[y0:rat][i:in(y0,s1)] t1:=estie(rat,[x:rat]urt(ksi,x),y0,i):urt(ksi,y0) [m:more"rt"(x01,y0)] t2:=lrtrpofrt(x01,y0,satz82(x01,y0,m)):lrt(rpofrt(x01),y0) t3:=isp(cut,[x:cut]lrt(x,y0),rpofrt(x01),ksi,t2,isrprt2(ksi,rtksi)):lrt(ksi,y0) 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)) t5:=satz81e(x01,y0,t4):lessis"rt"(x01,y0) rtksi@t6:=[x:rat][t:in(x,s1)]t5(x,t):lb(s1,x01) t7:=urtrpofrt(x01,x01,moreisi2"rt"(x01,x01,refis(rat,x01))):urt(rpofrt(x01),x01) t8:=isp(cut,[x:cut]urt(x,x01),rpofrt(x01),ksi,t7,isrprt2(ksi,rtksi)):urt(ksi,x01) t9:=estii(rat,[x:rat]urt(ksi,x),x01,t8):in(x01,s1) t10:=andi(lb(s1,x01),in(x01,s1),t6,t9):min(s1,x01) -5157 satz157a:=t10".5157":min(setof(rat,[x:rat]urt(ksi,x)),rtofrp(ksi,rtksi)) 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)) ksi@[x0:rat][m:min(setof(rat,[x:rat]urt(ksi,x)),x0)] +*5157 m"rp"@t11:=ande1(lb(s1,x0),in(x0,s1),m):lb(s1,x0) t12:=ande2(lb(s1,x0),in(x0,s1),m):in(x0,s1) t13:=estie(rat,[x:rat]urt(ksi,x),x0,t12):urt(ksi,x0) [y0:rat][ly:lrt(ksi,y0)] t14:=cutapp2a(ksi,y0,ly,x0,t13):less"rt"(y0,x0) t15:=lrtrpofrt(x0,y0,t14):lrt(rpofrt(x0),y0) y0@[uy:urt(ksi,y0)] t17:=estii(rat,[x:rat]urt(ksi,x),y0,uy):in(y0,s1) t18:=satz85(x0,y0,t11):moreis"rt"(y0,x0) t19:=urtrpofrt(x0,y0,t18):urt(rpofrt(x0),y0) y0@t20:=cp(lrt(rpofrt(x0),y0),lrt(ksi,y0),[t:urt(ksi,y0)]t19(t)):imp(lrt(rpofrt(x0),y0),lrt(ksi,y0)) -5157 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)) ksi@[s:some"rt"([x:rat]min(setof(rat,[y:rat]urt(ksi,y)),x))] +*5157 s@[x0:rat][m:min(s1,x0)] t21:=somei(rat,[x:rat]is(ksi,rpofrt(x)),x0,satz157c(x0,m)):ratrp(ksi) -5157 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) ksi@[x0:rat][lx:lrt(ksi,x0)] +5158 x0@xr:=rpofrt(x0):cut lx@t1:=urtrpofrt(x0,x0,moreisi2"rt"(x0,x0,refis(rat,x0))):urt(xr,x0) t2:=andi(urt(xr,x0),lrt(ksi,x0),t1,lx):and(urt(xr,x0),lrt(ksi,x0)) -5158 satz158a:=somei(rat,[x:rat]and(urt(rpofrt(x0),x),lrt(ksi,x)),x0,t2".5158"):less(rpofrt(x0),ksi) x0@[ux:urt(ksi,x0)] +*5158 ux@s1:=setof(rat,[x:rat]urt(ksi,x)):set(rat) [m:min(s1,x0)] t3:=symis(cut,ksi,xr,satz157c(ksi,x0,m)):is(xr,ksi) t4:=moreisi2(xr,ksi,t3):moreis(xr,ksi) ux@[n:not(min(s1,x0))] t5:=estii(rat,[x:rat]urt(ksi,x),x0,ux):in(x0,s1) t6:=th4"l.and"(lb(s1,x0),in(x0,s1),n,t5):not(lb(s1,x0)) 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)))) [y0:rat][o:not(imp(in(y0,s1),lessis"rt"(x0,y0)))] t8:=th5"l.imp"(in(y0,s1),lessis"rt"(x0,y0),o):in(y0,s1) t9:=estie(rat,[x:rat]urt(ksi,x),y0,t8):urt(ksi,y0) t10:=th6"l.imp"(in(y0,s1),lessis"rt"(x0,y0),o):not(lessis"rt"(x0,y0)) t11:=satz82(x0,y0,satz81k(x0,y0,t10)):less"rt"(y0,x0) t12:=lrtrpofrt(x0,y0,t11):lrt(xr,y0) t13:=andi(lrt(xr,y0),urt(ksi,y0),t12,t9):and(lrt(xr,y0),urt(ksi,y0)) t14:=somei(rat,[x:rat]and(lrt(xr,x),urt(ksi,x)),y0,t13):more(xr,ksi) 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) t16:=moreisi1(xr,ksi,t15):moreis(xr,ksi) -5158 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) x0@[l:less(rpofrt(x0),ksi)] +*5158 l@t17:=satz123h(xr,ksi,l):not(moreis(xr,ksi)) t18:=th3"l.imp"(urt(ksi,x0),moreis(xr,ksi),t17,[t:urt(ksi,x0)]satz158b(t)):not(urt(ksi,x0)) -5158 l@satz158c:=et(lrt(ksi,x0),t18".5158"):lrt(ksi,x0) x0@[m:moreis(rpofrt(x0),ksi)] +*5158 m"rp"@t19:=satz123c(xr,ksi,m):not(less(xr,ksi)) -5158 m@satz158d:=th3"l.imp"(lrt(ksi,x0),less(rpofrt(x0),ksi),t19".5158",[t:lrt(ksi,x0)]satz158a(t)):urt(ksi,x0) ksi@[eta:cut][l:less(ksi,eta)] +5159 [x0:rat] xr:=rpofrt(x0):cut [ux:urt(ksi,x0)][lx:lrt(eta,x0)][z0:rat] zr:=rpofrt(z0):cut [lz:lrt(eta,z0)][k:less"rt"(x0,z0)] t1:=satz127a(ksi,xr,zr,satz124(xr,ksi,satz158b(ksi,x0,ux)),satz154c(x0,z0,k)):less(ksi,zr) t2:=andi(less(ksi,zr),less(zr,eta),t1,satz158a(eta,z0,lz)):and(less(ksi,zr),less(zr,eta)) 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))) 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))) -5159 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))) +*5159 x0@[a:and(less(ksi,xr),less(xr,eta))] t5:=andi(ratrp(xr),and(less(ksi,xr),less(xr,eta)),ratrpi(x0),a):and3(ratrp(xr),less(ksi,xr),less(xr,eta)) 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))) -5159 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))) [p:'prop'][p1:[x:rat][t:less(ksi,rpofrt(x))][u:less(rpofrt(x),eta)]p] +*5159 p1@[y0:rat] yr:=rpofrt(y0):cut [a:and(less(ksi,yr),less(yr,eta))] t7:=ande1(less(ksi,yr),less(yr,eta),a):less(ksi,yr) t8:=ande2(less(ksi,yr),less(yr,eta),a):less(yr,eta) t9:=p1:p -5159 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 eta@[z0:rat][m:more(rpofrt(z0),ts(ksi,eta))] +5160 zr:=rpofrt(z0):cut nm:=mn(zr,ts(ksi,eta),m):cut dn:=pl(pl(ksi,eta),1rp):cut fr:=ov(nm,dn):cut zeta:=ite(less(fr,1rp),cut,fr,1rp):cut [l:less(fr,1rp)] t1:=itet(less(fr,1rp),cut,fr,1rp,l):is(zeta,fr) t2:=lessisi2(zeta,fr,t1):lessis(zeta,fr) t3:=lessisi1(zeta,1rp,satz127a(zeta,fr,1rp,t2,l)):lessis(zeta,1rp) m@[n:not(less(fr,1rp))] t4:=itef(less(fr,1rp),cut,fr,1rp,n):is(zeta,1rp) t5:=lessisi2(zeta,1rp,t4):lessis(zeta,1rp) t6:=trlessis(zeta,1rp,fr,t5,satz124(fr,1rp,satz123f(fr,1rp,n))):lessis(zeta,fr) 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) 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) [z1:rat] zr1:=rpofrt(z1):cut [l1:less(ksi,zr1)][l2:less(zr1,pl(ksi,zeta))][z2:rat] zr2:=rpofrt(z2):cut [l3:less(eta,zr2)][l4:less(zr2,pl(eta,zeta))] 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))) 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))) 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)) 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))) 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))) 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) 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)) 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))) 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))) 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) 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)) 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)) t21:=isless2(pl(ts(ksi,eta),nm),zr,ts(zr1,zr2),satz140c(zr,ts(ksi,eta),m),t20):less(ts(zr1,zr2),zr) 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) x0:=ov"rt"(z0,z2):rat xr:=rpofrt(x0):cut y0:=z2:rat yr:=rpofrt(y0):cut t23:=satz110e(z0,z2):is"rt"(ts"rt"(x0,y0),z0) 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)) t25:=satz106a(x0,z1,z2,t24):more"rt"(x0,z1) t26:=trmore(xr,zr1,ksi,satz154a(x0,z1,t25),satz122(ksi,zr1,l1)):more(xr,ksi) t27:=satz122(eta,yr,l3):more(yr,eta) z0@[u0:rat] ur:=rpofrt(u0):cut [v0:rat] vr:=rpofrt(v0):cut prop1:=and3(more(ur,ksi),more(vr,eta),is"rt"(ts"rt"(u0,v0),z0)):'prop' z0@prop2:=some"rt"([x:rat]some"rt"([y:rat]prop1(x,y))):'prop' l4@t28:=and3i(more(xr,ksi),more(yr,eta),is"rt"(ts"rt"(x0,y0),z0),t26,t27,t23):prop1(x0,y0) t29:=somei(rat,[y:rat]prop1(x0,y),y0,t28):some"rt"([y:rat]prop1(x0,y)) t30:=somei(rat,[x:rat]some"rt"([y:rat]prop1(x,y)),x0,t29):prop2 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 -5160 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)))) [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] +*5160 p1@[x1:rat] xr1:=rpofrt(x1):cut [px:some"rt"([y:rat]prop1(x1,y))][y1:rat] yr1:=rpofrt(y1):cut [py:prop1(x1,y1)] t32:=and3e1(more(xr1,ksi),more(yr1,eta),is"rt"(ts"rt"(x1,y1),z0),py):more(xr1,ksi) t33:=and3e2(more(xr1,ksi),more(yr1,eta),is"rt"(ts"rt"(x1,y1),z0),py):more(yr1,eta) t34:=and3e3(more(xr1,ksi),more(yr1,eta),is"rt"(ts"rt"(x1,y1),z0),py):is"rt"(ts"rt"(x1,y1),z0) t35:=p1:p px@t36:=someapp(rat,[y:rat]prop1(x1,y),px,p,[y:rat][v:prop1(x1,y)]t35(y,v)):p -5160 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 +5161 @[ksi:cut][eta:cut] min:=ite(less(ksi,eta),cut,ksi,eta):cut max:=ite(more(ksi,eta),cut,ksi,eta):cut [u0:rat] ur:=rpofrt(u0):cut [lu:lrt(min,u0)] t1:=satz158a(min,u0,lu):less(ur,min) [l:less(ksi,eta)] t2:=isless2(min,ksi,ur,itet(less(ksi,eta),cut,ksi,eta,l),t1):less(ur,ksi) t3:=trless(ur,ksi,eta,t2,l):less(ur,eta) lu@[n:not(less(ksi,eta))] t4:=isless2(min,eta,ur,itef(less(ksi,eta),cut,ksi,eta,n),t1):less(ur,eta) t5:=satz127b(ur,eta,ksi,t4,satz124(ksi,eta,satz123f(ksi,eta,n))):less(ur,ksi) 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) 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) u0@[uu:urt(max,u0)] t8:=satz158b(max,u0,uu):moreis(ur,max) [m:more(ksi,eta)] t9:=ismoreis2(max,ksi,ur,itet(more(ksi,eta),cut,ksi,eta,m),t8):moreis(ur,ksi) t10:=trmoreis(ur,ksi,eta,t9,moreisi1(ksi,eta,m)):moreis(ur,eta) uu@[n:not(more(ksi,eta))] t11:=ismoreis2(max,eta,ur,itef(more(ksi,eta),cut,ksi,eta,n),t8):moreis(ur,eta) t12:=trmoreis(ur,eta,ksi,t11,satz125(ksi,eta,satz123e(ksi,eta,n))):moreis(ur,ksi) 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) 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) -5161 @[zeta:cut] +*5161 zeta@[ksi1:cut][ksi2:cut][m:more(ksi1,ksi2)] t15:=satz147(ksi1,ksi2,ksi1,ksi2,m,m):more(ts(ksi1,ksi1),ts(ksi2,ksi2)) ksi2@sq1:=ts(ksi1,ksi1):cut sq2:=ts(ksi2,ksi2):cut m@t16:=ec3e21(is(sq1,sq2),more(sq1,sq2),less(sq1,sq2),satz123b(sq1,sq2),t15):nis(sq1,sq2) ksi2@[i:is(sq1,zeta)][j:is(sq2,zeta)] t17:=tris2(cut,sq1,sq2,zeta,i,j):is(sq1,sq2) t18:=[t:more(ksi1,ksi2)]t16(t):not(more(ksi1,ksi2)) t19:=[t:less(ksi1,ksi2)]t16(ksi2,ksi1,satz122(ksi1,ksi2,t)):not(less(ksi1,ksi2)) t20:=or3e1(is(ksi1,ksi2),more(ksi1,ksi2),less(ksi1,ksi2),satz123a(ksi1,ksi2),t18,t19):is(ksi1,ksi2) 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)) sqrtset:=setof(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta)):set(rat) [x0:rat] xr:=rpofrt(x0):cut [lx:lrt(min(1rp,zeta),x0)] t22:=t6(1rp,zeta,x0,lx):less(xr,1rp) t23:=t7(1rp,zeta,x0,lx):less(xr,zeta) t24:=isless1(xr,ts(xr,1rp),zeta,satz151a(xr),t23):less(ts(xr,1rp),zeta) 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) t26:=estii(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),x0,t25):in(x0,sqrtset) x0@[ux:urt(max(1rp,zeta),x0)] t27:=t13(1rp,zeta,x0,ux):moreis(xr,1rp) t28:=t14(1rp,zeta,x0,ux):moreis(xr,zeta) t29:=ismoreis1(xr,ts(xr,1rp),zeta,satz151a(xr),t28):moreis(ts(xr,1rp),zeta) 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) t31:=satz123c(ts(xr,xr),zeta,t30):not(less(ts(xr,xr),zeta)) 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)) x0@[i:in(x0,sqrtset)][y0:rat] yr:=rpofrt(y0):cut [l:less"rt"(y0,x0)] i@t33:=estie(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),x0,i):less(ts(xr,xr),zeta) l@t34:=satz154c(y0,x0,l):less(yr,xr) t35:=trless(ts(yr,yr),ts(xr,xr),zeta,satz147a(yr,xr,yr,xr,t34,t34),t33):less(ts(yr,yr),zeta) t36:=estii(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),y0,t35):in(y0,sqrtset) i@t37:=satz122(ts(xr,xr),zeta,t33):more(zeta,ts(xr,xr)) nm:=mn(zeta,ts(xr,xr),t37):cut dn:=pl(xr,pl(xr,1rp)):cut fr:=ov(nm,dn):cut [z0:rat] zr:=rpofrt(z0):cut [lz:lrt(min(1rp,fr),z0)] t38:=t6(1rp,fr,z0,lz):less(zr,1rp) t39:=t7(1rp,fr,z0,lz):less(zr,fr) t40:=satz94(x0,z0):more"rt"(pl"rt"(x0,z0),x0) 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))) 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)))) 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))) 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)) 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))) 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))) 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))) 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) 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)) 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) 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) t52:=estii(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),pl"rt"(x0,z0),t51):in(pl"rt"(x0,z0),sqrtset) 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)) 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))) 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))) x0@[lx:lrt(min(1rp,zeta),x0)][y0:rat][uy:urt(max(1rp,zeta),y0)] 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) lx@t57:=cutapp1b(max(1rp,zeta),cutprop(sqrtset),[y:rat][t:urt(max(1rp,zeta),y)]t56(y,t)):cutprop(sqrtset) zeta@t58:=cutapp1a(min(1rp,zeta),cutprop(sqrtset),[x:rat][t:lrt(min(1rp,zeta),x)]t57(x,t)):cutprop(sqrtset) rtc:=cutof(sqrtset,t58):cut @[x0:rat][y0:rat][l:lessis"rt"(x0,y0)] 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)) y0@[m:moreis"rt"(x0,y0)] t60:=satz125(rpofrt(y0),rpofrt(x0),t59(y0,x0,satz84(x0,y0,m))):moreis(rpofrt(x0),rpofrt(y0)) zeta@[m:more(ts(rtc,rtc),zeta)] t61:=satz121(ts(rtc,rtc),zeta,m):less(zeta,ts(rtc,rtc)) [z1:rat] zr1:=rpofrt(z1):cut [l1:less(zeta,zr1)][l2:less(zr1,ts(rtc,rtc))] t62:=satz158c(ts(rtc,rtc),z1,l2):lrt(ts(rtc,rtc),z1) [x1:rat] xr1:=rpofrt(x1):cut [lx1:lrt(rtc,x1)][x2:rat] xr2:=rpofrt(x2):cut [lx2:lrt(rtc,x2)][i:is"rt"(z1,ts"rt"(x1,x2))] xm:=ite(more"rt"(x1,x2),rat,x1,x2):rat xrm:=rpofrt(xm):cut [o:more"rt"(x1,x2)] t63:=symis(rat,xm,x1,itet(more"rt"(x1,x2),rat,x1,x2,o)):is"rt"(x1,xm) t64:=isp(rat,[x:rat]lrt(rtc,x),x1,xm,lx1,t63):lrt(rtc,xm) t65:=lessisi2"rt"(x1,xm,t63):lessis"rt"(x1,xm) t66:=lessisi1"rt"(x2,xm,satz87b(x2,x1,xm,satz82(x1,x2,o),t65)):lessis"rt"(x2,xm) i@[n:not(more"rt"(x1,x2))] t67:=symis(rat,xm,x2,itef(more"rt"(x1,x2),rat,x1,x2,n)):is"rt"(x2,xm) t68:=isp(rat,[x:rat]lrt(rtc,x),x2,xm,lx2,t67):lrt(rtc,xm) t69:=lessisi2"rt"(x2,xm,t67):lessis"rt"(x2,xm) t70:=satz88(x1,x2,xm,satz81e(x1,x2,n),t69):lessis"rt"(x1,xm) 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) 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) 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) t74:=ini(sqrtset,t58,xm,t71):in(xm,sqrtset) t75:=t59(x1,xm,t72):lessis(xr1,xrm) t76:=t59(x2,xm,t73):lessis(xr2,xrm) 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)) 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)) t79:=estie(rat,[x:rat]less(ts(rpofrt(x),rpofrt(x)),zeta),xm,t74):less(ts(xrm,xrm),zeta) t80:=satz127a(zr1,ts(xrm,xrm),zeta,t78,t79):less(zr1,zeta) t81:=ec3e23(is(zr1,zeta),more(zr1,zeta),less(zr1,zeta),satz123b(zr1,zeta),satz122(zeta,zr1,l1)):con 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 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 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 zeta@[l:less(ts(rtc,rtc),zeta)][z2:rat] zr2:=rpofrt(z2):cut [l3:less(ts(rtc,rtc),zr2)][l4:less(zr2,zeta)] t84:=satz122(ts(rtc,rtc),zr2,l3):more(zr2,ts(rtc,rtc)) [y1:rat] yr1:=rpofrt(y1):cut [m1:more(yr1,rtc)][y2:rat] yr2:=rpofrt(y2):cut [m2:more(yr2,rtc)][i:is"rt"(ts"rt"(y1,y2),z2)] ym:=ite(less"rt"(y1,y2),rat,y1,y2):rat yrm:=rpofrt(ym):cut [k:less"rt"(y1,y2)] t85:=symis(rat,ym,y1,itet(less"rt"(y1,y2),rat,y1,y2,k)):is"rt"(y1,ym) t86:=satz154b(y1,ym,t85):is(yr1,yrm) t87:=ismore1(yr1,yrm,rtc,t86,m1):more(yrm,rtc) t88:=moreisi2(yr1,yrm,t86):moreis(yr1,yrm) t89:=moreisi1(yr2,yrm,satz127d(yr2,yr1,yrm,satz122(yr1,yr2,satz154c(y1,y2,k)),t88)):moreis(yr2,yrm) i@[n:not(less"rt"(y1,y2))] t90:=symis(rat,ym,y2,itef(less"rt"(y1,y2),rat,y1,y2,n)):is"rt"(y2,ym) t91:=satz154b(y2,ym,t90):is(yr2,yrm) t92:=ismore1(yr2,yrm,rtc,t91,m2):more(yrm,rtc) t93:=moreisi2(yr2,yrm,t91):moreis(yr2,yrm) t94:=trmoreis(yr1,yr2,yrm,t60(y1,y2,satz81f(y1,y2,n)),t93):moreis(yr1,yrm) 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) 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) 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) t98:=satz158d(rtc,ym,moreisi1(yrm,rtc,t95)):urt(rtc,ym) t99:=th3"l.imp"(in(ym,sqrtset),lrt(rtc,ym),t98,[t:in(ym,sqrtset)]ine(sqrtset,t58,ym,t)):not(in(ym,sqrtset)) 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)) t101:=satz123f(ts(yrm,yrm),zeta,t100):moreis(ts(yrm,yrm),zeta) t101a:=satz149(yr1,yrm,yr2,yrm,t96,t97):moreis(ts(yr1,yr2),ts(yrm,yrm)) 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)) t103:=trmoreis(zr2,ts(yrm,yrm),zeta,t102,t101):moreis(zr2,zeta) t104:=satz123c(zr2,zeta,t103):con 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 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 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) t108:=somei(cut,[a:cut]is(ts(a,a),zeta),rtc,t107):some([a:cut]is(ts(a,a),zeta)) -5161 zeta@satz161:=onei(cut,[a:cut]is(ts(a,a),zeta),t21".5161",t108".5161"):one([a:cut]is(ts(a,a),zeta)) @[ksi:cut] irratrp:=not(ratrp(ksi)):'prop' -rp -rt +5162 @[v:nat] 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)) t2:=isless2(pl(v,v),ts(<1>suc,v),v,t1,satz18a(v,v)):less(v,ts(<1>suc,v)) [w:nat][l:less(ts(v,v),ts(w,w))] 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) 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))) 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))) 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)))) 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))) 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))) prop1:=eq(tf(fr(w,v),fr(w,v)),fr(<1>suc,1)):'prop' v@prop2:=some([t:nat]prop1(t)):'prop' @prop3:=some([u:nat]prop2(u)):'prop' [p:prop3] y:=ind(nat,[t:nat]min([u:nat]prop2(u),t),satz27a([u:nat]prop2(u),p)):nat t7:=oneax(nat,[t:nat]min([u:nat]prop2(u),t),satz27a([u:nat]prop2(u),p)):min([u:nat]prop2(u),y) t8:=ande1(lb([u:nat]prop2(u),y),prop2(y),t7):lb([u:nat]prop2(u),y) t9:=ande2(lb([u:nat]prop2(u),y),prop2(y),t7):prop2(y) [x:nat][q:prop1(y,x)] 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))) 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)) 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)) 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))) t14:=t3(y,x,t12):less(y,x) t15:=t3(x,ts(<1>suc,y),t13):less(x,ts(<1>suc,y)) [u:nat][i:is(x,pl(y,u))] 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)) t17:=satz20f(u,y,y,t16):less(u,y) [t:nat][j:is(y,pl(u,t))] t18:=symis(nat,y,pl(u,t),j):is(pl(u,t),y) 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))) 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)))) 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))) 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)))) 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)))) 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))))) 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)) 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)))) 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)) 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)))) 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))) 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))) 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) t32:=somei(nat,[v:nat]prop1(u,v),t,t31):prop2(u) t33:=t8:lessis(y,u) t34:=satz10g(y,u,satz12(u,y,t17)):con i@t35:=someapp(nat,[v:nat]diffprop(y,u,v),t17,con,[v:nat][w:diffprop(y,u,v)]t34(v,w)):con q@t36:=someapp(nat,[v:nat]diffprop(x,y,v),t14,con,[v:nat][w:diffprop(x,y,v)]t35(v,w)):con p@t37:=someapp(nat,[v:nat]prop1(y,v),t9,con,[v:nat][w:prop1(y,v)]t36(v,w)):con -5162 +*rt +5162 @[x0:rat][i:is(ts(x0,x0),rtofn(<1>suc))][x:frac][xix0:inf(x,class(x0))] 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)) t39:=refeq1(fr(num(x),den(x)),x,fris(x)):eq"n"(fr(num(x),den(x)),x) 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)) 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)) t42:=somei(nat,[t:nat]prop1"n.5162"(den(x),t),num(x),t41):prop2"n.5162"(den(x)) t43:=somei(nat,[t:nat]prop2"n.5162"(t),den(x),t42):prop3"n.5162" t44:=t37"n.5162"(t43):con i@t45:=ratapp1(x0,con,[x:frac][t:inf(x,class(x0))]t44(x,t)):con -5162 +*rp +5162 @ksi:=ind(cut,[a:cut]is(ts(a,a),rpofnt(<1>suc)),satz161(rpofnt(<1>suc))):cut t46:=oneax(cut,[a:cut]is(ts(a,a),rpofnt(<1>suc)),satz161(rpofnt(<1>suc))):is(ts(ksi,ksi),rpofnt(<1>suc)) [r:ratrp(ksi)] x0:=rtofrp(ksi,r):rat 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)) t48:=isrtirp(ts"rt"(x0,x0),rtofn(<1>suc),t47):is"rt"(ts"rt"(x0,x0),rtofn(<1>suc)) t49:=t45"rt.5162"(x0,t48):con -5162 @satz162:=somei(cut,[a:cut]irratrp(a),ksi".5162",[t:ratrp(ksi".5162")]t49".5162"(t)):some([a:cut]irratrp(a)) [zeta:cut] sqrt:=ind(cut,[a:cut]is(ts(a,a),zeta),satz161(zeta)):cut thsqrt1:=oneax(cut,[a:cut]is(ts(a,a),zeta),satz161(zeta)):is(ts(sqrt(zeta),sqrt(zeta)),zeta) [ksi:cut][i:is(ts(ksi,ksi),zeta)] thsqrt2:=t20".5161"(zeta,ksi,sqrt,i,thsqrt1):is(ksi,sqrt) @[ksi:cut][eta:cut][i:is(ksi,eta)] issqrt:=isf(cut,cut,[t:cut]sqrt(t),ksi,eta,i):is(sqrt(ksi),sqrt(eta)) @[ksi:cut][nx:natrp(ksi)][eta:cut][ny:natrp(eta)] +iiia x:=ntofrp(ksi,nx):nat y:=ntofrp(eta,ny):nat t1:=isrpnt1(ksi,nx):is(ksi,rpofnt(x)) t2:=isrpnt1(eta,ny):is(eta,rpofnt(y)) t3:=ispl12(ksi,rpofnt(x),eta,rpofnt(y),t1,t2):is(pl(ksi,eta),pl(rpofnt(x),rpofnt(y))) x0:=rtofn(x):rat y0:=rtofn(y):rat t4:=natrti(x):natrt(x0) t5:=natrti(y):natrt(y0) 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))) t7:=satz112d(x0,t4,y0,t5):natrt(pl"rt"(x0,y0)) xpy:=nofrt(pl"rt"(x0,y0),t7):nat t8:=isrtn1(pl"rt"(x0,y0),t7):is"rt"(pl"rt"(x0,y0),rtofn(xpy)) t9:=isrterp(pl"rt"(x0,y0),rtofn(xpy),t8):is(rpofrt(pl"rt"(x0,y0)),rpofnt(xpy)) 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)) -iiia natpl:=somei(nat,[t:nat]is(pl(ksi,eta),rpofnt(t)),xpy".iiia",t10".iiia"):natrp(pl(ksi,eta)) +*iiia ny@t11:=ists12(ksi,rpofnt(x),eta,rpofnt(y),t1,t2):is(ts(ksi,eta),ts(rpofnt(x),rpofnt(y))) 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))) t13:=satz112f(x0,t4,y0,t5):natrt(ts"rt"(x0,y0)) xty:=nofrt(ts"rt"(x0,y0),t13):nat t14:=isrtn1(ts"rt"(x0,y0),t13):is"rt"(ts"rt"(x0,y0),rtofn(xty)) t15:=isrterp(ts"rt"(x0,y0),rtofn(xty),t14):is(rpofrt(ts"rt"(x0,y0)),rpofnt(xty)) 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)) -iiia ny@natts:=somei(nat,[t:nat]is(ts(ksi,eta),rpofnt(t)),xty".iiia",t16".iiia"):natrp(ts(ksi,eta)) [m:more(ksi,eta)] +*iiia m@t17:=ismore12(ksi,rpofnt(x),eta,rpofnt(y),t1,t2,m):more(rpofnt(x),rpofnt(y)) t18:=satz154d(x0,y0,t17):more"rt"(x0,y0) 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))) 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))) t22:=satz112g(x0,t4,y0,t5,t18):natrt(mn"rt"(x0,y0,t18)) xmy:=nofrt(mn"rt"(x0,y0,t18),t22):nat t23:=isrtn1(mn"rt"(x0,y0,t18),t22):is"rt"(mn"rt"(x0,y0,t18),rtofn(xmy)) t24:=isrterp(mn"rt"(x0,y0,t18),rtofn(xmy),t23):is(rpofrt(mn"rt"(x0,y0,t18)),rpofnt(xmy)) 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)) -iiia m@natmn:=somei(nat,[t:nat]is(mn(ksi,eta,m),rpofnt(t)),xmy".iiia",t25".iiia"):natrp(mn(ksi,eta,m)) @[p:cut][q:cut][r:cut] 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))) [s:cut] 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))) 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))) 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))) 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)) 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)) @dif:=pair1type(cut):'type' [a1:cut][a2:cut] df:=pair1(cut,a1,a2):dif @[a:dif] stm:=first1(cut,a):cut std:=second1(cut,a):cut a2@stmis:=first1is1(cut,a1,a2):is(stm(df(a1,a2)),a1) isstm:=first1is2(cut,a1,a2):is(a1,stm(df(a1,a2))) stdis:=second1is1(cut,a1,a2):is(std(df(a1,a2)),a2) isstd:=second1is2(cut,a1,a2):is(a2,std(df(a1,a2))) a@1a:=stm(a):cut 2a:=std(a):cut dfis:=pair1is1(cut,a):is"e"(dif,df(1a,2a),a) isdf:=pair1is2(cut,a):is"e"(dif,a,df(1a,2a)) a2@[b1:cut][b2:cut] 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)))) 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)) a@[r1:cut][r2:cut] 1sdissmsd:=ispl1(r1,stm(df(r1,r2)),2a,isstm(r1,r2)):is(pl(r1,2a),pl(stm(df(r1,r2)),2a)) smsdis1sd:=symis(cut,pl(r1,2a),pl(stm(df(r1,r2)),2a),1sdissmsd):is(pl(stm(df(r1,r2)),2a),pl(r1,2a)) sm2issmsd:=ispl2(r2,std(df(r1,r2)),1a,isstd(r1,r2)):is(pl(1a,r2),pl(1a,std(df(r1,r2)))) smsdissm2:=symis(cut,pl(1a,r2),pl(1a,std(df(r1,r2))),sm2issmsd):is(pl(1a,std(df(r1,r2))),pl(1a,r2)) a2@[r:cut][i:is(a1,r)] issm:=isf(cut,dif,[t:cut]df(t,a2),a1,r,i):is"e"(dif,df(a1,a2),df(r,a2)) r@[i:is(a2,r)] issd:=isf(cut,dif,[t:cut]df(a1,t),a2,r,i):is"e"(dif,df(a1,a2),df(a1,r)) b2@[i:is(a1,b1)][j:is(a2,b2)] 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)) a@[b:dif] 1b:=stm(b):cut 2b:=std(b):cut eq:=is(pl(1a,2b),pl(1b,2a)):'prop' b2@[i:is(pl(a1,b2),pl(b1,a2))] 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)) r2@[i:is(pl(1a,r2),pl(r1,2a))] 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)) r2@[i:is(pl(r1,2a),pl(1a,r2))] 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) b2@[e:eq(df(a1,a2),df(b1,b2))] 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)) a@satzd163:=refis(cut,pl(1a,2a)):eq(a,a) refeq:=satzd163:eq(a,a) b@[i:is"e"(dif,a,b)] refeq1:=isp(dif,[x:dif]eq(a,x),a,b,refeq,i):eq(a,b) refeq2:=isp(dif,[x:dif]eq(x,a),a,b,refeq,i):eq(b,a) b2@[i:is(a1,b1)][j:is(a2,b2)] eqsmsd:=refeq1(df(a1,a2),df(b1,b2),issmsd(i,j)):eq(df(a1,a2),df(b1,b2)) r@[i:is(a1,r)] eqsm:=refeq1(df(a1,a2),df(r,a2),issm(i)):eq(df(a1,a2),df(r,a2)) r@[i:is(a2,r)] eqsd:=refeq1(df(a1,a2),df(a1,r),issd(i)):eq(df(a1,a2),df(a1,r)) b@[e:eq(a,b)] satzd164:=symis(cut,pl(1a,2b),pl(1b,2a),e):eq(b,a) symeq:=satzd164:eq(b,a) b@[c:dif] 1c:=stm(c):cut 2c:=std(c):cut [e:eq(a,b)][f:eq(b,c)] +1d165 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))) 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))) -1d165 satzd165:=satz136b(pl(1a,2c),pl(1c,2a),pl(1b,2b),t2".1d165"):eq(a,c) treq:=satzd165:eq(a,c) c@[e:eq(c,a)][f:eq(c,b)] treq1:=treq(a,c,b,symeq(c,a,e),f):eq(a,b) c@[e:eq(a,c)][f:eq(b,c)] treq2:=treq(a,c,b,e,symeq(b,c,f)):eq(a,b) c@[d:dif][e1:eq(a,b)][e2:eq(b,c)][e3:eq(c,d)] tr3eq:=treq(a,b,d,e1,treq(b,c,d,e2,e3)):eq(a,d) d@[e:dif][e1:eq(a,b)][e2:eq(b,c)][e3:eq(c,d)][e4:eq(d,e)] tr4eq:=tr3eq(a,b,c,e,e1,e2,treq(c,d,e,e3,e4)):eq(a,e) a@posd:=more(1a,2a):'prop' zero:=is(1a,2a):'prop' negd:=less(1a,2a):'prop' a2@[m:more(a1,a2)] posdi:=ismore12(a1,stm(df(a1,a2)),a2,std(df(a1,a2)),isstm(a1,a2),isstd(a1,a2),m):posd(df(a1,a2)) a2@[i:is(a1,a2)] zeroi:=tr3is(cut,stm(df(a1,a2)),a1,a2,std(df(a1,a2)),stmis(a1,a2),i,isstd(a1,a2)):zero(df(a1,a2)) a2@[l:less(a1,a2)] negdi:=isless12(a1,stm(df(a1,a2)),a2,std(df(a1,a2)),isstm(a1,a2),isstd(a1,a2),l):negd(df(a1,a2)) a@axrde:=satz123b(1a,2a):ec3(zero(a),posd(a),negd(a)) axrdo:=satz123a(1a,2a):or3(zero(a),posd(a),negd(a)) axrd:=orec3i(zero(a),posd(a),negd(a),axrdo,axrde):orec3(zero(a),posd(a),negd(a)) [p:'prop'][p1:[t:posd(a)]p][p2:[t:zero(a)]p][p3:[t:negd(a)]p] rappd:=or3app(zero(a),posd(a),negd(a),p,axrdo,p2,p1,p3):p a@[p:posd(a)] pnot0d:=ec3e21(zero(a),posd(a),negd(a),axrde,p):not(zero(a)) pnotnd:=ec3e23(zero(a),posd(a),negd(a),axrde,p):not(negd(a)) a@[z:zero(a)] 0notpd:=ec3e12(zero(a),posd(a),negd(a),axrde,z):not(posd(a)) 0notnd:=ec3e13(zero(a),posd(a),negd(a),axrde,z):not(negd(a)) a@[n:negd(a)] nnotpd:=ec3e32(zero(a),posd(a),negd(a),axrde,n):not(posd(a)) nnot0d:=ec3e31(zero(a),posd(a),negd(a),axrde,n):not(zero(a)) b@[e:eq(a,b)][p:posd(a)] +iv1d 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)) -iv1d eqposd:=satz136a(1b,2b,2a,t1".iv1d"):posd(b) e@[z:zero(a)] +*iv1d 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)) -iv1d z@eqzero:=satz136b(1b,2b,2a,t2".iv1d"):zero(b) e@[n:negd(a)] +*iv1d 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)) -iv1d n@eqnegd:=satz136c(1b,2b,2a,t3".iv1d"):negd(b) b@[z:zero(a)][y:zero(b)] 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) @[r:cut] pdofrp:=df(pl(r,1rp),1rp):dif ndofrp:=df(1rp,pl(r,1rp)):dif [s:cut][i:is(r,s)] isrpepd:=refeq1(pdofrp(r),pdofrp(s),isf(cut,dif,[x:cut]pdofrp(x),r,s,i)):eq(pdofrp(r),pdofrp(s)) isrpend:=refeq1(ndofrp(r),ndofrp(s),isf(cut,dif,[x:cut]ndofrp(x),r,s,i)):eq(ndofrp(r),ndofrp(s)) s@[e:eq(pdofrp(r),pdofrp(s))] +*iv1d 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)) -iv1d e@isrpipd:=satz136b(r,s,1rp,t4".iv1d"):is(r,s) s@[e:eq(ndofrp(r),ndofrp(s))] +*iv1d 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)) -iv1d e@isrpind:=symis(cut,s,r,satz136b(s,r,1rp,t5".iv1d")):is(r,s) r@posdirp:=posdi(pl(r,1rp),1rp,ismore1(pl(1rp,r),pl(r,1rp),1rp,compl(1rp,r),satz133(1rp,r))):posd(pdofrp(r)) negdirp:=negdi(1rp,pl(r,1rp),isless2(pl(1rp,r),pl(r,1rp),1rp,compl(1rp,r),satz133a(1rp,r))):negd(ndofrp(r)) a@[p:posd(a)] rpofpd:=mn(1a,2a,p):cut +*iv1d 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)) -iv1d p@eqpdrp1:=eqi1(a,pl(rpofpd,1rp),1rp,t6".iv1d"):eq(a,pdofrp(rpofpd(a,p))) eqpdrp2:=symeq(a,pdofrp(rpofpd(a,p)),eqpdrp1):eq(pdofrp(rpofpd(a,p)),a) a@[n:negd(a)] rpofnd:=mn(2a,1a,satz122(1a,2a,n)):cut +*iv1d 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)) -iv1d n@eqndrp1:=eqi1(a,1rp,pl(rpofnd,1rp),t7".iv1d"):eq(a,ndofrp(rpofnd(a,n))) eqndrp2:=symeq(a,ndofrp(rpofnd(a,n)),eqndrp1):eq(ndofrp(rpofnd(a,n)),a) @[h:dif][p:posd(h)][k:dif][q:posd(k)][e:eq(h,k)] +*iv1d 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))) -iv1d e@eqpderp:=isrpipd(rpofpd(h,p),rpofpd(k,q),t8".iv1d"):is(rpofpd(h,p),rpofpd(k,q)) q@[i:is(rpofpd(h,p),rpofpd(k,q))] +*iv1d i@t9:=isrpepd(rpofpd(h,p),rpofpd(k,q),i):eq(pdofrp(rpofpd(h,p)),pdofrp(rpofpd(k,q))) -iv1d i@eqpdirp:=tr3eq(h,pdofrp(rpofpd(h,p)),pdofrp(rpofpd(k,q)),k,eqpdrp1(h,p),t9".iv1d",eqpdrp2(k,q)):eq(h,k) h@[n:negd(h)][k:dif][o:negd(k)][e:eq(h,k)] +*iv1d 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))) -iv1d e@eqnderp:=isrpind(rpofnd(h,n),rpofnd(k,o),t10".iv1d"):is(rpofnd(h,n),rpofnd(k,o)) o@[i:is(rpofnd(h,n),rpofnd(k,o))] +*iv1d i@t11:=isrpend(rpofnd(h,n),rpofnd(k,o),i):eq(ndofrp(rpofnd(h,n)),ndofrp(rpofnd(k,o))) -iv1d i@eqndirp:=tr3eq(h,ndofrp(rpofnd(h,n)),ndofrp(rpofnd(k,o)),k,eqndrp1(h,n),t11".iv1d",eqndrp2(k,o)):eq(h,k) @[r:cut] +*iv1d r@t12:=eqpdrp1(pdofrp(r),posdirp(r)):eq(pdofrp(r),pdofrp(rpofpd(pdofrp(r),posdirp(r)))) -iv1d r@isrppd1:=isrpipd(r,rpofpd(pdofrp(r),posdirp(r)),t12".iv1d"):is(r,rpofpd(pdofrp(r),posdirp(r))) isrppd2:=symis(cut,r,rpofpd(pdofrp(r),posdirp(r)),isrppd1):is(rpofpd(pdofrp(r),posdirp(r)),r) +*iv1d r@t13:=eqndrp1(ndofrp(r),negdirp(r)):eq(ndofrp(r),ndofrp(rpofnd(ndofrp(r),negdirp(r)))) -iv1d r@isrpnd1:=isrpind(r,rpofnd(ndofrp(r),negdirp(r)),t13".iv1d"):is(r,rpofnd(ndofrp(r),negdirp(r))) isrpnd2:=symis(cut,r,rpofnd(ndofrp(r),negdirp(r)),isrpnd1):is(rpofnd(ndofrp(r),negdirp(r)),r) a2@[r:cut] 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))) lemmad2:=symeq(df(a1,a2),df(pl(a1,r),pl(a2,r)),lemmad1):eq(df(pl(a1,r),pl(a2,r)),df(a1,a2)) a@[r:cut] 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))) lemmad4:=symeq(a,df(pl(1a,r),pl(2a,r)),lemmad3):eq(df(pl(1a,r),pl(2a,r)),a) a@absd:=ite(negd(a),dif,df(2a,1a),a):dif [n:negd(a)] absnd:=refeq1(absd(a),df(2a,1a),itet(negd(a),dif,df(2a,1a),a,n)):eq(absd(a),df(2a,1a)) a@[n:not(negd(a))] absnnd:=refeq1(absd(a),a,itef(negd(a),dif,df(2a,1a),a,n)):eq(absd(a),a) a2@[l:less(a1,a2)] 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)) a2@[m:moreis(a1,a2)] 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)) b@[e:eq(a,b)] +iv2d [n:negd(a)] 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)) 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)) e@[n:not(negd(a))] 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)) -iv2d 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)) a@[p:posd(a)] satzd166a:=eqposd(a,absd(a),symeq(absd(a),a,absnnd(a,pnotnd(a,p))),p):posd(absd(a)) a@[n:negd(a)] +2d166 t1:=posdi(2a,1a,satz122(1a,2a,n)):posd(df(2a,1a)) -2d166 satzd166b:=eqposd(df(2a,1a),absd(a),symeq(absd(a),df(2a,1a),absnd(a,n)),t1".2d166"):posd(absd(a)) b@[p:posd(a)][q:posd(b)][e:eq(absd(a),absd(b))] 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) b@[n:negd(a)][o:negd(b)][e:eq(absd(a),absd(b))] +*2d166 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)) -2d166 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) a@[n:not(zero(a))] 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)) a@[z:zero(a)] satzd166f:=eqzero(a,absd(a),symeq(absd(a),a,absnnd(a,0notnd(a,z))),z):zero(absd(a)) b@mored:=more(pl(1a,2b),pl(1b,2a)):'prop' b2@[m:more(pl(a1,b2),pl(b1,a2))] 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)) r2@[m:more(pl(1a,r2),pl(r1,2a))] 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)) r2@[m:more(pl(r1,2a),pl(1a,r2))] 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) b2@[m:mored(df(a1,a2),df(b1,b2))] 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)) b@lessd:=less(pl(1a,2b),pl(1b,2a)):'prop' [m:mored(a,b)] lemmad5:=satz121(pl(1a,2b),pl(1b,2a),m):lessd(b,a) b@[l:lessd(a,b)] lemmad6:=satz122(pl(1a,2b),pl(1b,2a),l):mored(b,a) b2@[l:less(pl(a1,b2),pl(b1,a2))] 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)) r2@[l:less(pl(1a,r2),pl(r1,2a))] lessdi1:=lemmad5(df(r1,r2),a,moredi2(a,r1,r2,satz122(pl(1a,r2),pl(r1,2a),l))):lessd(a,df(r1,r2)) r2@[l:less(pl(r1,2a),pl(1a,r2))] lessdi2:=lemmad5(a,df(r1,r2),moredi1(a,r1,r2,satz122(pl(r1,2a),pl(1a,r2),l))):lessd(df(r1,r2),a) b2@[l:lessd(df(a1,a2),df(b1,b2))] 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)) b@satzd167:=satz123(pl(1a,2b),pl(1b,2a)):orec3(eq(a,b),mored(a,b),lessd(a,b)) satzd167a:=satz123a(pl(1a,2b),pl(1b,2a)):or3(eq(a,b),mored(a,b),lessd(a,b)) satzd167b:=satz123b(pl(1a,2b),pl(1b,2a)):ec3(eq(a,b),mored(a,b),lessd(a,b)) d@1d:=stm(d):cut 2d:=std(d):cut [e:eq(a,b)][f:eq(c,d)][m:mored(a,c)] +*iv2d 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))) 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))) -iv2d m@eqmored12:=satz136a(pl(1b,2d),pl(1d,2b),pl(1a,2c),t5".iv2d"):mored(b,d) f@[l:lessd(a,c)] eqlessd12:=lemmad5(d,b,eqmored12(c,d,a,b,f,e,lemmad6(a,c,l))):lessd(b,d) c@[e:eq(a,b)][m:mored(a,c)] eqmored1:=eqmored12(a,b,c,c,e,refeq(c),m):mored(b,c) e@[m:mored(c,a)] eqmored2:=eqmored12(c,c,a,b,refeq(c),e,m):mored(c,b) e@[l:lessd(a,c)] eqlessd1:=eqlessd12(a,b,c,c,e,refeq(c),l):lessd(b,c) e@[l:lessd(c,a)] eqlessd2:=eqlessd12(c,c,a,b,refeq(c),e,l):lessd(c,b) b@moreq:=or(mored(a,b),eq(a,b)):'prop' lesseq:=or(lessd(a,b),eq(a,b)):'prop' [m:moreq(a,b)] 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) b@[l:lesseq(a,b)] 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) c@[e:eq(a,b)][m:moreq(a,c)] 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) e@[m:moreq(c,a)] 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) e@[l:lesseq(a,c)] eqlesseq1:=satzd168a(c,b,eqmoreq2(a,b,c,e,satzd168b(a,c,l))):lesseq(b,c) e@[l:lesseq(c,a)] eqlesseq2:=satzd168a(b,c,eqmoreq1(a,b,c,e,satzd168b(c,a,l))):lesseq(c,b) d@[e:eq(a,b)][f:eq(c,d)][m:moreq(a,c)] eqmoreq12:=eqmoreq1(a,b,d,e,eqmoreq2(c,d,a,f,m)):moreq(b,d) f@[l:lesseq(a,c)] eqlesseq12:=eqlesseq1(a,b,d,e,eqlesseq2(c,d,a,f,l)):lesseq(b,d) b@[m:mored(a,b)] moreqi1:=ori1(mored(a,b),eq(a,b),m):moreq(a,b) b@[l:lessd(a,b)] lesseqi1:=ori1(lessd(a,b),eq(a,b),l):lesseq(a,b) b@[e:eq(a,b)] moreqi2:=ori2(mored(a,b),eq(a,b),e):moreq(a,b) lesseqi2:=ori2(lessd(a,b),eq(a,b),e):lesseq(a,b) b@[m:moreq(a,b)] 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)) b@[l:lesseq(a,b)] satzd167d:=th9"l.ec3"(eq(a,b),mored(a,b),lessd(a,b),satzd167b,l):not(mored(a,b)) b@[n:not(mored(a,b))] satzd167e:=th2"l.or3"(eq(a,b),mored(a,b),lessd(a,b),satzd167a,n):lesseq(a,b) b@[n:not(lessd(a,b))] 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) b@[m:mored(a,b)] 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)) b@[l:lessd(a,b)] 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)) b@[n:not(moreq(a,b))] 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) b@[n:not(lesseq(a,b))] 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) b@[z:zero(b)][p:posd(a)] 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) z@[m:mored(a,b)] 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) z@[n:negd(a)] 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) z@[l:lessd(a,b)] 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) +2d170 z@[p:posd(a)] t1:=moreqi1(absd(a),b,satzd169a(absd(a),b,z,satzd166a(a,p))):moreq(absd(a),b) z@[y:zero(a)] t2:=moreqi2(absd(a),b,treq(absd(a),a,b,absnnd(a,0notnd(a,y)),zeroeq(a,b,y,z))):moreq(absd(a),b) z@[n:negd(a)] t3:=moreqi1(absd(a),b,satzd169a(absd(a),b,z,satzd166b(a,n))):moreq(absd(a),b) -2d170 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) c@[l:lessd(a,b)][k:lessd(b,c)] +2d171 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))) 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))) -2d171 satzd171:=satz136c(pl(1a,2c),pl(1c,2a),pl(1b,2b),t2".2d171"):lessd(a,c) trlessd:=satzd171:lessd(a,c) c@[m:mored(a,b)][n:mored(b,c)] trmored:=lemmad6(c,a,trlessd(c,b,a,lemmad5(b,c,n),lemmad5(a,b,m))):mored(a,c) c@[l:lesseq(a,b)][k:lessd(b,c)] 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) c@[l:lessd(a,b)][k:lesseq(b,c)] 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) c@[m:moreq(a,b)][n:mored(b,c)] satzd172c:=lemmad6(c,a,satzd172b(c,b,a,lemmad5(b,c,n),satzd168a(a,b,m))):mored(a,c) c@[m:mored(a,b)][n:moreq(b,c)] satzd172d:=lemmad6(c,a,satzd172a(c,b,a,satzd168a(b,c,n),lemmad5(a,b,m))):mored(a,c) c@[l:lesseq(a,b)][k:lesseq(b,c)] +2d173 [j:lessd(a,b)] t1:=lesseqi1(a,c,satzd172b(j,k)):lesseq(a,c) k@[e:eq(a,b)] t2:=eqlesseq1(b,a,c,symeq(a,b,e),k):lesseq(a,c) -2d173 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) trlesseq:=satzd173:lesseq(a,c) c@[m:moreq(a,b)][n:moreq(b,c)] trmoreq:=satzd168b(c,a,trlesseq(c,b,a,satzd168a(b,c,n),satzd168a(a,b,m))):moreq(a,c) a@ratd:=[t:not(zero(a))]ratrp(rpofpd(absd(a),satzd166e(a,t))):'prop' irratd:=not(ratd(a)):'prop' b@[e:eq(a,b)][r:ratd(a)] +*iv2d r@[n:not(zero(b))] t6:=th3"l.imp"(zero(a),zero(b),n,[t:zero(a)]eqzero(a,b,e,t)):not(zero(a)) 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))) t8:=isp(cut,[t:cut]ratrp(t),rpofpd(absd(a),satzd166e(a,t6)),rpofpd(absd(b),satzd166e(b,n)),r,t7):ratrp(rpofpd(absd(b),satzd166e(b,n))) -iv2d r@eqratd:=[t:not(zero(b))]t8".iv2d"(t):ratd(b) e@[i:irratd(a)] eqirratd:=th3"l.imp"(ratd(b),ratd(a),i,[t:ratd(b)]eqratd(b,a,symeq(a,b,e),t)):irratd(b) a@[z:zero(a)] 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) @[r:cut][i:irratrp(r)][x0:rat] +*iv2d x0@[s:ratrp(pl(r,rpofrt(x0)))][y0:rat][j:is(pl(r,rpofrt(x0)),rpofrt(y0))] 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)) t10:=ismore1(pl(rpofrt(x0),r),rpofrt(y0),rpofrt(x0),t9,satz133(rpofrt(x0),r)):more(rpofrt(y0),rpofrt(x0)) t11:=satz154d(y0,x0,t10):more"rt"(y0,x0) t12:=satz155b(y0,x0,t11):is(rpofrt(mn"rt"(y0,x0,t11)),mn(rpofrt(y0),rpofrt(x0),satz154a(y0,x0,t11))) t13:=satz140g(rpofrt(y0),rpofrt(x0),r,satz154a(y0,x0,t11),t9):is(r,mn(rpofrt(y0),rpofrt(x0),satz154a(y0,x0,t11))) 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))) t15:=somei(rat,[x:rat]is(r,rpofrt(x)),mn"rt"(y0,x0,t11),t14):ratrp(r) 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))]i):con -iv2d x0@remark1:=[t:ratrp(pl(r,rpofrt(x0)))]t16".iv2d"(t):irratrp(pl(r,rpofrt(x0))) +*iv2d r@rp:=pdofrp(r):dif rn:=ndofrp(r):dif t17:=posdirp(r):posd(rp) t18:=pnot0d(rp,t17):not(zero(rp)) t19:=nnot0d(rn,negdirp(r)):not(zero(rn)) [n:not(zero(rp))] 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))) 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) [n:not(zero(rn))] 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))) r@[s:cut][i:is(r,s)][rr:ratrp(r)] t23:=isp(cut,[x:cut]ratrp(x),r,s,rr,i):ratrp(s) i@[rs:ratrp(s)] t24:=isp1(cut,[x:cut]ratrp(x),s,r,rs,i):ratrp(r) -iv2d r@[rr:ratrp(r)] remark2a:=[t:not(zero(pdofrp(r)))]t23".iv2d"(rpofpd(absd(pdofrp(r)),satzd166e(pdofrp(r),t)),t20".iv2d"(t),rr):ratd(pdofrp(r)) remark2b:=t17".iv2d":posd(pdofrp(r)) remark3a:=[t:not(zero(ndofrp(r)))]t23".iv2d"(rpofpd(absd(ndofrp(r)),satzd166e(ndofrp(r),t)),t22".iv2d"(t),rr):ratd(ndofrp(r)) remark3b:=negdirp(r):negd(ndofrp(r)) r@[i:irratrp(r)] 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"),t)):irratd(pdofrp(r)) remark4b:=t17".iv2d":posd(pdofrp(r)) 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"),t)):irratd(ndofrp(r)) remark5b:=negdirp(r):negd(ndofrp(r)) a@natd:=and(posd(a),[t:posd(a)]natrp(rpofpd(a,t))):'prop' [n:natd(a)] natposd:=ande1(posd(a),[t:posd(a)]natrp(rpofpd(a,t)),n):posd(a) natderp:=ande2"l.r"(posd(a),[t:posd(a)]natrp(rpofpd(a,t)),n):natrp(rpofpd(a,natposd(a,n))) b@[e:eq(a,b)][n:natd(a)] +*iv2d n"rp"@t25:=eqposd(a,b,e,natposd(a,n)):posd(b) [p:posd(b)] t26:=eqpderp(a,natposd(a,n),b,p,e):is(rpofpd(a,natposd(a,n)),rpofpd(b,p)) t27:=isp(cut,[t:cut]natrp(t),rpofpd(a,natposd(a,n)),rpofpd(b,p),natderp(a,n),t26):natrp(rpofpd(b,p)) -iv2d n@eqnatd:=andi(posd(b),[t:posd(b)]natrp(rpofpd(b,t)),t25".iv2d",[t:posd(b)]t27".iv2d"(t)):natd(b) @[x:nat] pdofnt:=pdofrp(rpofnt(x)):dif +*iv2d x@t28:=posdirp(rpofnt(x)):posd(pdofnt(x)) [p:posd(pdofnt(x))] t29:=isrppd1(rpofnt(x)):is(rpofnt(x),rpofpd(pdofnt(x),t28)) t30:=eqpderp(pdofnt(x),t28,pdofnt(x),p,refeq(pdofnt(x))):is(rpofpd(pdofnt(x),t28),rpofpd(pdofnt(x),p)) t31:=tris(cut,rpofnt(x),rpofpd(pdofnt(x),t28),rpofpd(pdofnt(x),p),t29,t30):is(rpofnt(x),rpofpd(pdofnt(x),p)) t32:=isp(cut,[t:cut]natrp(t),rpofnt(x),rpofpd(pdofnt(x),p),natrpi(x),t31):natrp(rpofpd(pdofnt(x),p)) -iv2d 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)) a@intd:=or(zero(a),natd(absd(a))):'prop' b@[e:eq(a,b)][i:intd(a)] +*iv2d i"rp"@[z:zero(a)] t33:=eqzero(a,b,e,z):zero(b) i"rp"@[n:natd(absd(a))] t34:=eqnatd(absd(a),absd(b),eqabsd(a,b,e),n):natd(absd(b)) -iv2d 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) a@[n:natd(a)] +*iv2d n"rp"@t34a:=symeq(absd(a),a,absnnd(a,pnotnd(a,natposd(a,n)))):eq(a,absd(a)) t35:=eqnatd(a,absd(a),t34a,n):natd(absd(a)) -iv2d n@natintd:=ori2(zero(a),natd(absd(a)),t35".iv2d"):intd(a) a@[p:posd(a)][i:intd(a)] +*iv2d i"rp"@t36:=ore2(zero(a),natd(absd(a)),i,pnot0d(a,p)):natd(absd(a)) -iv2d i@posintnatd:=eqnatd(absd(a),a,absnnd(a,pnotnd(a,p)),t36".iv2d"):natd(a) a@[z:zero(a)] intdi0:=ori1(zero(a),natd(absd(a)),z):intd(a) r@[n:natrp(r)] +*iv2d n"rp"@t37:=posdirp(r):posd(pdofrp(r)) [p:posd(pdofrp(r))] 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)) t39:=isp(cut,[t:cut]natrp(t),r,rpofpd(pdofrp(r),p),n,t38):natrp(rpofpd(pdofrp(r),p)) -iv2d 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)) remark6:=natintd(pdofrp(r),remark6a):intd(pdofrp(r)) +*iv2d 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)) t41:=eqnatd(pdofrp(r),absd(ndofrp(r)),symeq(absd(ndofrp(r)),pdofrp(r),t40),remark6a):natd(absd(ndofrp(r))) -iv2d n@remark7:=ori2(zero(ndofrp(r)),natd(absd(ndofrp(r))),t41".iv2d"):intd(ndofrp(r)) a@[i:intd(a)] +2d174 [n:not(zero(a))] t1:=ore2(zero(a),natd(absd(a)),i,n):natd(absd(a)) t2:=ande2(posd(absd(a)),[t:posd(absd(a))]natrp(rpofpd(absd(a),t)),t1):[t:posd(absd(a))]natrp(rpofpd(absd(a),t)) t3:=lemmaiii5(rpofpd(absd(a),satzd166e(a,n)),t2):ratrp(rpofpd(absd(a),satzd166e(a,n))) -2d174 satzd174:=[t:not(zero(a))]t3".2d174"(t):ratd(a) b@pd:=df(pl(1a,1b),pl(2a,2b)):dif 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))) 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))) 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))) 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))) 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))) 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))) 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))) 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))) 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)) 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)) compd:=satzd175:eq(pd(a,b),pd(b,a)) c@[e:eq(a,b)] +iv3d 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))) -iv3d eqpd1:=eqi12(pl(1a,1c),pl(2a,2c),pl(1b,1c),pl(2b,2c),t1".iv3d"):eq(pd(a,c),pd(b,c)) 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)) d@[e:eq(a,b)][f:eq(c,d)] 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)) b@[z:zero(a)] +*iv3d 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))) -iv3d z@pd01:=eqi2(b,pl(1a,1b),pl(2a,2b),t2".iv3d"):eq(pd(a,b),b) b@[z:zero(b)] pd02:=treq(pd(a,b),pd(b,a),a,compd(a,b),pd01(b,a,z)):eq(pd(a,b),a) b@[p:posd(a)][q:posd(b)] ppd:=posdi(pl(1a,1b),pl(2a,2b),satz137(1a,2a,1b,2b,p,q)):posd(pd(a,b)) b@[n:negd(a)][o:negd(b)] npd:=negdi(pl(1a,1b),pl(2a,2b),satz137a(1a,2a,1b,2b,n,o)):negd(pd(a,b)) a@m0d:=df(2a,1a):dif 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)) m0deqb:=symeq(m0d(df(a1,a2)),df(a2,a1),m0deqa):eq(df(a2,a1),m0d(df(a1,a2))) b@[e:eq(a,b)] +*iv3d 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)) -iv3d e@eqm0d:=eqi12(2a,1a,2b,1b,t3".iv3d"):eq(m0d(a),m0d(b)) a@[p:posd(a)] satzd176a:=negdi(2a,1a,satz121(1a,2a,p)):negd(m0d(a)) a@[z:zero(a)] satzd176b:=zeroi(2a,1a,symis(cut,1a,2a,z)):zero(m0d(a)) a@[n:negd(a)] satzd176c:=posdi(2a,1a,satz122(1a,2a,n)):posd(m0d(a)) a@[n:negd(m0d(a))] satzd176d:=satz122(2a,1a,isless12(stm(m0d(a)),2a,std(m0d(a)),1a,stmis(2a,1a),stdis(2a,1a),n)):posd(a) a@[z:zero(m0d(a))] 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) a@[p:posd(m0d(a))] satzd176f:=satz121(2a,1a,ismore12(stm(m0d(a)),2a,std(m0d(a)),1a,stmis(2a,1a),stdis(2a,1a),p)):negd(a) a@[z:zero(a)] m0d0:=zeroeq(m0d(a),a,satzd176b(a,z),z):eq(m0d(a),a) +3d177 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) -3d177 a@satzd177:=refeq1(m0d(m0d(a)),a,t1".3d177"):eq(m0d(m0d(a)),a) satzd177a:=symeq(m0d(m0d(a)),a,satzd177):eq(a,m0d(m0d(a))) b@[e:eq(a,m0d(b))] satzd177b:=treq(m0d(a),m0d(m0d(b)),b,eqm0d(a,m0d(b),e),satzd177(b)):eq(m0d(a),b) satzd177c:=symeq(m0d(a),b,satzd177b):eq(b,m0d(a)) b@[e:eq(m0d(a),b)] satzd177d:=satzd177c(b,a,symeq(m0d(a),b,e)):eq(a,m0d(b)) satzd177e:=symeq(a,m0d(b),satzd177d):eq(m0d(b),a) +3d178 a@[p:posd(a)] 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)) a@[z:zero(a)] 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)) a@[n:negd(a)] 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)) -3d178 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)) satzd178a:=symeq(absd(m0d(a)),absd(a),satzd178):eq(absd(a),absd(m0d(a))) +3d179 t1:=pdeq1b(a,2a,1a):eq(df(pl(1a,2a),pl(2a,1a)),pd(a,m0d(a))) t2:=zeroi(pl(1a,2a),pl(2a,1a),compl(1a,2a)):zero(df(pl(1a,2a),pl(2a,1a))) -3d179 satzd179:=eqzero(df(pl(1a,2a),pl(2a,1a)),pd(a,m0d(a)),t1".3d179",t2".3d179"):zero(pd(a,m0d(a))) satzd179a:=eqzero(pd(a,m0d(a)),pd(m0d(a),a),compd(a,m0d(a)),satzd179):zero(pd(m0d(a),a)) 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))) satzd180a:=symeq(m0d(pd(a,b)),pd(m0d(a),m0d(b)),satzd180):eq(pd(m0d(a),m0d(b)),m0d(pd(a,b))) md:=pd(a,m0d(b)):dif 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))) 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))) 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))) 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))) mdeq2a:=pdeq12a(r1,r2,2a,1a):eq(md(df(r1,r2),a),df(pl(r1,2a),pl(r2,1a))) mdeq2b:=pdeq12b(r1,r2,2a,1a):eq(df(pl(r1,2a),pl(r2,1a)),md(df(r1,r2),a)) c@[e:eq(a,b)] eqmd1:=eqpd1(a,b,m0d(c),e):eq(md(a,c),md(b,c)) eqmd2:=eqpd2(m0d(a),m0d(b),c,eqm0d(a,b,e)):eq(md(c,a),md(c,b)) d@[e:eq(a,b)][f:eq(c,d)] 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)) 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)) satzd181a:=symeq(m0d(md(b,a)),md(a,b),satzd181(b,a)):eq(md(a,b),m0d(md(b,a))) +3d182 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))) t2:=symeq(md(a,b),df(pl(1a,2b),pl(1b,2a)),t1):eq(df(pl(1a,2b),pl(1b,2a)),md(a,b)) t3:=stmis(pl(1a,2b),pl(1b,2a)):is(stm(df(pl(1a,2b),pl(1b,2a))),pl(1a,2b)) t4:=stdis(pl(1a,2b),pl(1b,2a)):is(std(df(pl(1a,2b),pl(1b,2a))),pl(1b,2a)) -3d182 [p:posd(md(a,b))] +*3d182 p@t5:=eqposd(md(a,b),df(pl(1a,2b),pl(1b,2a)),t1,p):posd(df(pl(1a,2b),pl(1b,2a))) -3d182 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) b@[z:zero(md(a,b))] +*3d182 z@t6:=eqzero(md(a,b),df(pl(1a,2b),pl(1b,2a)),t1,z):zero(df(pl(1a,2b),pl(1b,2a))) -3d182 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) b@[n:negd(md(a,b))] +*3d182 n@t7:=eqnegd(md(a,b),df(pl(1a,2b),pl(1b,2a)),t1,n):negd(df(pl(1a,2b),pl(1b,2a))) -3d182 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) b@[m:mored(a,b)] +*3d182 m@t8:=posdi(pl(1a,2b),pl(1b,2a),m):posd(df(pl(1a,2b),pl(1b,2a))) -3d182 m@satzd182d:=eqposd(df(pl(1a,2b),pl(1b,2a)),md(a,b),t2".3d182",t8".3d182"):posd(md(a,b)) b@[e:eq(a,b)] +*3d182 e@t9:=zeroi(pl(1a,2b),pl(1b,2a),e):zero(df(pl(1a,2b),pl(1b,2a))) -3d182 e@satzd182e:=eqzero(df(pl(1a,2b),pl(1b,2a)),md(a,b),t2".3d182",t9".3d182"):zero(md(a,b)) b@[l:lessd(a,b)] +*3d182 l@t10:=negdi(pl(1a,2b),pl(1b,2a),l):negd(df(pl(1a,2b),pl(1b,2a))) -3d182 l@satzd182f:=eqnegd(df(pl(1a,2b),pl(1b,2a)),md(a,b),t2".3d182",t10".3d182"):negd(md(a,b)) +3d183 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)))) t2:=t1(b,a):is(pl(1b,2a),pl(stm(m0d(a)),std(m0d(b)))) -3d183 b@[m:mored(a,b)] 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)) b@[e:eq(a,b)] staz183b:=eqm0d(a,b,e):eq(m0d(a),m0d(b)) b@[l:lessd(a,b)] 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)) b@[l:lessd(m0d(a),m0d(b))] satzd183d:=eqmored12(m0d(m0d(a)),a,m0d(m0d(b)),b,satzd177(a),satzd177(b),satzd183c(m0d(a),m0d(b),l)):mored(a,b) b@[e:eq(m0d(a),m0d(b))] satzd183e:=tr3eq(a,m0d(m0d(a)),m0d(m0d(b)),b,satzd177a(a),eqm0d(m0d(a),m0d(b),e),satzd177(b)):eq(a,b) b@[m:mored(m0d(a),m0d(b))] satzd183f:=eqlessd12(m0d(m0d(a)),a,m0d(m0d(b)),b,satzd177(a),satzd177(b),satzd183a(m0d(a),m0d(b),m)):lessd(a,b) +3d184 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))) 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)))) 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)))) -3d184 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))))) 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))) asspd2:=symeq(pd(pd(a,b),c),pd(a,pd(b,c)),asspd1):eq(pd(a,pd(b,c)),pd(pd(a,b),c)) 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)) 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))) 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) 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) 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))) c@satzd186:=asspd1:eq(pd(pd(a,b),c),pd(a,pd(b,c))) 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) [x:dif][e:eq(pd(b,x),a)] 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) satzd187d:=symeq(md(a,b),x,satzd187c):eq(x,md(a,b)) x@[e:eq(pd(x,b),a)] satzd187e:=satzd187c(treq(pd(b,x),pd(x,b),a,compd(b,x),e)):eq(md(a,b),x) satzd187f:=symeq(md(a,b),x,satzd187e):eq(x,md(a,b)) +3d188 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)) t2:=symeq(md(pd(a,c),pd(b,c)),md(a,b),t1):eq(md(a,b),md(pd(a,c),pd(b,c))) -3d188 c@[m:mored(pd(a,c),pd(b,c))] +*3d188 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)) -3d188 m@satzd188a:=satzd182a(a,b,t3".3d188"):mored(a,b) c@[e:eq(pd(a,c),pd(b,c))] +*3d188 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)) -3d188 e@satzd188b:=satzd182b(a,b,t4".3d188"):eq(a,b) c@[l:lessd(pd(a,c),pd(b,c))] +*3d188 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)) -3d188 l@satzd188c:=satzd182c(a,b,t5".3d188"):lessd(a,b) c@[m:mored(a,b)] +*3d188 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))) -3d188 m@satzd188d:=satzd182a(pd(a,c),pd(b,c),t6".3d188"):mored(pd(a,c),pd(b,c)) c@[e:eq(a,b)] satzd188e:=eqpd1(a,b,c,e):eq(pd(a,c),pd(b,c)) c@[l:lessd(a,b)] +*3d188 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))) -3d188 l@satzd188f:=satzd182c(pd(a,c),pd(b,c),t7".3d188"):lessd(pd(a,c),pd(b,c)) c@[m:mored(pd(c,a),pd(c,b))] 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) c@[e:eq(pd(c,a),pd(c,b))] 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) [l:lessd(pd(c,a),pd(c,b))] 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) c@[m:mored(a,b)] 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)) c@[e:eq(a,b)] satzd188l:=eqpd2(a,b,c,e):eq(pd(c,a),pd(c,b)) c@[l:lessd(a,b)] 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)) d@[e:eq(a,b)][m:mored(c,d)] 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)) 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)) e@[l:lessd(c,d)] 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)) 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)) d@[m:mored(a,b)][n:mored(c,d)] 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)) d@[l:lessd(a,b)][k:lessd(c,d)] 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)) d@[m:moreq(a,b)][n:mored(c,d)] 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)) d@[m:mored(a,b)][n:moreq(c,d)] 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)) d@[l:lesseq(a,b)][k:lessd(c,d)] 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)) d@[l:lessd(a,b)][k:lesseq(c,d)] 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)) d@[m:moreq(a,b)][n:moreq(c,d)] +3d191 [e:eq(a,b)][f:eq(c,d)] t1:=moreqi2(pd(a,c),pd(b,d),eqpd12(a,b,c,d,e,f)):moreq(pd(a,c),pd(b,d)) e@[o:mored(c,d)] t2:=moreqi1(pd(a,c),pd(b,d),satzd190a(m,o)):moreq(pd(a,c),pd(b,d)) 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)) n@[o:mored(a,b)] t4:=moreqi1(pd(a,c),pd(b,d),satzd190b(o,n)):moreq(pd(a,c),pd(b,d)) -3d191 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)) d@[l:lesseq(a,b)][k:lesseq(c,d)] 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)) b@td:=df(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b))):dif +iv4d a2@[r:cut] t1:=ists1(stm(df(a1,a2)),a1,r,stmis(a1,a2)):is(ts(stm(df(a1,a2)),r),ts(a1,r)) t2:=ists2(stm(df(a1,a2)),a1,r,stmis(a1,a2)):is(ts(r,stm(df(a1,a2))),ts(r,a1)) t3:=ists1(std(df(a1,a2)),a2,r,stdis(a1,a2)):is(ts(std(df(a1,a2)),r),ts(a2,r)) t4:=ists2(std(df(a1,a2)),a2,r,stdis(a1,a2)):is(ts(r,std(df(a1,a2))),ts(r,a2)) [s:cut] 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))) 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))) 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))) 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))) 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))) 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))) -iv4d 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)))) 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)))) 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)))) 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)))) 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))) 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)))) 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))) 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)))) 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)) +4d194 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))) 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))) -4d194 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)) comtd:=satzd194:eq(td(a,b),td(b,a)) c@[e:eq(a,b)] +*iv4d e@[r:cut] 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))) 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)))) -iv4d 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)) 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)) d@[e:eq(a,b)][f:eq(c,d)] 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)) b@[z:zero(a)] +4d192 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))) -4d192 satzd192a:=zeroi(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b)),t1".4d192"):zero(td(a,b)) b@[z:zero(b)] satzd192b:=eqzero(td(b,a),td(a,b),comtd(b,a),satzd192a(b,a,z)):zero(td(a,b)) b@[z:zero(a)] td01:=satzd192a(z):zero(td(a,b)) b@[z:zero(b)] td02:=satzd192b(z):zero(td(a,b)) 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))) 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))) 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))) satzd197d:=symeq(td(m0d(a),b),td(a,m0d(b)),satzd197c):eq(td(a,m0d(b)),td(m0d(a),b)) satzd197e:=symeq(td(m0d(a),b),m0d(td(a,b)),satzd197a):eq(m0d(td(a,b)),td(m0d(a),b)) satzd197f:=symeq(td(a,m0d(b)),m0d(td(a,b)),satzd197b):eq(m0d(td(a,b)),td(a,m0d(b))) 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)) satzd198a:=symeq(td(m0d(a),m0d(b)),td(a,b),satzd198):eq(td(a,b),td(m0d(a),m0d(b))) [p:posd(a)][q:posd(b)] +*iv4d q@[r:cut] 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)) [s:cut] 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)) 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))) 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)))) 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))) -iv4d q@ptdpp:=posdi(pl(ts(1a,1b),ts(2a,2b)),pl(ts(1a,2b),ts(2a,1b)),t17".iv4d"):posd(td(a,b)) b@[p:posd(a)][n:negd(b)] +*iv4d 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))) -iv4d n@ntdpn:=satzd176f(td(a,b),t18".iv4d"):negd(td(a,b)) b@[n:negd(a)][p:posd(b)] ntdnp:=eqnegd(td(b,a),td(a,b),comtd(b,a),ntdpn(b,a,p,n)):negd(td(a,b)) b@[n:negd(a)][o:negd(b)] 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)) b@[n:not(zero(a))][o:not(zero(b))] +*4d192 o@[p:posd(a)][q:posd(b)] t2:=pnot0d(td(a,b),ptdpp(a,b,p,q)):not(zero(td(a,b))) p@[m:negd(b)] t3:=nnot0d(td(a,b),ntdpn(a,b,p,m)):not(zero(td(a,b))) 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))) o@[m:negd(a)][p:posd(b)] t5:=nnot0d(td(a,b),ntdnp(a,b,m,p)):not(zero(td(a,b))) m@[l:negd(b)] t6:=pnot0d(td(a,b),ptdnn(a,b,m,l)):not(zero(td(a,b))) 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))) -4d192 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))) b@[z:zero(td(a,b))] +*4d192 z@[n:not(zero(a))] 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) -4d192 z@satzd192c:=[t:not(zero(a))]t8".4d192"(t):or(zero(a),zero(b)) +4d193 b@[p:posd(a)][q:posd(b)] 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))) p@[n:negd(b)] 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))) 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))) b@[z:zero(a)] 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))) b@[z:zero(b)] 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))) b@[n:negd(a)][p:posd(b)] 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))) n@[o:negd(b)] 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)) 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))) b@[p:posd(a)] 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))) b@[n:negd(a)] 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))) -4d193 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))) satzd103a:=symeq(absd(td(a,b)),td(absd(a),absd(b)),satzd193):eq(td(absd(a),absd(b)),absd(td(a,b))) @1df:=pdofrp(1rp):dif +4d195 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))) 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))) 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))))) -4d195 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) satzd195a:=symeq(td(a,1df),a,satzd195):eq(a,td(a,1df)) satzd195b:=treq(td(1df,a),td(a,1df),a,comtd(1df,a),satzd195):eq(td(1df,a),a) satzd195c:=symeq(td(1df,a),a,satzd195b):eq(a,td(1df,a)) b@[p:posd(a)][q:posd(b)] 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))) b@[n:negd(a)][o:negd(b)] 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))) b@[p:posd(a)][n:negd(b)] 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)))) b@[n:negd(a)][p:posd(b)] 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)))) +4d196 b@p1p2:=and(posd(a),posd(b)):'prop' p1n2:=and(posd(a),negd(b)):'prop' n1p2:=and(negd(a),posd(b)):'prop' n1n2:=and(negd(a),negd(b)):'prop' -4d196 b@[n:not(zero(a))][o:not(zero(b))][e:eq(td(a,b),td(absd(a),absd(b)))] +*4d196 o@t1:=ptdpp(absd(a),absd(b),satzd166e(a,n),satzd166e(b,o)):posd(td(absd(a),absd(b))) 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))) [p:posd(a)] t3:=th3"l.imp"(negd(b),negd(td(a,b)),t2,[t:negd(b)]ntdpn(a,b,p,t)):not(negd(b)) t4:=or3e2(zero(b),posd(b),negd(b),axrdo(b),t3,o):posd(b) t5:=andi(posd(a),posd(b),p,t4):p1p2 t6:=ori1(p1p2,n1n2,t5):or(p1p2,n1n2) e@[m:negd(a)] t7:=th3"l.imp"(posd(b),negd(td(a,b)),t2,[t:posd(b)]ntdnp(a,b,m,t)):not(posd(b)) t8:=or3e3(zero(b),posd(b),negd(b),axrdo(b),o,t7):negd(b) t9:=andi(negd(a),negd(b),m,t8):n1n2 t10:=ori2(p1p2,n1n2,t9):or(p1p2,n1n2) -4d196 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))) o@[e:eq(td(a,b),m0d(td(absd(a),absd(b))))] +*4d196 o@t11:=satzd176a(td(absd(a),absd(b)),t1):negd(m0d(td(absd(a),absd(b)))) 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))) [p:posd(a)] t13:=th3"l.imp"(posd(b),posd(td(a,b)),t12,[t:posd(b)]ptdpp(a,b,p,t)):not(posd(b)) t14:=or3e3(zero(b),posd(b),negd(b),axrdo(b),o,t13):negd(b) t15:=andi(posd(a),negd(b),p,t14):p1n2 t16:=ori1(p1n2,n1p2,t15):or(p1n2,n1p2) e@[m:negd(a)] t17:=th3"l.imp"(negd(b),posd(td(a,b)),t12,[t:negd(b)]ptdnn(a,b,m,t)):not(negd(b)) t18:=or3e2(zero(b),posd(b),negd(b),axrdo(b),t17,o):posd(b) t19:=andi(negd(a),posd(b),m,t18):n1p2 t20:=ori2(p1n2,n1p2,t19):or(p1n2,n1p2) -4d196 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))) +4d199 @[p:cut][q:cut][r:cut][s:cut][t:cut][u:cut] 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)))) 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)))) 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))))) -4d199 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))) asstd1:=satzd199:eq(td(td(a,b),c),td(a,td(b,c))) asstd2:=symeq(td(td(a,b),c),td(a,td(b,c)),satzd199):eq(td(a,td(b,c)),td(td(a,b),c)) +4d201 @[p:cut][q:cut][r:cut][s:cut][t:cut][u:cut] 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)))) -4d201 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))) 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))) disttpd2:=satzd201:eq(td(a,pd(b,c)),pd(td(a,b),td(a,c))) 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)) 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))) 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))) 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))) disttmd2:=satzd202:eq(td(a,md(b,c)),md(td(a,b),td(a,c))) 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)) 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))) satzd200:=satzd202:eq(td(a,md(b,c)),md(td(a,b),td(a,c))) [m:mored(a,b)] +4d203 t1:=satzd182d(a,b,m):posd(md(a,b)) -4d203 [p:posd(c)] +*4d203 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))) -4d203 p@satzd203a:=satzd182a(td(a,c),td(b,c),t2".4d203"):mored(td(a,c),td(b,c)) m@[z:zero(c)] satzd203b:=zeroeq(td(a,c),td(b,c),td02(a,c,z),td02(b,c,z)):eq(td(a,c),td(b,c)) m@[n:negd(c)] +*4d203 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))) -4d203 n@satzd203c:=satzd182c(td(a,c),td(b,c),t3".4d203"):lessd(td(a,c),td(b,c)) 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)) z@satzd203e:=zeroeq(td(c,a),td(c,b),td01(c,a,z),td01(c,b,z)):eq(td(c,a),td(c,b)) 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)) c@[l:lessd(a,b)][p:posd(c)] satzd203g:=lemmad5(td(b,c),td(a,c),satzd203a(b,a,c,lemmad6(a,b,l),p)):lessd(td(a,c),td(b,c)) l@[z:zero(c)] satzd203h:=zeroeq(td(a,c),td(b,c),td02(a,c,z),td02(b,c,z)):eq(td(a,c),td(b,c)) l@[n:negd(c)] satzd203j:=lemmad6(td(b,c),td(a,c),satzd203c(b,a,c,lemmad6(a,b,l),n)):mored(td(a,c),td(b,c)) 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)) z@satzd203l:=zeroeq(td(c,a),td(c,b),td01(c,a,z),td01(c,b,z)):eq(td(c,a),td(c,b)) 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)) +*iv4d @[p:cut][q:cut] 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)) [r:cut] 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))) 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))) a@[p:posd(a)] arp:=rpofpd(a,p):cut arpi:=ov(1rp,arp):cut ai:=pdofrp(arpi):dif 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))) 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)) 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))) t25:=eqi12(ts(1a,arpi),ts(2a,arpi),pl(1rp,1rp),1rp,t24):eq(df(ts(1a,arpi),ts(2a,arpi)),1df) t26:=treq(td(a,ai),df(ts(1a,arpi),ts(2a,arpi)),1df,t22,t25):eq(td(a,ai),1df) t27:=somei(dif,[x:dif]eq(td(a,x),1df),ai,t26):some"l"(dif,[x:dif]eq(td(a,x),1df)) a@[n:negd(a)] t28:=satzd176c(a,n):posd(m0d(a)) [h:dif][e:eq(td(m0d(a),h),1df)] t29:=treq(td(a,m0d(h)),td(m0d(a),h),1df,satzd197d(a,h),e):eq(td(a,m0d(h)),1df) t30:=somei(dif,[x:dif]eq(td(a,x),1df),m0d(h),t29):some"l"(dif,[x:dif]eq(td(a,x),1df)) 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)) -iv4d a@[n:not(zero(a))] 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)) b@[n:not(zero(b))][h:dif][k:dif][e:eq(td(b,h),a)][f:eq(td(b,k),a)] +4d204 t1:=treq2(td(b,h),td(b,k),a,e,f):eq(td(b,h),td(b,k)) 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))) t3:=ore2(zero(b),zero(md(h,k)),satzd192c(b,md(h,k),t2),n):zero(md(h,k)) -4d204 satzd204b:=satzd182b(h,k,t3".4d204"):eq(h,k) +*4d204 n@[h:dif][e:eq(td(b,h),1df)] 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) t5:=somei(dif,[x:dif]eq(td(b,x),a),td(h,a),t4):some"l"(dif,[x:dif]eq(td(b,x),a)) -4d204 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)) @[r:cut][s:cut][m:more(r,s)] +iv5d 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)) -iv5d morerpepd:=moredi12(pl(r,1rp),1rp,pl(s,1rp),1rp,t1".iv5d"):mored(pdofrp(r),pdofrp(s)) s@[m:mored(pdofrp(r),pdofrp(s))] +*iv5d m@t2:=morede12(pl(r,1rp),1rp,pl(s,1rp),1rp,m):more(pl(pl(r,1rp),1rp),pl(pl(s,1rp),1rp)) -iv5d 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) s@[l:less(r,s)] lessrpepd:=lemmad5(pdofrp(s),pdofrp(r),morerpepd(s,r,satz122(r,s,l))):lessd(pdofrp(r),pdofrp(s)) s@[l:lessd(pdofrp(r),pdofrp(s))] lessrpipd:=satz121(s,r,morerpipd(s,r,lemmad6(pdofrp(r),pdofrp(s),l))):less(r,s) +*iv5d @i:=1rp:cut 2:=pl(i,i):cut r@rp1:=pl(r,i):cut s@sp1:=pl(s,i):cut rps:=pl(r,s):cut rs:=ts(r,s):cut 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)) 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)) -iv5d s@lemmad8:=t4".iv5d":eq(pd(pdofrp(r),pdofrp(s)),pdofrp(pl(r,s))) +*iv5d 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)) 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)) 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))) 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)) 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)) 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)))) 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)) -iv5d s@lemmad9:=t11".iv5d":eq(td(pdofrp(r),pdofrp(s)),pdofrp(ts(r,s))) @[r:cut][s0:set(cut)] in:=esti(cut,r,s0):'prop' @[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)))] +5p205 t0@[r:cut] prop1:=all([x:cut][t:less(x,r)]in(x,s0)):'prop' prop2:=all([x:cut][t:more(x,r)]in(x,t0)):'prop' prop3:=and(prop1,prop2):'prop' p2@[r1:cut][r2:cut][pr1:prop3(r1)][pr2:prop3(r2)] t1:=ande2(prop1(r1),prop2(r1),pr1):prop2(r1) t2:=ande1(prop1(r2),prop2(r2),pr2):prop1(r2) [l:less(r1,r2)][x0:rat] rx:=rpofrt(x0):cut [l1:less(r1,rx)][l2:less(rx,r2)] t3:=t2:in(rx,s0) t4:=t1:in(rx,t0) t5:=ec3e31(is(rx,rx),more(rx,rx),less(rx,rx),satz123b(rx,rx),p2):con l@t6:=satz159app(r1,r2,l,con,[x:rat][t:less(r1,rpofrt(x))][u:less(rpofrt(x),r2)]t5(x,t,u)):con pr2@t7:=[t:less(r1,r2)]t6(t):not(less(r1,r2)) t8:=[t:more(r1,r2)]t6(r2,r1,pr2,pr1,satz121(r1,r2,t)):not(more(r1,r2)) t9:=or3e1(is(r1,r2),more(r1,r2),less(r1,r2),satz123a(r1,r2),t8,t7):is(r1,r2) p2@t10:=[x:cut][y:cut][t:prop3(x)][u:prop3(y)]t9(x,y,t,u):amone(cut,[x:cut]prop3(x)) [x0:rat] schnittprop:=some([y:cut]and(in(y,s0),lrt(y,x0))):'prop' p2@schnittset:=setof(rat,[x:rat]schnittprop(x)):set(rat) [r:cut][i:in(r,s0)][x0:rat][lx:lrt(r,x0)] t11:=andi(in(r,s0),lrt(r,x0),i,lx):and(in(r,s0),lrt(r,x0)) t12:=somei(cut,[y:cut]and(in(y,s0),lrt(y,x0)),r,t11):schnittprop(x0) t13:=estii(rat,[x:rat]schnittprop(x),x0,t12):in"rt"(x0,schnittset) r@[i:in(r,t0)][x0:rat][ux:urt(r,x0)][s:cut][j:in(s,s0)] t14:=satz122(s,r,p2):more(r,s) t15:=satz158b(r,x0,ux):moreis(rpofrt(x0),r) t16:=moreisi1(rpofrt(x0),s,satz127c(rpofrt(x0),r,s,t15,t14)):moreis(rpofrt(x0),s) t17:=satz158d(s,x0,t16):urt(s,x0) s@t18:=weli(ec(in(s,s0),lrt(s,x0)),[t:in(s,s0)]t17(t)):not(and(in(s,s0),lrt(s,x0))) ux@t19:=th5"l.some"(cut,[y:cut]and(in(y,s0),lrt(y,x0)),[y:cut]t18(y)):not(schnittprop(x0)) 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)) p2@[x0:rat][i:in"rt"(x0,schnittset)][y0:rat][l:less"rt"(y0,x0)] i@t21:=estie(rat,[x:rat]schnittprop(x),x0,i):schnittprop(x0) l@[r:cut][a:and(in(r,s0),lrt(r,x0))] t22:=ande1(in(r,s0),lrt(r,x0),a):in(r,s0) t23:=ande2(in(r,s0),lrt(r,x0),a):lrt(r,x0) t24:=satz120(r,x0,t23,y0,l):lrt(r,y0) t25:=andi(in(r,s0),lrt(r,y0),t22,t24):and(in(r,s0),lrt(r,y0)) t26:=somei(cut,[y:cut]and(in(y,s0),lrt(y,y0)),r,t25):schnittprop(y0) 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) t28:=estii(rat,[x:rat]schnittprop(x),y0,t27):in"rt"(y0,schnittset) i@[r:cut][a:and(in(r,s0),lrt(r,x0))] t29:=ande1(in(r,s0),lrt(r,x0),a):in(r,s0) t30:=ande2(in(r,s0),lrt(r,x0),a):lrt(r,x0) [y0:rat][ly:lrt(r,y0)][l:less"rt"(x0,y0)] t31:=andi(in(r,s0),lrt(r,y0),t29,ly):and(in(r,s0),lrt(r,y0)) t32:=somei(cut,[y:cut]and(in(y,s0),lrt(y,y0)),r,t31):schnittprop(y0) t33:=estii(rat,[x:rat]schnittprop(x),y0,t32):in"rt"(y0,schnittset) t34:=satz83(x0,y0,l):more"rt"(y0,x0) t35:=andi(in"rt"(y0,schnittset),more"rt"(y0,x0),t33,t34):and(in"rt"(y0,schnittset),more"rt"(y0,x0)) 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))) 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))) 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))) 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)] 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) j@t40:=cutapp1b(s,cutprop(schnittset),[x:rat][t:urt(s,x)]t39(x,t)):cutprop(schnittset) lx@t41:=nonemptyapp(cut,t0,p1b,cutprop(schnittset),[y:cut][t:in(y,t0)]t40(y,t)):cutprop(schnittset) i@t42:=cutapp1a(r,cutprop(schnittset),[x:rat][t:lrt(r,x)]t41(x,t)):cutprop(schnittset) p2@t43:=nonemptyapp(cut,s0,p1a,cutprop(schnittset),[y:cut][t:in(y,s0)]t42(y,t)):cutprop(schnittset) snt:=cutof(schnittset,t43):cut [r:cut][l:less(r,snt)][x0:rat][ux:urt(r,x0)][lx:lrt(snt,x0)] t44:=ini(schnittset,t43,x0,lx):in"rt"(x0,schnittset) t45:=estie(rat,[x:rat]schnittprop(x),x0,t44):schnittprop(x0) [s:cut][a:and(in(s,s0),lrt(s,x0))] t46:=ande1(in(s,s0),lrt(s,x0),a):in(s,s0) t47:=ande2(in(s,s0),lrt(s,x0),a):lrt(s,x0) t48:=andi(urt(r,x0),lrt(s,x0),ux,t47):and(urt(r,x0),lrt(s,x0)) t49:=somei(rat,[x:rat]and(urt(r,x),lrt(s,x)),x0,t48):less(r,s) t50:=ec3e23(is(s,r),more(s,r),less(s,r),satz123b(s,r),satz122(r,s,t49)):not(less(s,r)) t51:=th3"l.imp"(in(r,t0),less(s,r),t50,p2):not(in(r,t0)) t52:=ore1(in(r,s0),in(r,t0),p0,t51):in(r,s0) 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) 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) r@[m:more(r,snt)][x0:rat][lx:lrt(r,x0)][ux:urt(snt,x0)][i:in(r,s0)] t55:=andi(in(r,s0),lrt(r,x0),i,lx):and(in(r,s0),lrt(r,x0)) t56:=somei(cut,[y:cut]and(in(y,s0),lrt(y,x0)),r,t55):schnittprop(x0) t57:=estii(rat,[x:rat]schnittprop(x),x0,t56):in"rt"(x0,schnittset) t58:=ine(schnittset,t43,x0,t57):lrt(snt,x0) ux@t59:=th3"l.imp"(in(r,s0),lrt(snt,x0),ux,[t:in(r,s0)]t58(t)):not(in(r,s0)) t60:=ore2(in(r,s0),in(r,t0),p0,t59):in(r,t0) 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) 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) t63:=somei(cut,[x:cut]prop3(x),snt,t62):some([x:cut]prop3(x)) -5p205 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)))) schnitt:=ind(cut,[x:cut]prop3".5p205"(x),satzp205):cut 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)) 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)) @[r:cut][s:cut] +ivad @i:=1rp:cut r@r1:=pl(r,i):cut s@s1:=pl(s,i):cut rps:=pl(r,s):cut @2:=pl(i,i):cut s@t1:=pdeq12a(r1,i,s1,i):eq(pd(pdofrp(r),pdofrp(s)),df(pl(r1,s1),2)) 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)) 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)) 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)) t5:=eqi12(pl(r1,s1),2,pl(rps,i),i,t4):eq(df(pl(r1,s1),2),pdofrp(rps)) -ivad 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))) +*ivad s@rs:=ts(r,s):cut 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)))) 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)) 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)) 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))) 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))) 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))) 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)))) 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)) -ivad 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))) [m:mored(pdofrp(r),pdofrp(s))] +*ivad m@t14:=morede12(r1,i,s1,i,m):more(pl(r1,i),pl(s1,i)) t15:=satz136a(r1,s1,i,t14):more(r1,s1) -ivad m@lemmaivad3:=satz136a(r,s,1rp,t15".ivad"):more(r,s) @[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)] +d161 t1:=treq2(td(a,a),td(b,b),c,e,f):eq(td(a,a),td(b,b)) 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)) 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))) 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))) t5:=satzd192c(pd(a,b),md(a,b),t4):or(zero(pd(a,b)),zero(md(a,b))) [z:zero(a)] t6:=eqzero(td(a,a),td(b,b),t1,td01(a,a,z)):zero(td(b,b)) t7:=th1"l.imp"(zero(b),zero(b),refimp(zero(b)),satzd192c(b,b,t6)):zero(b) t8:=zeroeq(a,b,z,t7):eq(a,b) f@[p:not(zero(a))] t9:=or3e2(zero(a),posd(a),negd(a),axrdo(a),n,p):posd(a) t10:=th3"l.imp"(zero(b),zero(a),p,[t:zero(b)]t7(b,a,o,n,f,e,t)):not(zero(b)) t11:=t9(b,a,o,n,f,e,t10):posd(b) t12:=pnot0d(pd(a,b),ppd(a,b,t9,t11)):not(zero(pd(a,b))) t13:=ore2(zero(pd(a,b)),zero(md(a,b)),t5,t12):zero(md(a,b)) t14:=satzd182b(a,b,t13):eq(a,b) -d161 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) c@[n:not(negd(c))] +*d161 n@[z:zero(c)] t15:=zeroeq(td(c,c),c,td01(c,c,z),z):eq(td(c,c),c) t16:=andi(not(negd(c)),eq(td(c,c),c),n,t15):and(not(negd(c)),eq(td(c,c),c)) 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))) n@[o:not(zero(c))] t18:=or3e2(zero(c),posd(c),negd(c),axrdo(c),n,o):posd(c) crp:=rpofpd(c,t18):cut srp:=sqrt(crp):cut s:=pdofrp(srp):dif 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) t20:=andi(not(negd(s)),eq(td(s,s),c),pnotnd(s,posdirp(srp)),t19):and(not(negd(s)),eq(td(s,s),c)) 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))) -d161 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))) @[a:dif][i:intd(a)] +intd [z:zero(a)] t1:=ori1(zero(absd(a)),natd(absd(absd(a))),satzd166f(a,z)):intd(absd(a)) i@[n:natd(absd(a))] t2:=natintd(absd(a),n):intd(absd(a)) -intd 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)) +*intd n@t4:=eqnatd(absd(a),absd(m0d(a)),satzd178a(a),n):natd(absd(m0d(a))) -intd 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)) a@[b:dif][i:intd(a)][j:intd(b)] +*intd j@[z:zero(a)] t5:=symeq(pd(a,b),b,pd01(a,b,z)):eq(b,pd(a,b)) t6:=eqintd(b,pd(a,b),t5,j):intd(pd(a,b)) j@[z:zero(b)] t7:=symeq(pd(a,b),a,pd02(a,b,z)):eq(a,pd(a,b)) t8:=eqintd(a,pd(a,b),t7,i):intd(pd(a,b)) a@[i:intd(a)][p:posd(a)] t9:=

ande2(posd(a),[t:posd(a)]natrp(rpofpd(a,t)),posintnatd(a,p,i)):natrp(rpofpd(a,p)) j@[pp:posd(pd(a,b))] apb1:=rpofpd(pd(a,b),pp):cut [p:posd(a)] a1:=rpofpd(a,p):cut [q:posd(b)] b1:=rpofpd(b,q):cut t10:=natpl(a1,t9(a,i,p),b1,t9(b,j,q)):natrp(pl(a1,b1)) t11:=eqpd12(a,pdofrp(a1),b,pdofrp(b1),eqpdrp1(a,p),eqpdrp1(b,q)):eq(pd(a,b),pd(pdofrp(a1),pdofrp(b1))) 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))) 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)) t14:=isp1(cut,[t:cut]natrp(t),pl(a1,b1),apb1,t10,t13):natrp(apb1) 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)) t16:=natintd(pd(a,b),t15):intd(pd(a,b)) p@[n:negd(b)] t17:=satzd176c(b,n):posd(m0d(b)) b2:=rpofpd(m0d(b),t17):cut t18:=eqpd2(b,m0d(m0d(b)),a,satzd177a(b)):eq(pd(a,b),md(a,m0d(b))) t19:=eqposd(pd(a,b),md(a,m0d(b)),t18,pp):posd(md(a,m0d(b))) t20:=satzd182a(a,m0d(b),t19):mored(a,m0d(b)) t21:=eqmored12(a,pdofrp(a1),m0d(b),pdofrp(b2),eqpdrp1(a,p),eqpdrp1(m0d(b),t17),t20):mored(pdofrp(a1),pdofrp(b2)) t22:=lemmaivad3(a1,b2,t21):more(a1,b2) t23:=natmn(a1,t9(a,i,p),b2,t9(m0d(b),intm0d(b,j),t17),t22):natrp(mn(a1,b2,t22)) 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))) 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))) 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) t27:=satz140g(a1,b2,apb1,t22,t26):is(apb1,mn(a1,b2,t22)) t28:=isp1(cut,[t:cut]natrp(t),mn(a1,b2,t22),apb1,t23,t27):natrp(apb1) 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)) t30:=natintd(pd(a,b),t29):intd(pd(a,b)) 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)) pp@[n:negd(a)] 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)) 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)) t33:=or3e2(zero(b),posd(b),negd(b),axrdo(b),t31a,t32):posd(b) t34:=eqposd(pd(a,b),pd(b,a),compd(a,b),pp):posd(pd(b,a)) t35:=t30(b,a,j,i,t34,t33,n):intd(pd(b,a)) t36:=eqintd(pd(b,a),pd(a,b),compd(b,a),t35):intd(pd(a,b)) 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)) j@[0p:zero(pd(a,b))] t38:=intdi0(pd(a,b),0p):intd(pd(a,b)) j@[np:negd(pd(a,b))] t39:=satzd176c(pd(a,b),np):posd(m0d(pd(a,b))) t40:=eqposd(m0d(pd(a,b)),pd(m0d(a),m0d(b)),satzd180(a,b),t39):posd(pd(m0d(a),m0d(b))) t41:=t37(m0d(a),m0d(b),intm0d(a,i),intm0d(b,j),t40):intd(pd(m0d(a),m0d(b))) t42:=eqintd(pd(m0d(a),m0d(b)),m0d(pd(a,b)),satzd180a(a,b),t41):intd(m0d(pd(a,b))) t43:=intm0d(m0d(pd(a,b)),t42):intd(m0d(m0d(pd(a,b)))) t44:=eqintd(m0d(m0d(pd(a,b))),pd(a,b),satzd177(pd(a,b)),t43):intd(pd(a,b)) -intd 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)) intmd:=intpd(a,m0d(b),i,intm0d(b,j)):intd(md(a,b)) +*intd j@[n:not(zero(td(a,b)))] t45:=th3"l.imp"(zero(a),zero(td(a,b)),n,[t:zero(a)]td01(a,b,t)):not(zero(a)) t46:=th3"l.imp"(zero(b),zero(td(a,b)),n,[t:zero(b)]td02(a,b,t)):not(zero(b)) t47:=satzd166e(a,t45):posd(absd(a)) a3:=rpofpd(absd(a),t47):cut t48:=satzd166e(b,t46):posd(absd(b)) b3:=rpofpd(absd(b),t48):cut t49:=natts(a3,t9(absd(a),intabsd(a,i),t47),b3,t9(absd(b),intabsd(b,j),t48)):natrp(ts(a3,b3)) t50:=satzd166e(td(a,b),n):posd(absd(td(a,b))) [p:posd(absd(td(a,b)))] atb3:=rpofpd(absd(td(a,b)),p):cut 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))) 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))) 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)) t54:=isp1(cut,[t:cut]natrp(t),ts(a3,b3),atb3,t49,t53):natrp(atb3) 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))) -intd j@inttd:=[t:not(zero(td(a,b)))]t55".intd"(t):intd(td(a,b)) +r @eq:=[x:dif][y:dif]eq"rp"(x,y):[x:dif][y:dif]'prop' refeq:=[x:dif]refeq"rp"(x):[x:dif]eq symeq:=[x:dif][y:dif][t:eq]symeq"rp"(x,y,t):[x:dif][y:dif][t:eq]eq treq:=[x:dif][y:dif][z:dif][t:eq][u:eq]treq"rp"(x,y,z,t,u):[x:dif][y:dif][z:dif][t:eq][u:eq]eq [a:dif][s:set(dif)] inn:=esti(dif,a,s):'prop' @real:=ect"eq"(dif,eq,refeq,symeq,treq):'type' [r:real][s:real] is:=is"e"(real,r,s):'prop' nis:=not(is(r,s)):'prop' @[p:[x:real]'prop'] some:=some"l"(real,p):'prop' all:=all"l"(real,p):'prop' one:=one"e"(real,p):'prop' r@[s0:set(real)] in:=esti(real,r,s0):'prop' a@realof:=ectelt"eq"(dif,eq,refeq,symeq,treq,a):real r@class:=ecect"eq"(dif,eq,refeq,symeq,treq,r):set(dif) a@innclass:=th5"eq.4"(dif,eq,refeq,symeq,treq,a):inn(a,class(realof(a))) r@[a:dif][b:dif][e:eq"rp"(a,b)][air:inn(a,class(r))] eqinn:=th8"eq.4"(dif,eq,refeq,symeq,treq,r,a,air,b,e):inn(b,class(r)) r@[p:'prop'][p1:[x:dif][xi:inn(x,class(r))]p] realapp1:=th3"eq.4"(dif,eq,refeq,symeq,treq,r,p,p1):p r@[s:real][p:'prop'][p1:[x:dif][y:dif][xi:inn(x,class(r))][yi:inn(y,class(s))]p] +ivr1 [a:dif][air:inn(a,class(r))] t1:=realapp1(s,p,[y:dif]p1):p -ivr1 realapp2:=realapp1(r,p,[x:dif][xi:inn(x,class(r))]t1".ivr1"(x,xi)):p 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] +*ivr1 p1@[a:dif][air:inn(a,class(r))] t2:=realapp2(s,t,p,[y:dif][z:dif]p1):p -ivr1 p1@realapp3:=realapp1(r,p,[x:dif][xi:inn(x,class(r))]t2".ivr1"(x,xi)):p 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] +*ivr1 p1@[a:dif][air:inn(a,class(r))] t3:=realapp3(s,t,u,p,[y:dif][z:dif][v:dif]p1):p -ivr1 p1@realapp4:=realapp1(r,p,[x:dif][xi:inn(x,class(r))]t3".ivr1"(x,xi)):p s@[a1:dif][b1:dif][a1ir:inn(a1,class(r))][b1is:inn(b1,class(s))][e:eq"rp"(a1,b1)] isin:=th3"eq.5"(dif,eq,refeq,symeq,treq,r,s,a1,a1ir,b1,b1is,e):is(r,s) b1is@[i:is(r,s)] isex:=th5"eq.5"(dif,eq,refeq,symeq,treq,r,s,a1,a1ir,b1,b1is,i):eq"rp"(a1,b1) b1is@[n:not(eq"rp"(a1,b1))] nisin:=th3"l.imp"(is(r,s),eq"rp"(a1,b1),n,[t:is(r,s)]isex(t)):nis(r,s) b1is@[n:nis(r,s)] nisex:=th3"l.imp"(eq"rp"(a1,b1),is(r,s),n,[t:eq"rp"(a1,b1)]isin(t)):not(eq"rp"(a1,b1)) @[alpha:'type'][f:[x:dif]alpha] fixf:=fixfu"eq"(dif,eq,refeq,symeq,treq,alpha,f):'prop' [ff:fixf(alpha,f)][r0:real] indreal:=indeq"eq"(dif,eq,refeq,symeq,treq,alpha,f,ff,r0):alpha [a:dif][air:inn(a,class(r0))] isindreal:=th2"eq.10"(dif,eq,refeq,symeq,treq,alpha,f,ff,r0,a,air):is"e"(alpha,f,indreal) alpha@[g:[x:dif][y:dif]alpha] fixf2:=fixfu2"eq"(dif,eq,refeq,symeq,treq,alpha,g):'prop' [ff2:fixf2(alpha,g)][r0:real][s0:real] indreal2:=indeq2"eq"(dif,eq,refeq,symeq,treq,alpha,g,ff2,r0,s0):alpha [a:dif][b:dif][air:inn(a,class(r0))][bis:inn(b,class(s0))] isindreal2:=th1"eq.11"(dif,eq,refeq,symeq,treq,alpha,g,ff2,r0,s0,a,air,b,bis):is"e"(alpha,g,indreal2) @0:=realof(df(1rp,1rp)):real r@[a0:dif][a0ir:inn(a0,class(r))][z:zero(a0)] 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) a0ir@[i:is(r,0)] 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) +*ivr1 a0@propp:=and(inn(a0,class(r)),posd(a0)):'prop' -ivr1 r@pos:=some"l"(dif,[x:dif]propp".ivr1"(x)):'prop' a0ir@[p:posd(a0)] +*ivr1 p@t4:=andi(inn(a0,class(r)),posd(a0),a0ir,p):propp(a0) -ivr1 p@posin:=somei(dif,[x:dif]propp".ivr1"(x),a0,t4".ivr1"):pos(r) a0ir@[p:pos(r)] +*ivr1 p@[a:dif][q1:propp(a)] t5:=ande1(inn(a,class(r)),posd(a),q1):inn(a,class(r)) t6:=ande2(inn(a,class(r)),posd(a),q1):posd(a) t7:=eqposd(a,a0,isex(r,r,a,a0,t5,a0ir,refis(real,r)),t6):posd(a0) -ivr1 p@posex:=someapp(dif,[x:dif]propp".ivr1"(x),p,posd(a0),[x:dif][t:propp".ivr1"(x)]t7".ivr1"(x,t)):posd(a0) +*ivr1 a0@propn:=and(inn(a0,class(r)),negd(a0)):'prop' -ivr1 r@neg:=some"l"(dif,[x:dif]propn".ivr1"(x)):'prop' a0ir@[n:negd(a0)] +*ivr1 n@t8:=andi(inn(a0,class(r)),negd(a0),a0ir,n):propn(a0) -ivr1 n@negin:=somei(dif,[x:dif]propn".ivr1"(x),a0,t8".ivr1"):neg(r) a0ir@[n:neg(r)] +*ivr1 n@[a:dif][pl:propn(a)] t9:=ande1(inn(a,class(r)),negd(a),pl):inn(a,class(r)) t10:=ande2(inn(a,class(r)),negd(a),pl):negd(a) t11:=eqnegd(a,a0,isex(r,r,a,a0,t9,a0ir,refis(real,r)),t10):negd(a0) -ivr1 n@negex:=someapp(dif,[x:dif]propn".ivr1"(x),n,negd(a0),[x:dif][t:propn".ivr1"(x)]t11".ivr1"(x,t)):negd(a0) +*ivr1 a0ir@[p:posd(a0)] t12:=or3i2(is(r,0),pos(r),neg(r),posin(p)):or3(is(r,0),pos(r),neg(r)) a0ir@[z:zero(a0)] t13:=or3i1(is(r,0),pos(r),neg(r),0in(z)):or3(is(r,0),pos(r),neg(r)) a0ir@[n:negd(a0)] t14:=or3i3(is(r,0),pos(r),neg(r),negin(n)):or3(is(r,0),pos(r),neg(r)) 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)) -ivr1 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)) +*ivr1 a0ir@[i:is(r,0)] t16:=th3"l.imp"(pos(r),posd(a0),0notpd(a0,0ex(i)),[t:pos(r)]posex(t)):not(pos(r)) a0ir@[p:pos(r)] t17:=th3"l.imp"(neg(r),negd(a0),pnotnd(a0,posex(p)),[t:neg(r)]negex(t)):not(neg(r)) a0ir@[n:neg(r)] t18:=th3"l.imp"(is(r,0),zero(a0),nnot0d(a0,negex(n)),[t:is(r,0)]0ex(t)):not(is(r,0)) 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)) -ivr1 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)) axrl:=orec3i(is(r,0),pos(r),neg(r),axrlo,axrle):orec3(is(r,0),pos(r),neg(r)) [p:'prop'][p1:[t:pos(r)]p][p2:[t:is(r,0)]p][p3:[t:neg(r)]p] rapp:=or3app(is(r,0),pos(r),neg(r),p,axrlo,p2,p1,p3):p r@[p:pos(r)] pnotn:=ec3e23(is(r,0),pos(r),neg(r),axrle,p):not(neg(r)) pnot0:=ec3e21(is(r,0),pos(r),neg(r),axrle,p):nis(r,0) r@[i:is(r,0)] 0notp:=ec3e12(is(r,0),pos(r),neg(r),axrle,i):not(pos(r)) 0notn:=ec3e13(is(r,0),pos(r),neg(r),axrle,i):not(neg(r)) r@[n:neg(r)] nnotp:=ec3e32(is(r,0),pos(r),neg(r),axrle,n):not(pos(r)) nnot0:=ec3e31(is(r,0),pos(r),neg(r),axrle,n):nis(r,0) s@[i:is(r,s)][p:pos(r)] ispos:=isp(real,[x:real]pos(x),r,s,p,i):pos(s) i@[n:neg(r)] isneg:=isp(real,[x:real]neg(x),r,s,n,i):neg(s) @[r0:cut] pofrp:=realof(pdofrp(r0)):real nofrp:=realof(ndofrp(r0)):real [s0:cut][i:is"rp"(r0,s0)] isrpep:=isf(cut,real,[x:cut]pofrp(x),r0,s0,i):is(pofrp(r0),pofrp(s0)) isrpen:=isf(cut,real,[x:cut]nofrp(x),r0,s0,i):is(nofrp(r0),nofrp(s0)) s0@[i:is(pofrp(r0),pofrp(s0))] +*ivr1 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)) -ivr1 i@isrpip:=isrpipd(r0,s0,t20".ivr1"):is"rp"(r0,s0) s0@[i:is(nofrp(r0),nofrp(s0))] +*ivr1 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)) -ivr1 i@isrpin:=isrpind(r0,s0,t21".ivr1"):is"rp"(r0,s0) r0@posi:=posin(pofrp(r0),pdofrp(r0),innclass(pdofrp(r0)),posdirp(r0)):pos(pofrp(r0)) negi:=negin(nofrp(r0),ndofrp(r0),innclass(ndofrp(r0)),negdirp(r0)):neg(nofrp(r0)) s@[r0:cut][s0:cut][i:is(r,pofrp(r0))][j:is(s,pofrp(s0))][k:is(r,s)] +*ivr1 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) 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))) a0ir@[p1:pos(r)] t24:=posex(p1):posd(a0) pr:=rpofpd(a0,t24):cut t25:=isin(r,pofrp(pr),a0,pdofrp(pr),a0ir,innclass(pdofrp(pr)),eqpdrp1(a0,t24)):is(r,pofrp(pr)) t26:=somei(cut,[x:cut]is(r,pofrp(x)),pr,t25):some"rp"([x:cut]is(r,pofrp(x))) -ivr1 r@[p:pos(r)] +*ivr1 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))) t28:=onei(cut,[x:cut]is(r,pofrp(x)),t23,t27):one"rp"([x:cut]is(r,pofrp(x))) -ivr1 p@rpofp:=ind(cut,[x:cut]is(r,pofrp(x)),t28".ivr1"):cut isprp1:=oneax(cut,[x:cut]is(r,pofrp(x)),t28".ivr1"):is(r,pofrp(rpofp(r,p))) isprp2:=symis(real,r,pofrp(rpofp(r,p)),isprp1):is(pofrp(rpofp(r,p)),r) @[r1:real][p:pos(r1)][s1:real][q:pos(s1)][i:is(r1,s1)] 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)) q@[i:is"rp"(rpofp(r1,p),rpofp(s1,q))] 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) @[r0:cut] 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))) isrpp2:=symis(cut,r0,rpofp(pofrp(r0),posi(r0)),isrpp1):is"rp"(rpofp(pofrp(r0),posi(r0)),r0) s@[r0:cut][s0:cut][i:is(r,nofrp(r0))][j:is(s,nofrp(s0))][k:is(r,s)] +*ivr1 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) 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))) a0ir@[n1:neg(r)] t31:=negex(n1):negd(a0) nr:=rpofnd(a0,t31):cut t32:=isin(r,nofrp(nr),a0,ndofrp(nr),a0ir,innclass(ndofrp(nr)),eqndrp1(a0,t31)):is(r,nofrp(nr)) t33:=somei(cut,[x:cut]is(r,nofrp(x)),nr,t32):some"rp"([x:cut]is(r,nofrp(x))) -ivr1 r@[n:neg(r)] +*ivr1 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))) t35:=onei(cut,[x:cut]is(r,nofrp(x)),t30,t34):one"rp"([x:cut]is(r,nofrp(x))) -ivr1 n@rpofn:=ind(cut,[x:cut]is(r,nofrp(x)),t35".ivr1"):cut isnrp1:=oneax(cut,[x:cut]is(r,nofrp(x)),t35".ivr1"):is(r,nofrp(rpofn(r,n))) isnrp2:=symis(real,r,nofrp(rpofn(r,n)),isnrp1):is(nofrp(rpofn(r,n)),r) @[r1:real][n:neg(r1)][s1:real][m:neg(s1)][i:is(r1,s1)] 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)) m@[i:is"rp"(rpofn(r1,n),rpofn(s1,m))] 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) @[r0:cut] 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))) isrpn2:=symis(cut,r0,rpofn(nofrp(r0),negi(r0)),isrpn1):is"rp"(rpofn(nofrp(r0),negi(r0)),r0) r@satz163:=refis(real,r):is(r,r) s@[i:is(r,s)] satz164:=symis(real,r,s,i):is(s,r) t@[i:is(r,s)][j:is(s,t)] satz165:=tris(real,r,s,t,i,j):is(r,t) @absdr:=[x:dif]realof(absd(x)):[x:dif]real +ivr2 [a:dif][b:dif][e:eq"rp"(a,b)] t1:=isin(realof(absd(a)),realof(absd(b)),absd(a),absd(b),innclass(absd(a)),innclass(absd(b)),eqabsd(a,b,e)):is(absdr,absdr) -ivr2 fabsdr:=[x:dif][y:dif][t:eq]t1".ivr2"(x,y,t):fixf(real,absdr) r@abs:=indreal(real,absdr,fabsdr,r):real +*ivr2 a0ir@t2:=isindreal(real,absdr,fabsdr,r,a0,a0ir):is(realof(absd(a0)),abs(r)) -ivr2 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))) s@[i:is(r,s)] isabs:=isf(real,real,[x:real]abs(x),r,s,i):is(abs(r),abs(s)) +2r166 a0ir@[p:pos(r)] t1:=satzd166a(a0,posex(p)):posd(absd(a0)) t2:=posin(abs(r),absd(a0),aica,t1):pos(abs(r)) -2r166 r@[p:pos(r)] satz166a:=realapp1(pos(abs(r)),[x:dif][t:inn(x,class(r))]t2".2r166"(x,t,p)):pos(abs(r)) +*2r166 a0ir@[n:neg(r)] t3:=satzd166b(a0,negex(n)):posd(absd(a0)) t4:=posin(abs(r),absd(a0),aica,t3):pos(abs(r)) -2r166 r@[n:neg(r)] satz166b:=realapp1(pos(abs(r)),[x:dif][t:inn(x,class(r))]t4".2r166"(x,t,n)):pos(abs(r)) +*2r166 b1is@[p:pos(r)][q:pos(s)][i:is(abs(r),abs(s))] 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) t6:=isin(t5):is(r,s) -2r166 s@[p:pos(r)][q:pos(s)][i:is(abs(r),abs(s))] 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) +*2r166 b1is@[n:neg(r)][o:neg(s)][i:is(abs(r),abs(s))] 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) t8:=isin(t7):is(r,s) -2r166 s@[n:neg(r)][o:neg(s)][i:is(abs(r),abs(s))] 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) r@[n:nis(r,0)] 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)) +*2r166 a0ir@[i:is(r,0)] t9:=satzd166f(a0,0ex(i)):zero(absd(a0)) t10:=0in(abs(r),absd(a0),aica,t9):is(abs(r),0) -2r166 r@[i:is(r,0)] satz166f:=realapp1(is(abs(r),0),[x:dif][t:inn(x,class(r))]t10".2r166"(x,t,i)):is(abs(r),0) 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' +*ivr2 b1@propm:=and3(inn(a1,class(r)),inn(b1,class(s)),mored(a1,b1)):'prop' -ivr2 b1is@[m:mored(a1,b1)] +*ivr2 m@t3:=and3i(inn(a1,class(r)),inn(b1,class(s)),mored(a1,b1),a1ir,b1is,m):propm(a1,b1) t4:=somei(dif,[x:dif]propm(a1,x),b1,t3):some"l"(dif,[x:dif]propm(a1,x)) -ivr2 m@morein:=somei(dif,[x:dif]some"l"(dif,[y:dif]propm".ivr2"(x,y)),a1,t4".ivr2"):more(r,s) b1is@[m:more(r,s)] +*ivr2 m@[a:dif][sa:some"l"(dif,[x:dif]propm(a,x))][b:dif][p2:propm(a,b)] t5:=and3e1(inn(a,class(r)),inn(b,class(s)),mored(a,b),p2):inn(a,class(r)) t6:=and3e2(inn(a,class(r)),inn(b,class(s)),mored(a,b),p2):inn(b,class(s)) t7:=and3e3(inn(a,class(r)),inn(b,class(s)),mored(a,b),p2):mored(a,b) 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) 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) -ivr2 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) 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' +*ivr2 b1@propl:=and3(inn(a1,class(r)),inn(b1,class(s)),lessd(a1,b1)):'prop' -ivr2 b1is@[l:lessd(a1,b1)] +*ivr2 l@t10:=and3i(inn(a1,class(r)),inn(b1,class(s)),lessd(a1,b1),a1ir,b1is,l):propl(a1,b1) t11:=somei(dif,[x:dif]propl(a1,x),b1,t10):some"l"(dif,[x:dif]propl(a1,x)) -ivr2 l@lessin:=somei(dif,[x:dif]some"l"(dif,[y:dif]propl".ivr2"(x,y)),a1,t11".ivr2"):less(r,s) b1is@[l:less(r,s)] +*ivr2 l@[a:dif][sa:some"l"(dif,[x:dif]propl(a,x))][b:dif][p2:propl(a,b)] t12:=and3e1(inn(a,class(r)),inn(b,class(s)),lessd(a,b),p2):inn(a,class(r)) t13:=and3e2(inn(a,class(r)),inn(b,class(s)),lessd(a,b),p2):inn(b,class(s)) t14:=and3e3(inn(a,class(r)),inn(b,class(s)),lessd(a,b),p2):lessd(a,b) 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) 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) -ivr2 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) t@[i:is(r,s)][m:more(r,t)] ismore1:=isp(real,[x:real]more(x,t),r,s,m,i):more(s,t) i@[m:more(t,r)] ismore2:=isp(real,[x:real]more(t,x),r,s,m,i):more(t,s) i@[l:less(r,t)] isless1:=isp(real,[x:real]less(x,t),r,s,l,i):less(s,t) i@[l:less(t,r)] isless2:=isp(real,[x:real]less(t,x),r,s,l,i):less(t,s) u@[i:is(r,s)][j:is(t,u)][m:more(r,t)] ismore12:=ismore2(t,u,s,j,ismore1(r,s,t,i,m)):more(s,u) j@[l:less(r,t)] isless12:=isless2(t,u,s,j,isless1(r,s,t,i,l)):less(s,u) +*ivr2 b1is@[m:more(r,s)] t17:=lemmad5(a1,b1,moreex(m)):lessd(b1,a1) t18:=lessin(s,r,b1,a1,b1is,a1ir,t17):less(s,r) -ivr2 s@[m:more(r,s)] 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) +*ivr2 b1is@[l:less(r,s)] t19:=lemmad6(a1,b1,lessex(l)):mored(b1,a1) t20:=morein(s,r,b1,a1,b1is,a1ir,t19):more(s,r) -ivr2 s@[l:less(r,s)] 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) +2r167 b1is@t1:=satzd167a(a1,b1):or3(eq"rp"(a1,b1),mored(a1,b1),lessd(a1,b1)) [e:eq"rp"(a1,b1)] t2:=or3i1(is(r,s),more(r,s),less(r,s),isin(e)):or3(is(r,s),more(r,s),less(r,s)) b1is@[m:mored(a1,b1)] t3:=or3i2(is(r,s),more(r,s),less(r,s),morein(m)):or3(is(r,s),more(r,s),less(r,s)) b1is@[l:lessd(a1,b1)] t4:=or3i3(is(r,s),more(r,s),less(r,s),lessin(l)):or3(is(r,s),more(r,s),less(r,s)) 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)) t6:=satzd167b(a1,b1):ec3(eq"rp"(a1,b1),mored(a1,b1),lessd(a1,b1)) [i:is(r,s)] 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)) b1is@[m:more(r,s)] 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)) b1is@[l:less(r,s)] 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)) 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)) t11:=orec3i(is(r,s),more(r,s),less(r,s),t5,t10):orec3(is(r,s),more(r,s),less(r,s)) -2r167 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)) satz167a:=orec3e1(is(r,s),more(r,s),less(r,s),satz167):or3(is(r,s),more(r,s),less(r,s)) satz167b:=orec3e2(is(r,s),more(r,s),less(r,s),satz167):ec3(is(r,s),more(r,s),less(r,s)) moreis:=or(more(r,s),is(r,s)):'prop' lessis:=or(less(r,s),is(r,s)):'prop' [m:moreis(r,s)] 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) s@[l:lessis(r,s)] 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) t@[i:is(r,s)][m:moreis(r,t)] ismoreis1:=isp(real,[x:real]moreis(x,t),r,s,m,i):moreis(s,t) i@[m:moreis(t,r)] ismoreis2:=isp(real,[x:real]moreis(t,x),r,s,m,i):moreis(t,s) i@[l:lessis(r,t)] islessis1:=isp(real,[x:real]lessis(x,t),r,s,l,i):lessis(s,t) i@[l:lessis(t,r)] islessis2:=isp(real,[x:real]lessis(t,x),r,s,l,i):lessis(t,s) u@[i:is(r,s)][j:is(t,u)][m:moreis(r,t)] ismoreis12:=ismoreis2(t,u,s,j,ismoreis1(r,s,t,i,m)):moreis(s,u) j@[l:lessis(r,t)] islessis12:=islessis2(t,u,s,j,islessis1(r,s,t,i,l)):lessis(s,u) s@[m:more(r,s)] moreisi1:=ori1(more(r,s),is(r,s),m):moreis(r,s) s@[l:less(r,s)] lessisi1:=ori1(less(r,s),is(r,s),l):lessis(r,s) s@[i:is(r,s)] moreisi2:=ori2(more(r,s),is(r,s),i):moreis(r,s) lessisi2:=ori2(less(r,s),is(r,s),i):lessis(r,s) b1is@[m:moreq(a1,b1)] 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) b1is@[m:moreis(r,s)] 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) b1is@[l:lesseq(a1,b1)] 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) b1is@[l:lessis(r,s)] 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) s@[m:moreis(r,s)] 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)) s@[l:lessis(r,s)] satz167d:=th9"l.ec3"(is(r,s),more(r,s),less(r,s),satz167b,l):not(more(r,s)) s@[n:not(more(r,s))] satz167e:=th2"l.or3"(is(r,s),more(r,s),less(r,s),satz167a,n):lessis(r,s) s@[n:not(less(r,s))] s@[n:not(less(r,s))] 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) s@[m:more(r,s)] 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)) s@[l:less(r,s)] 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)) s@[n:not(moreis(r,s))] 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) s@[n:not(lessis(r,s))] 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) r@[p:pos(r)] +2r169 [a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))] t1:=satzd169a(a,b,0ex(0,b,bi0,refis(real,0)),posex(a,air,p)):mored(a,b) t2:=morein(r,0,a,b,air,bi0,t1):more(r,0) -2r169 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) r@[m:more(r,0)] +*2r169 m@[a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))] t3:=satzd169b(a,b,0ex(0,b,bi0,refis(real,0)),moreex(r,0,a,b,air,bi0,m)):posd(a) t4:=posin(r,a,air,t3):pos(r) -2r169 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) r@[n:neg(r)] +*2r169 n@[a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))] t5:=satzd169c(a,b,0ex(0,b,bi0,refis(real,0)),negex(a,air,n)):lessd(a,b) t6:=lessin(r,0,a,b,air,bi0,t5):less(r,0) -2r169 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) r@[l:less(r,0)] +*2r169 l@[a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))] t7:=satzd169d(a,b,0ex(0,b,bi0,refis(real,0)),lessex(r,0,a,b,air,bi0,l)):negd(a) t8:=negin(r,a,air,t7):neg(r) -2r169 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) +2r170 r@[a:dif][b:dif][air:inn(a,class(r))][bi0:inn(b,class(0))] t1:=satzd170(a,b,0ex(0,b,bi0,refis(real,0))):moreq(absd(a),b) t2:=moreisin(abs(r),0,absd(a),b,aica(r,a,air),bi0,t1):moreis(abs(r),0) -2r170 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) 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))) t@[l:less(r,s)][k:less(s,t)] +2r171 [a:dif][b:dif][c:dif][air:inn(a,class(r))][bis:inn(b,class(s))][cit:inn(c,class(t))] t1:=satzd171(a,b,c,lessex(r,s,a,b,air,bis,l),lessex(s,t,b,c,bis,cit,k)):lessd(a,c) t2:=lessin(r,t,a,c,air,cit,t1):less(r,t) -2r171 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) trless:=satz171:less(r,t) t@[m:more(r,s)][n:more(s,t)] trmore:=lemma2(t,r,trless(t,s,r,lemma1(s,t,n),lemma1(r,s,m))):more(r,t) t@[a2:dif][b2:dif][c2:dif][a2ir:inn(a2,class(r))][b2is:inn(b2,class(s))][c2it:inn(c2,class(t))] +2r172 [l:lessis(r,s)][k:less(s,t)] t1:=satzd172a(a2,b2,c2,lessisex(a2,b2,a2ir,b2is,l),lessex(s,t,b2,c2,b2is,c2it,k)):lessd(a2,c2) t2:=lessin(r,t,a2,c2,a2ir,c2it,t1):less(r,t) -2r172 t@[l:lessis(r,s)][k:less(s,t)] 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) +*2r172 c2it@[l:less(r,s)][k:lessis(s,t)] t3:=satzd172b(a2,b2,c2,lessex(a2,b2,a2ir,b2is,l),lessisex(s,t,b2,c2,b2is,c2it,k)):lessd(a2,c2) t4:=lessin(r,t,a2,c2,a2ir,c2it,t3):less(r,t) -2r172 t@[l:less(r,s)][k:lessis(s,t)] 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) t@[m:moreis(r,s)][n:more(s,t)] satz172c:=lemma2(t,r,satz172b(t,s,r,lemma1(s,t,n),satz168a(m))):more(r,t) t@[m:more(r,s)][n:moreis(s,t)] satz172d:=lemma2(t,r,satz172a(t,s,r,satz168a(s,t,n),lemma1(m))):more(r,t) +2r173 c2it@[l:lessis(r,s)][k:lessis(s,t)] t1:=satzd173(a2,b2,c2,lessisex(a2,b2,a2ir,b2is,l),lessisex(s,t,b2,c2,b2is,c2it,k)):lesseq(a2,c2) t2:=lessisin(r,t,a2,c2,a2ir,c2it,t1):lessis(r,t) -2r173 t@[l:lessis(r,s)][k:lessis(s,t)] 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) trlessis:=satz173:lessis(r,t) t@[m:moreis(r,s)][n:moreis(s,t)] trmoreis:=satz168b(t,r,trlessis(t,s,r,satz168a(s,t,n),satz168a(m))):moreis(r,t) r@ratrl:=some"l"(dif,[x:dif]and(inn(x,class(r)),ratd(x))):'prop' a0ir@[r1:ratd(a0)] +*ivr2 r1@t21:=andi(inn(a0,class(r)),ratd(a0),a0ir,r1):and(inn(a0,class(r)),ratd(a0)) -ivr2 r1@ratrlin:=somei(dif,[x:dif]and(inn(x,class(r)),ratd(x)),a0,t21".ivr2"):ratrl(r) a0ir@[rr:ratrl(r)] +*ivr2 rr@[a:dif][b:and(inn(a,class(r)),ratd(a))] t22:=ande1(inn(a,class(r)),ratd(a),b):inn(a,class(r)) t23:=ande2(inn(a,class(r)),ratd(a),b):ratd(a) t24:=eqratd(a,a0,isex(r,r,a,a0,t22,a0ir,refis(real,r)),t23):ratd(a0) -ivr2 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) r@irratrl:=not(ratrl(r)):'prop' @[r0:cut][rr:ratrp(r0)] remark2:=ratrlin(pofrp(r0),pdofrp(r0),innclass(pdofrp(r0)),remark2a(r0,rr)):ratrl(pofrp(r0)) remark3:=ratrlin(nofrp(r0),ndofrp(r0),innclass(ndofrp(r0)),remark3a(r0,rr)):ratrl(nofrp(r0)) r0@[ir:irratrp(r0)] 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)) 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)) r@natrl:=some"l"(dif,[x:dif]and(inn(x,class(r)),natd(x))):'prop' a0ir@[n:natd(a0)] +*ivr2 n@t25:=andi(inn(a0,class(r)),natd(a0),a0ir,n):and(inn(a0,class(r)),natd(a0)) -ivr2 n@natrlin:=somei(dif,[x:dif]and(inn(x,class(r)),natd(x)),a0,t25".ivr2"):natrl(r) a0ir@[n:natrl(r)] +*ivr2 n@[a:dif][b:and(inn(a,class(r)),natd(a))] t26:=ande1(inn(a,class(r)),natd(a),b):inn(a,class(r)) t27:=ande2(inn(a,class(r)),natd(a),b):natd(a) t28:=eqnatd(a,a0,isex(r,r,a,a0,t26,a0ir,refis(real,r)),t27):natd(a0) -ivr2 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) +*ivr2 a0ir@[n:natrl(r)] t29:=natposd(a0,natrlex(n)):posd(a0) t30:=posin(t29):pos(r) -ivr2 r@[n:natrl(r)] natpos:=realapp1(pos(r),[x:dif][t:inn(x,class(r))]t30".ivr2"(x,t,n)):pos(r) @[x:nat] rlofnt:=realof(pdofnt(x)):real natrli:=natrlin(rlofnt(x),pdofnt(x),innclass(pdofnt(x)),natdi(x)):natrl(rlofnt(x)) [y:nat][i:is"n"(x,y)] isnterl:=isf(nat,real,[u:nat]rlofnt(u),x,y,i):is(rlofnt(x),rlofnt(y)) y@[i:is(rlofnt(x),rlofnt(y))] isntirl:=isntirp(x,y,isrpip(rpofnt(x),rpofnt(y),i)):is"n"(x,y) +*ivr2 @t31:=[x:nat][y:nat][t:is(rlofnt(x),rlofnt(y))]isntirl(x,y,t):injective(nat,real,[x:nat]rlofnt(x)) a0ir@[n:natrl(r)] t32:=natposd(a0,natrlex(n)):posd(a0) ap:=rpofpd(a0,t32):cut t33:=natderp(a0,natrlex(n)):natrp(ap) x0:=ntofrp(ap,t33):nat t34:=isrpepd(ap,rpofnt(x0),isrpnt1(ap,t33)):eq"rp"(pdofrp(ap),pdofnt(x0)) t35:=treq"rp"(a0,pdofrp(ap),pdofnt(x0),eqpdrp1(a0,t32),t34):eq"rp"(a0,pdofnt(x0)) t36:=isin(r,rlofnt(x0),a0,pdofnt(x0),a0ir,innclass(pdofnt(x0)),t35):is(r,rlofnt(x0)) t37:=somei(nat,[x:nat]is(r,rlofnt(x)),x0,t36):image(nat,real,[x:nat]rlofnt(x),r) -ivr2 r@[n:natrl(r)] 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) r@[i:image(nat,real,[x:nat]rlofnt(x),r)] +*ivr2 i"r"@[x:nat][j:is(r,rlofnt(x))] t38:=isp1(real,[u:real]natrl(u),rlofnt(x),r,natrli(x),j):natrl(r) -ivr2 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) r@[n:natrl(r)] ntofrl:=soft(nat,real,[x:nat]rlofnt(x),t31".ivr2",r,natimage(r,n)):nat @[r1:real][n:natrl(r1)][s1:real][m:natrl(s1)][i:is(r1,s1)] 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)) m@[i:is"n"(ntofrl(r1,n),ntofrl(s1,m))] isrlint:=isinve(nat,real,[x:nat]rlofnt(x),t31".ivr2",r1,natimage(r1,n),s1,natimage(s1,m),i):is(r1,s1) r@[n:natrl(r)] isrlnt1:=ists1"e"(nat,real,[x:nat]rlofnt(x),t31".ivr2",r,natimage(r,n)):is(r,rlofnt(ntofrl(r,n))) isrlnt2:=symis(real,r,rlofnt(ntofrl(r,n)),isrlnt1):is(rlofnt(ntofrl(r,n)),r) @[x:nat] +*ivr2 x"r"@xn:=soft(nat,real,[u:nat]rlofnt(u),t31,rlofnt(x),imagei(nat,real,[u:nat]rlofnt(u),x)):nat 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))) -ivr2 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))) isntrl2:=symis(nat,x,ntofrl(rlofnt(x),natrli(x)),isntrl1):is"n"(ntofrl(rlofnt(x),natrli(x)),x) r@intrl:=some"l"(dif,[x:dif]and(inn(x,class(r)),intd(x))):'prop' a0ir@[i:intd(a0)] +*ivr2 i@t40:=andi(inn(a0,class(r)),intd(a0),a0ir,i):and(inn(a0,class(r)),intd(a0)) -ivr2 i@intrlin:=somei(dif,[x:dif]and(inn(x,class(r)),intd(x)),a0,t40".ivr2"):intrl(r) a0ir@[i:intrl(r)] +*ivr2 i@[a:dif][b:and(inn(a,class(r)),intd(a))] t41:=ande1(inn(a,class(r)),intd(a),b):inn(a,class(r)) t42:=ande2(inn(a,class(r)),intd(a),b):intd(a) t43:=eqintd(a,a0,isex(r,r,a,a0,t41,a0ir,refis(real,r)),t42):intd(a0) -ivr2 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) +*ivr2 a0ir@[n:natrl(r)] t44:=natintd(a0,natrlex(n)):intd(a0) t45:=intrlin(t44):intrl(r) -ivr2 r@[n:natrl(r)] natintrl:=realapp1(intrl(r),[x:dif][t:inn(x,class(r))]t45".ivr2"(x,t,n)):intrl(r) +*ivr2 a0ir@[p:pos(r)][i:intrl(r)] t46:=posintnatd(a0,posex(p),intrlex(i)):natd(a0) t47:=natrlin(t46):natrl(r) -ivr2 r@[p:pos(r)][i:intrl(r)] posintnatrl:=realapp1(natrl(r),[x:dif][t:inn(x,class(r))]t47".ivr2"(x,t,p,i)):natrl(r) +*ivr2 a0ir@[i2:is(r,0)] t48:=intdi0(a0,0ex(i2)):intd(a0) t49:=intrlin(t48):intrl(r) -ivr2 r@[i:is(r,0)] intrli0:=realapp1(intrl(r),[x:dif][t:inn(x,class(r))]t49".ivr2"(x,t,i)):intrl(r) r0@[n:natrp(r0)] remark6:=intrlin(pofrp(r0),pdofrp(r0),innclass(pdofrp(r0)),remark6"rp"(r0,n)):intrl(pofrp(r0)) remark7:=intrlin(nofrp(r0),ndofrp(r0),innclass(ndofrp(r0)),remark7"rp"(r0,n)):intrl(nofrp(r0)) +2r174 a0ir@[i:intrl(r)] t1:=satzd174(a0,intrlex(i)):ratd(a0) t2:=ratrlin(t1):ratrl(r) -2r174 r@[i:intrl(r)] satz174:=realapp1(ratrl(r),[x:dif][t:inn(x,class(r))]t2".2r174"(x,t,i)):ratrl(r) @plusdr:=[x:dif][y:dif]realof(pd(x,y)):[x:dif][y:dif]real +ivr3 [a:dif][b:dif][c:dif][d:dif][e:eq"rp"(a,b)][f:eq"rp"(c,d)] 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(plusdr,plusdr) -ivr3 fplusdr:=[x:dif][y:dif][z:dif][v:dif][t:eq][u:eq]t1".ivr3"(x,y,z,v,t,u):fixf2(real,plusdr) s@pl:=indreal2(real,plusdr,fplusdr,r,s):real +*ivr3 b1is@t2:=isindreal2(real,plusdr,fplusdr,r,s,a1,b1,a1ir,b1is):is(realof(pd(a1,b1)),pl(r,s)) -ivr3 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))) t@[i:is(r,s)] ispl1:=isf(real,real,[x:real]pl(x,t),r,s,i):is(pl(r,t),pl(s,t)) ispl2:=isf(real,real,[x:real]pl(t,x),r,s,i):is(pl(t,r),pl(t,s)) u@[i:is(r,s)][j:is(t,u)] 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)) +3r175 b1is@t1:=satzd175(a1,b1):eq"rp"(pd(a1,b1),pd(b1,a1)) 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)) -3r175 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)) compl:=satz175:is(pl(r,s),pl(s,r)) +*ivr3 b1is@[i:is(r,0)] t3:=pd01(a1,b1,0ex(r,a1,a1ir,i)):eq"rp"(pd(a1,b1),b1) t4:=isin(pl(r,s),s,pd(a1,b1),b1,picp,b1is,t3):is(pl(r,s),s) -ivr3 s@[i:is(r,0)] 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) s@[i:is(s,0)] pl02:=tris(real,pl(r,s),pl(s,r),r,compl,pl01(s,r,i)):is(pl(r,s),r) +*ivr3 b1is@[p:pos(r)][q:pos(s)] t5:=ppd(a1,b1,posex(r,a1,a1ir,p),posex(s,b1,b1is,q)):posd(pd(a1,b1)) t6:=posin(pl(r,s),pd(a1,b1),picp,t5):pos(pl(r,s)) -ivr3 s@[p:pos(r)][q:pos(s)] 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)) +*ivr3 b1is@[n:neg(r)][o:neg(s)] t7:=npd(a1,b1,negex(r,a1,a1ir,n),negex(s,b1,b1is,o)):negd(pd(a1,b1)) t8:=negin(pl(r,s),pd(a1,b1),picp,t7):neg(pl(r,s)) -ivr3 s@[n:neg(r)][o:neg(s)] 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)) @m0dr:=[x:dif]realof(m0d(x)):[x:dif]real +*ivr3 @[a:dif][b:dif][e:eq"rp"(a,b)] t5a:=isin(realof(m0d(a)),realof(m0d(b)),m0d(a),m0d(b),innclass(m0d(a)),innclass(m0d(b)),eqm0d(a,b,e)):is(m0dr,m0dr) -ivr3 @fm0dr:=[x:dif][y:dif][t:eq]t5a".ivr3"(x,y,t):fixf(real,m0dr) r@m0:=indreal(real,m0dr,fm0dr,r):real +*ivr3 a0ir@t6a:=isindreal(real,m0dr,fm0dr,r,a0,a0ir):is(realof(m0d(a0)),m0(r)) -ivr3 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))) s@[i:is(r,s)] ism0:=isf(real,real,[x:real]m0(x),r,s,i):is(m0(r),m0(s)) +*ivr3 a0ir@[n:neg(r)] t7a:=absnd(a0,negex(n)):eq"rp"(absd(a0),m0d(a0)) t8a:=isin(abs(r),m0(r),absd(a0),m0d(a0),aica,micm0,t7a):is(abs(r),m0(r)) -ivr3 r@[n:neg(r)] absn:=realapp1(is(abs(r),m0(r)),[x:dif][t:inn(x,class(r))]t8a".ivr3"(x,t,n)):is(abs(r),m0(r)) +*ivr3 a0ir@[nn:not(neg(r))] t9:=absnnd(a0,th3"l.imp"(negd(a0),neg(r),nn,[t:negd(a0)]negin(t))):eq"rp"(absd(a0),a0) t10:=isin(abs(r),r,absd(a0),a0,aica,a0ir,t9):is(abs(r),r) -ivr3 r@[nn:not(neg(r))] absnn:=realapp1(is(abs(r),r),[x:dif][t:inn(x,class(r))]t10".ivr3"(x,t,nn)):is(abs(r),r) r@[p:pos(r)] absp:=absnn(r,pnotn(r,p)):is(abs(r),r) r@[i:is(r,0)] abs0:=tris(real,abs(r),r,0,absnn(r,0notn(r,i)),i):is(abs(r),0) +3r176 a0ir@[p:pos(r)] t1:=satzd176a(a0,posex(p)):negd(m0d(a0)) t2:=negin(m0(r),m0d(a0),micm0,t1):neg(m0(r)) -3r176 r@[p:pos(r)] satz176a:=realapp1(neg(m0(r)),[x:dif][t:inn(x,class(r))]t2".3r176"(x,t,p)):neg(m0(r)) +*3r176 a0ir@[i:is(r,0)] t3:=satzd176b(a0,0ex(i)):zero(m0d(a0)) t4:=0in(m0(r),m0d(a0),micm0,t3):is(m0(r),0) -3r176 r@[i:is(r,0)] satz176b:=realapp1(is(m0(r),0),[x:dif][t:inn(x,class(r))]t4".3r176"(x,t,i)):is(m0(r),0) +*3r176 a0ir@[n:neg(r)] t5:=satzd176c(a0,negex(n)):posd(m0d(a0)) t6:=posin(m0(r),m0d(a0),micm0,t5):pos(m0(r)) -3r176 r@[n:neg(r)] satz176c:=realapp1(pos(m0(r)),[x:dif][t:inn(x,class(r))]t6".3r176"(x,t,n)):pos(m0(r)) +*3r176 a0ir@[n:neg(m0(r))] t7:=satzd176d(a0,negex(m0(r),m0d(a0),micm0,n)):posd(a0) t8:=posin(t7):pos(r) -3r176 r@[n:neg(m0(r))] satz176d:=realapp1(pos(r),[x:dif][t:inn(x,class(r))]t8".3r176"(x,t,n)):pos(r) +*3r176 a0ir@[i:is(m0(r),0)] t9:=satzd176e(a0,0ex(m0(r),m0d(a0),micm0,i)):zero(a0) t10:=0in(t9):is(r,0) -3r176 r@[i:is(m0(r),0)] satz176e:=realapp1(is(r,0),[x:dif][t:inn(x,class(r))]t10".3r176"(x,t,i)):is(r,0) +*3r176 a0ir@[p:pos(m0(r))] t11:=satzd176f(a0,posex(m0(r),m0d(a0),micm0,p)):negd(a0) t12:=negin(t11):neg(r) -3r176 r@[p:pos(m0(r))] satz176f:=realapp1(neg(r),[x:dif][t:inn(x,class(r))]t12".3r176"(x,t,p)):neg(r) +3r177 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) -3r177 r@satz177:=realapp1(is(m0(m0(r)),r),[x:dif][t:inn(x,class(r))]t1".3r177"(x,t)):is(m0(m0(r)),r) satz177a:=symis(real,m0(m0(r)),r,satz177):is(r,m0(m0(r))) s@[i:is(r,m0(s))] satz177b:=tris(real,m0(r),m0(m0(s)),s,ism0(r,m0(s),i),satz177(s)):is(m0(r),s) satz177c:=symis(real,m0(r),s,satz177b):is(s,m0(r)) s@[i:is(m0(r),s)] satz177d:=satz177c(s,r,symis(real,m0(r),s,i)):is(r,m0(s)) satz177e:=symis(real,r,m0(s),satz177d):is(m0(s),r) +3r178 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)) -3r178 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)) satz178a:=symis(real,abs(m0(r)),abs(r),satz178):is(abs(r),abs(m0(r))) +3r179 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) -3r179 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) satz179a:=tris(real,pl(m0(r),r),pl(r,m0(r)),0,compl(m0(r),r),satz179):is(pl(m0(r),r),0) +3r180 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))) -3r180 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))) satz180a:=symis(real,m0(pl(r,s)),pl(m0(r),m0(s)),satz180):is(pl(m0(r),m0(s)),m0(pl(r,s))) mn:=pl(r,m0(s)):real b1is@micmn:=picp(r,m0(s),a1,m0d(b1),a1ir,micm0(s,b1,b1is)):inn(md(a1,b1),class(mn(r,s))) t@[i:is(r,s)] ismn1:=ispl1(r,s,m0(t),i):is(mn(r,t),mn(s,t)) ismn2:=ispl2(m0(r),m0(s),t,ism0(r,s,i)):is(mn(t,r),mn(t,s)) u@[i:is(r,s)][j:is(t,u)] ismn12:=ispl12(r,s,m0(t),m0(u),i,ism0(t,u,j)):is(mn(r,t),mn(s,u)) 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)) satz181a:=symis(real,m0(mn(s,r)),mn(r,s),satz181(s,r)):is(mn(r,s),m0(mn(s,r))) +3r182 b1is@[p:pos(mn(r,s))] t1:=satzd182a(a1,b1,posex(mn(r,s),md(a1,b1),micmn,p)):mored(a1,b1) t2:=morein(t1):more(r,s) -3r182 [p:pos(mn(r,s))] 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) +*3r182 b1is@[i:is(mn(r,s),0)] t3:=satzd182b(a1,b1,0ex(mn(r,s),md(a1,b1),micmn,i)):eq"rp"(a1,b1) t4:=isin(t3):is(r,s) -3r182 s@[i:is(mn(r,s),0)] 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) +*3r182 b1is@[n:neg(mn(r,s))] t5:=satzd182c(a1,b1,negex(mn(r,s),md(a1,b1),micmn,n)):lessd(a1,b1) t6:=lessin(t5):less(r,s) -3r182 s@[n:neg(mn(r,s))] 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) +*3r182 b1is@[m:more(r,s)] t7:=satzd182d(a1,b1,moreex(m)):posd(md(a1,b1)) t8:=posin(mn(r,s),md(a1,b1),micmn,t7):pos(mn(r,s)) -3r182 s@[m:more(r,s)] 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)) +*3r182 b1is@[i:is(r,s)] t9:=satzd182e(a1,b1,isex(i)):zero(md(a1,b1)) t10:=0in(mn(r,s),md(a1,b1),micmn,t9):is(mn(r,s),0) -3r182 s@[i:is(r,s)] 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) +*3r182 b1is@[l:less(r,s)] t11:=satzd182f(a1,b1,lessex(l)):negd(md(a1,b1)) t12:=negin(mn(r,s),md(a1,b1),micmn,t11):neg(mn(r,s)) -3r182 s@[l:less(r,s)] 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)) +3r183 b1is@[m:more(r,s)] t1:=satzd183a(a1,b1,moreex(m)):lessd(m0d(a1),m0d(b1)) t2:=lessin(m0(r),m0(s),m0d(a1),m0d(b1),micm0(r,a1,a1ir),micm0(s,b1,b1is),t1):less(m0(r),m0(s)) -3r183 s@[m:more(r,s)] 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)) s@[i:is(r,s)] satz183b:=ism0(r,s,i):is(m0(r),m0(s)) +*3r183 b1is@[l:less(r,s)] t3:=satzd183c(a1,b1,lessex(l)):mored(m0d(a1),m0d(b1)) t4:=morein(m0(r),m0(s),m0d(a1),m0d(b1),micm0(r,a1,a1ir),micm0(s,b1,b1is),t3):more(m0(r),m0(s)) -3r183 s@[l:less(r,s)] 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)) s@[l:less(m0(r),m0(s))] satz183d:=ismore12(m0(m0(r)),r,m0(m0(s)),s,satz177(r),satz177(s),satz183c(m0(r),m0(s),l)):more(r,s) s@[i:is(m0(r),m0(s))] satz183e:=tr3is(real,r,m0(m0(r)),m0(m0(s)),s,satz177a(r),ism0(m0(r),m0(s),i),satz177(s)):is(r,s) s@[m:more(m0(r),m0(s))] satz183f:=isless12(m0(m0(r)),r,m0(m0(s)),s,satz177(r),satz177(s),satz183a(m0(r),m0(s),m)):less(r,s) +3r184 t@prop1:=and3(pos(s),pos(t),is(r,mn(s,t))):'prop' s@prop2:=some([x:real]prop1(x)):'prop' r@prop3:=some([x:real]prop2(x)):'prop' a0ir@[a:dif][b:dif] prop1d:=and3(posd(a),posd(b),eq"rp"(a0,md(a,b))):'prop' a@prop2d:=some"l"(dif,[x:dif]prop1d(x)):'prop' [p2:prop2d(a)][b:dif][p1:prop1d(a,b)] t1:=and3e1(posd(a),posd(b),eq"rp"(a0,md(a,b)),p1):posd(a) t2:=and3e2(posd(a),posd(b),eq"rp"(a0,md(a,b)),p1):posd(b) t3:=and3e3(posd(a),posd(b),eq"rp"(a0,md(a,b)),p1):eq"rp"(a0,md(a,b)) p2@ra:=realof(a):real p1@rb:=realof(b):real t4:=innclass(a):inn(a,class(ra)) t5:=innclass(b):inn(b,class(rb)) t6:=isin(r,mn(ra,rb),a0,md(a,b),a0ir,micmn(ra,rb,a,b,t4,t5),t3):is(r,mn(ra,rb)) 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) t8:=somei(real,[x:real]prop1(ra,x),rb,t7):prop2(ra) p2@t9:=someapp(dif,[x:dif]prop1d(a,x),p2,prop2(ra),[x:dif][t:prop1d(a,x)]t8(x,t)):prop2(ra) t10:=somei(real,[x:real]prop2(x),ra,t9):prop3 a0ir@t11:=someapp(dif,[x:dif]prop2d(x),satzd184(a0),prop3,[x:dif][t:prop2d(x)]t10(x,t)):prop3 -3r184 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))))) 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))] +3r185 t1:=satzd185(a3,b3,c3,d3):eq"rp"(pd(md(a3,b3),md(c3,d3)),md(pd(a3,c3),pd(b3,d3))) 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))) -3r185 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))) +3r186 c2it@t1:=satzd186(a2,b2,c2):eq"rp"(pd(pd(a2,b2),c2),pd(a2,pd(b2,c2))) 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))) -3r186 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))) asspl1:=satz186:is(pl(pl(r,s),t),pl(r,pl(s,t))) 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)) 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) 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) 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) satz187b:=somei(real,[x:real]is(pl(s,x),r),mn(r,s),satz187a):some([x:real]is(pl(s,x),r)) [x:real][i:is(pl(s,x),r)] 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) satz187d:=symis(real,mn(r,s),x,satz187c):is(x,mn(r,s)) x@[i:is(pl(x,s),r)] satz187e:=satz187c(tris(real,pl(s,x),pl(x,s),r,compl(s,x),i)):is(mn(r,s),x) satz187f:=symis(real,mn(r,s),x,satz187e):is(x,mn(r,s)) +3r187 s@[x:real][y:real][i:is(pl(s,x),r)][j:is(pl(s,y),r)] t1:=tris1(real,x,y,mn(r,s),satz187c(x,i),satz187c(y,j)):is(x,y) 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)) -3r187 s@satz187:=onei(real,[x:real]is(pl(s,x),r),t2".3r187",satz187b):one([x:real]is(pl(s,x),r)) +3r188 c2it@[m:more(pl(r,t),pl(s,t))] 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) t2:=morein(r,s,a2,b2,a2ir,b2is,t1):more(r,s) -3r188 t@[m:more(pl(r,t),pl(s,t))] 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) +*3r188 c2it@[i:is(pl(r,t),pl(s,t))] 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) t4:=isin(r,s,a2,b2,a2ir,b2is,t3):is(r,s) -3r188 t@[i:is(pl(r,t),pl(s,t))] 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) +*3r188 c2it@[l:less(pl(r,t),pl(s,t))] 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) t6:=lessin(r,s,a2,b2,a2ir,b2is,t5):less(r,s) -3r188 t@[l:less(pl(r,t),pl(s,t))] 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) +*3r188 c2it@[m:more(r,s)] t7:=satzd188d(a2,b2,c2,moreex(r,s,a2,b2,a2ir,b2is,m)):mored(pd(a2,c2),pd(b2,c2)) 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)) -3r188 t@[m:more(r,s)] 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)) t@[i:is(r,s)] satz188e:=ispl1(r,s,t,i):is(pl(r,t),pl(s,t)) +*3r188 c2it@[l:less(r,s)] t9:=satzd188f(a2,b2,c2,lessex(r,s,a2,b2,a2ir,b2is,l)):lessd(pd(a2,c2),pd(b2,c2)) 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)) -3r188 t@[l:less(r,s)] 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)) t@[m:more(pl(t,r),pl(t,s))] 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) t@[i:is(pl(t,r),pl(t,s))] 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) t@[l:less(pl(t,r),pl(t,s))] 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) t@[m:more(r,s)] 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)) t@[i:is(r,s)] satz188l:=ispl2(r,s,t,i):is(pl(t,r),pl(t,s)) t@[l:less(r,s)] 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)) u@[i:is(r,s)][m:more(t,u)] 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)) 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)) i@[l:less(t,u)] 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)) 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)) u@[m:more(r,s)][n:more(t,u)] 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)) u@[l:less(r,s)][k:less(t,u)] 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)) u@[m:moreis(r,s)][n:more(t,u)] 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)) u@[m:more(r,s)][n:moreis(t,u)] 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)) u@[l:lessis(r,s)][k:less(t,u)] 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)) u@[l:less(r,s)][k:lessis(t,u)] 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)) +3r191 d3iu@[m:moreis(r,s)][n:moreis(t,u)] 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)) 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)) -3r191 u@[m:moreis(r,s)][n:moreis(t,u)] 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)) u@[l:lessis(r,s)][k:lessis(t,u)] 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)) @timesdr:=[x:dif][y:dif]realof(td(x,y)):[x:dif][y:dif]real +ivr4 [a:dif][b:dif][c:dif][d:dif][e:eq"rp"(a,b)][f:eq"rp"(c,d)] 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(timesdr,timesdr) -ivr4 ftimesdr:=[x:dif][y:dif][z:dif][v:dif][t:eq][u:eq]t1".ivr4"(x,y,z,v,t,u):fixf2(real,timesdr) s@ts:=indreal2(real,timesdr,ftimesdr,r,s):real +*ivr4 b1is@t2:=isindreal2(real,timesdr,ftimesdr,r,s,a1,b1,a1ir,b1is):is(realof(td(a1,b1)),ts(r,s)) -ivr4 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))) t@[i:is(r,s)] ists1:=isf(real,real,[x:real]ts(x,t),r,s,i):is(ts(r,t),ts(s,t)) ists2:=isf(real,real,[x:real]ts(t,x),r,s,i):is(ts(t,r),ts(t,s)) u@[i:is(r,s)][j:is(t,u)] 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)) +4r192 b1is@[i:is(r,0)] t1:=satzd192a(a1,b1,0ex(r,a1,a1ir,i)):zero(td(a1,b1)) t2:=0in(ts(r,s),td(a1,b1),tict,t1):is(ts(r,s),0) -4r192 s@[i:is(r,0)] 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) +*4r192 b1is@[i:is(s,0)] t3:=satzd192b(a1,b1,0ex(s,b1,b1is,i)):zero(td(a1,b1)) t4:=0in(ts(r,s),td(a1,b1),tict,t3):is(ts(r,s),0) -4r192 s@[i:is(s,0)] 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) +*4r192 b1is@[i:is(ts(r,s),0)] t5:=satzd192c(a1,b1,0ex(ts(r,s),td(a1,b1),tict,i)):or(zero(a1),zero(b1)) 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)) -4r192 s@[i:is(ts(r,s),0)] 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)) s@[n:nis(r,0)][o:nis(s,0)] 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) s@[i:is(r,0)] ts01:=satz192a(i):is(ts(r,s),0) s@[i:is(s,0)] ts02:=satz192b(i):is(ts(r,s),0) +4r193 b1is@t1:=satzd193(a1,b1):eq"rp"(absd(td(a1,b1)),td(absd(a1),absd(b1))) 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))) -4r193 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))) satz193a:=symis(real,abs(ts(r,s)),ts(abs(r),abs(s)),satz193):is(ts(abs(r),abs(s)),abs(ts(r,s))) +4r194 b1is@t1:=satzd194(a1,b1):eq"rp"(td(a1,b1),td(b1,a1)) 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)) -4r194 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)) comts:=satz194:is(ts(r,s),ts(s,r)) @1rl:=realof(1df):real pos1:=posin(1rl,1df,innclass(1df),posdirp(1rp)):pos(1rl) natrl1:=natrli(1):natrl(1rl) intrl1:=natintrl(1rl,natrl1):intrl(1rl) +4r195 a0ir@t1:=satzd195(a0):eq"rp"(td(a0,1df),a0) 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) -4r195 r@satz195:=realapp1(is(ts(r,1rl),r),[x:dif][t:inn(x,class(r))]t2".4r195"(x,t)):is(ts(r,1rl),r) satz195a:=symis(real,ts(r,1rl),r,satz195):is(r,ts(r,1rl)) satz195b:=tris(real,ts(1rl,r),ts(r,1rl),r,comts(1rl,r),satz195):is(ts(1rl,r),r) satz195c:=symis(real,ts(1rl,r),r,satz195b):is(r,ts(1rl,r)) s@[p:pos(r)][q:pos(s)] 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))) +4r196 b1is@[n:neg(r)][o:neg(s)] t1:=satzd196b(a1,b1,negex(r,a1,a1ir,n),negex(s,b1,b1is,o)):eq"rp"(td(a1,b1),td(absd(a1),absd(b1))) 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))) -4r196 s@[n:neg(r)][o:neg(s)] 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))) +*4r196 b1is@[p:pos(r)][n:neg(s)] 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)))) 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)))) -4r196 s@[p:pos(r)][n:neg(s)] 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)))) s@[n:neg(r)][p:pos(s)] 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)))) +*4r196 a0ir@[n:not(is(r,0))] t3:=th3"l.imp"(zero(a0),is(r,0),n,[t:zero(a0)]0in(t)):not(zero(a0)) b1is@[n:not(is(r,0))][o:not(is(s,0))][i:is(ts(r,s),ts(abs(r),abs(s)))] 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))) [a:and(posd(a1),posd(b1))] 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)) i@[a:and(negd(a1),negd(b1))] 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)) 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))) -4r196 s@[n:not(is(r,0))][o:not(is(s,0))][i:is(ts(r,s),ts(abs(r),abs(s)))] 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))) +*4r196 o@[i:is(ts(r,s),m0(ts(abs(r),abs(s))))] 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))) [a:and(posd(a1),negd(b1))] 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)) i@[a:and(negd(a1),posd(b1))] 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)) 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))) -4r196 o@[i:is(ts(r,s),m0(ts(abs(r),abs(s))))] 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))) s@[p:pos(ts(r,s))] +*4r196 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) 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) 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))) -4r196 p@satz196g:=satz196e(t12".4r196",t13".4r196",t14".4r196"):or(and(pos(r),pos(s)),and(neg(r),neg(s))) s@[n:neg(ts(r,s))] +*4r196 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) 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) 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)))) -4r196 n@satz196h:=satz196f(t15".4r196",t16".4r196",t17".4r196"):or(and(pos(r),neg(s)),and(neg(r),pos(s))) +4r197 b1is@t1:=satzd197a(a1,b1):eq"rp"(td(m0d(a1),b1),m0d(td(a1,b1))) 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))) -4r197 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))) 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))) 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))) satz197d:=symis(real,ts(m0(r),s),ts(r,m0(s)),satz197c):is(ts(r,m0(s)),ts(m0(r),s)) satz197e:=symis(real,ts(m0(r),s),m0(ts(r,s)),satz197a):is(m0(ts(r,s)),ts(m0(r),s)) satz197f:=symis(real,ts(r,m0(s)),m0(ts(r,s)),satz197b):is(m0(ts(r,s)),ts(r,m0(s))) 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)) satz198a:=symis(real,ts(m0(r),m0(s)),ts(r,s),satz198):is(ts(r,s),ts(m0(r),m0(s))) +*ivr4 b1is@[p:pos(r)][q:pos(s)] t3:=ptdpp(a1,b1,posex(r,a1,a1ir,p),posex(s,b1,b1is,q)):posd(td(a1,b1)) t4:=posin(ts(r,s),td(a1,b1),tict,t3):pos(ts(r,s)) -ivr4 s@[p:pos(r)][q:pos(s)] 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)) +*ivr4 p@[n:neg(s)] t5:=ntdpn(a1,b1,posex(r,a1,a1ir,p),negex(s,b1,b1is,n)):negd(td(a1,b1)) t6:=negin(ts(r,s),td(a1,b1),tict,t5):neg(ts(r,s)) -ivr4 s@[p:pos(r)][n:neg(s)] 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)) s@[n:neg(r)][p:pos(s)] negtsnp:=isneg(ts(s,r),ts(r,s),comts(s,r),negtspn(s,r,p,n)):neg(ts(r,s)) s@[n:neg(r)][o:neg(s)] 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)) r@[n:nis(r,0)] 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)) 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))) +4r199 c2it@t1:=satzd199(a2,b2,c2):eq"rp"(td(td(a2,b2),c2),td(a2,td(b2,c2))) 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))) -4r199 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))) assts1:=satz199:is(ts(ts(r,s),t),ts(r,ts(s,t))) 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)) +4r201 c2it@t1:=satzd201(a2,b2,c2):eq"rp"(td(a2,pd(b2,c2)),pd(td(a2,b2),td(a2,c2))) 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))) -4r201 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))) 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))) disttp2:=satz201:is(ts(r,pl(s,t)),pl(ts(r,s),ts(r,t))) 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)) 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))) 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))) 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))) disttm2:=satz202:is(ts(r,mn(s,t)),mn(ts(r,s),ts(r,t))) 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)) 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))) satz200:=satz202:is(ts(r,mn(s,t)),mn(ts(r,s),ts(r,t))) +4r203 c2it@[m:more(r,s)][p:pos(t)] 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)) 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)) -4r203 [m:more(r,s)][p:pos(t)] 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)) m@[i:is(t,0)] 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)) +*4r203 m@[n:neg(t)] 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)) 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)) -4r203 m@[n:neg(t)] 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)) 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)) 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)) 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)) t@[l:less(r,s)][p:pos(t)] satz203g:=lemma1(ts(s,t),ts(r,t),satz203a(s,r,t,lemma2(r,s,l),p)):less(ts(r,t),ts(s,t)) l@[i:is(t,0)] 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)) l@[n:neg(t)] satz203j:=lemma2(ts(s,t),ts(r,t),satz203c(s,r,t,lemma2(r,s,l),n)):more(ts(r,t),ts(s,t)) 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)) 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)) 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)) +4r204 a0ir@[n1:nis(r,0)] t1:=th3"l.imp"(zero(a0),is(r,0),n1,[t:zero(a0)]0in(t)):not(zero(a0)) d3iu@[n1:nis(s,0)][i:is(ts(s,t),r)][j:is(ts(s,u),r)] 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) t3:=isin(t,u,c3,d3,c3it,d3iu,t2):is(t,u) -4r204 s@[n:nis(s,0)][x:real][y:real][i:is(ts(s,x),r)][j:is(ts(s,y),r)] 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) +*4r204 b1is@[n1:nis(s,0)] t4:=satzd204a(a1,b1,t1(s,b1,b1is,n1)):some"l"(dif,[x:dif]eq"rp"(td(b1,x),a1)) [a:dif][e:eq"rp"(td(b1,a),a1)] ar:=realof(a):real 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) t6:=somei(real,[x:real]is(ts(s,x),r),ar,t5):some([x:real]is(ts(s,x),r)) 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)) -4r204 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)) 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)) ov:=ind(real,[x:real]is(ts(s,x),r),satz204):real satz204c:=oneax(real,[x:real]is(ts(s,x),r),satz204):is(ts(s,ov(r,s,n)),r) satz204d:=symis(real,ts(s,ov(r,s,n)),r,satz204c):is(r,ts(s,ov(r,s,n))) 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) satz204f:=symis(real,ts(ov(r,s,n),s),r,satz204e):is(r,ts(ov(r,s,n),s)) s@[x:real][n:nis(s,0)][i:is(ts(s,x),r)] satz204g:=satz204b(n,x,ov(r,s,n),i,satz204c(n)):is(x,ov(r,s,n)) s@[n:nis(s,0)][p:pos(r)][q:pos(s)] +*4r204 n@ros:=ov(r,s,n):real p@t8:=ispos(r,ts(s,ros),satz204d(n),p):pos(ts(s,ros)) q@t9:=th1"l.and"(neg(s),neg(ros),pnotn(s,q)):not(and(neg(s),neg(ros))) t10:=ore1(and(pos(s),pos(ros)),and(neg(s),neg(ros)),satz196g(s,ros,t8),t9):and(pos(s),pos(ros)) -4r204 q@posovpp:=ande2(pos(s),pos(ov(r,s,n)),t10".4r204"):pos(ov(r,s,n)) p@[m:neg(s)] +*4r204 m@t11:=th1"l.and"(pos(s),pos(ros),nnotp(s,m)):not(and(pos(s),pos(ros))) t12:=ore2(and(pos(s),pos(ros)),and(neg(s),neg(ros)),satz196g(s,ros,t8),t11):and(neg(s),neg(ros)) -4r204 m@negovpn:=ande2(neg(s),neg(ov(r,s,n)),t12".4r204"):neg(ov(r,s,n)) n@[m:neg(r)][p:pos(s)] +*4r204 m@t13:=isneg(r,ts(s,ros),satz204d(n),m):neg(ts(s,ros)) p@t14:=th1"l.and"(neg(s),pos(ros),pnotn(s,p)):not(and(neg(s),pos(ros))) t15:=ore1(and(pos(s),neg(ros)),and(neg(s),pos(ros)),satz196h(s,ros,t13),t14):and(pos(s),neg(ros)) -4r204 p@negovnp:=ande2(pos(s),neg(ov(r,s,n)),t15".4r204"):neg(ov(r,s,n)) m@[l:neg(s)] +*4r204 l@t16:=th1"l.and"(pos(s),neg(ros),nnotp(s,l)):not(and(pos(s),neg(ros))) t17:=ore2(and(pos(s),neg(ros)),and(neg(s),pos(ros)),satz196h(s,ros,t13),t16):and(neg(s),pos(ros)) -4r204 l@posovnn:=ande2(neg(s),pos(ov(r,s,n)),t17".4r204"):pos(ov(r,s,n)) @[r0:cut][s0:cut][m:more"rp"(r0,s0)] 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)) s0@[m:more(pofrp(r0),pofrp(s0))] morerpip:=morerpipd(r0,s0,moreex(pofrp(r0),pofrp(s0),pdofrp(r0),pdofrp(s0),innclass(pdofrp(r0)),innclass(pdofrp(s0)),m)):more"rp"(r0,s0) s0@[l:less"rp"(r0,s0)] lessrpep:=lemma1(pofrp(s0),pofrp(r0),morerpep(s0,r0,satz122(r0,s0,l))):less(pofrp(r0),pofrp(s0)) s0@[l:less(pofrp(r0),pofrp(s0))] lessrpip:=satz121(s0,r0,morerpip(s0,r0,lemma2(pofrp(r0),pofrp(s0),l))):less"rp"(r0,s0) s@[p:pos(r)][q:pos(s)][m:more(r,s)] q@[m:more(r,s)] +ivr5 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))) -ivr5 moreperp:=morerpip(rpofp(r,p),rpofp(s,q),t1".ivr5"):more"rp"(rpofp(r,p),rpofp(s,q)) q@[m:more"rp"(rpofp(r,p),rpofp(s,q))] +*ivr5 m@t2:=morerpep(rpofp(r,p),rpofp(s,q),m):more(pofrp(rpofp(r,p)),pofrp(rpofp(s,q))) -ivr5 m@morepirp:=ismore12(pofrp(rpofp(r,p)),r,pofrp(rpofp(s,q)),s,isprp2(r,p),isprp2(s,q),t2".ivr5"):more(r,s) q@[l:less(r,s)] 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)) q@[l:less"rp"(rpofp(r,p),rpofp(s,q))] lesspirp:=lemma1(s,r,morepirp(s,r,q,p,satz122(rpofp(r,p),rpofp(s,q),l))):less(r,s) r@s01:=setof(real,[x:real]lessis(x,r)):set(real) s02:=setof(real,[x:real]more(x,r)):set(real) +5r205 s@[n:not(in(s,s01))] 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)) t2:=estii(real,[x:real]more(x,r),s,satz167k(s,r,t1)):in(s,s02) -5r205 vb00:=[x:real][t:not(in(x,s01))]t2".5r205"(x,t):all([x:real]or(in(x,s01),in(x,s02))) +*5r205 r@t3:=estii(real,[x:real]lessis(x,r),r,lessisi2(r,r,refis(real,r))):in(r,s01) -5r205 r@vb01a:=nonemptyi(real,s01,r,t3".5r205"):nonempty(real,s01) +*5r205 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) t5:=estii(real,[x:real]more(x,r),pl(r,1rl),t4):in(pl(r,1rl),s02) -5r205 r@vb01b:=nonemptyi(real,s02,pl(r,1rl),t5".5r205"):nonempty(real,s02) +*5r205 s@[i:in(s,s01)][t:real][j:in(t,s02)] 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) -5r205 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))) s@[l:less(s,r)] vb03a:=estii(real,[x:real]lessis(x,r),s,lessisi1(s,r,l)):in(s,s01) s@[m:more(s,r)] vb03b:=estii(real,[x:real]more(x,r),s,m):in(s,s02) r@s11:=setof(real,[x:real]less(x,r)):set(real) s12:=setof(real,[x:real]moreis(x,r)):set(real) +*5r205 s@[n:not(in(s,s11))] 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)) t8:=estii(real,[x:real]moreis(x,r),s,satz167f(s,r,t7)):in(s,s12) -5r205 r@vb10:=[x:real][t:not(in(x,s11))]t8".5r205"(x,t):all([x:real]or(in(x,s11),in(x,s12))) +*5r205 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) t10:=estii(real,[x:real]less(x,r),mn(r,1rl),t9):in(mn(r,1rl),s11) -5r205 r@vb11a:=nonemptyi(real,s11,mn(r,1rl),t10".5r205"):nonempty(real,s11) +*5r205 r@t11:=estii(real,[x:real]moreis(x,r),r,moreisi2(r,r,refis(real,r))):in(r,s12) -5r205 r@vb11b:=nonemptyi(real,s12,r,t11".5r205"):nonempty(real,s12) +*5r205 s@[i:in(s,s11)][t:real][j:in(t,s12)] 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) -5r205 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))) s@[l:less(s,r)] vb13a:=estii(real,[x:real]less(x,r),s,l):in(s,s11) s@[m:more(s,r)] vb13b:=estii(real,[x:real]moreis(x,r),s,moreisi1(s,r,m)):in(s,s12) @2rl:=pl(1rl,1rl):real pos2:=pospl(1rl,1rl,pos1,pos1):pos(2rl) half:=ov(1rl,2rl,pnot0(2rl,pos2)):real poshalf:=posovpp(1rl,2rl,pnot0(2rl,pos2),pos1,pos2):pos(half) +*ivr5 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)) 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) -ivr5 s@[l:less(r,s)] +*ivr5 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))) -ivr5 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))) +*ivr5 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))) -ivr5 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) [p:pos(r)] lemma5:=satz169b(s,trmore(s,r,0,lemma2(r,s,l),satz169a(r,p))):pos(s) @[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)))] +*5r205 s2@[r:real] prop1:=all([x:real][t:less(x,r)]in(x,s1)):'prop' prop2:=all([x:real][t:more(x,r)]in(x,s2)):'prop' prop3:=and(prop1,prop2):'prop' p2@[x:real][y:real][px:prop3(x)][py:prop3(y)][l:less(x,y)] mxy:=ts(half,pl(x,y)):real t13:=lemma2(x,mxy,lemma3(x,y,l)):more(mxy,x) t14:=lemma4(x,y,l):less(mxy,y) t15:=ande1(prop1(y),prop2(y),py):in(mxy,s1) t16:=ande2(prop1(x),prop2(x),px):in(mxy,s2) t17:=p2:less(mxy,mxy) t18:=ec3e31(is(mxy,mxy),more(mxy,mxy),less(mxy,mxy),satz167b(mxy,mxy),t17):con py@t19:=[t:less(x,y)]t18(t):not(less(x,y)) t20:=[t:more(x,y)]t18(y,x,py,px,lemma1(x,y,t)):not(more(x,y)) t21:=or3e1(is(x,y),more(x,y),less(x,y),satz167a(x,y),t20,t19):is(x,y) p2@t22:=[x:real][y:real][t:prop3(x)][u:prop3(y)]t21(x,y,t,u):amone(real,[x:real]prop3(x)) [case1:some([x:real]and(pos(x),in(x,s1)))][r:real][a:and(pos(r),in(r,s1))] t23:=ande1(pos(r),in(r,s1),a):pos(r) t24:=ande2(pos(r),in(r,s1),a):in(r,s1) sc1:=setof(cut,[x:cut]in(pofrp(x),s1)):set(cut) sc2:=setof(cut,[x:cut]in(pofrp(x),s2)):set(cut) [r0:cut][i:in(pofrp(r0),s1)] t25:=estii(cut,[x:cut]in(pofrp(x),s1),r0,i):in"rp"(r0,sc1) r0@[i:in"rp"(r0,sc1)] t26:=estie(cut,[x:cut]in(pofrp(x),s1),r0,i):in(pofrp(r0),s1) r0@[i:in(pofrp(r0),s2)] t27:=estii(cut,[x:cut]in(pofrp(x),s2),r0,i):in"rp"(r0,sc2) r0@[i:in"rp"(r0,sc2)] t28:=estie(cut,[x:cut]in(pofrp(x),s2),r0,i):in(pofrp(r0),s2) r0@t29:=th9"l.or"(in(pofrp(r0),s1),in(pofrp(r0),s2),in"rp"(r0,sc1),in"rp"(r0,sc2),p0,[t:in(pofrp(r0),s1)]t25(t),[t:in(pofrp(r0),s2)]t27(t)):or(in"rp"(r0,sc1),in"rp"(r0,sc2)) a@pr1:=rpofp(r,t23):cut t30:=isp(real,[x:real]in(x,s1),r,pofrp(pr1),t24,isprp1(r,t23)):in(pofrp(pr1),s1) t31:=nonemptyi(cut,sc1,pr1,t25(pr1,t30)):nonempty(cut,sc1) [s:real][i:in(s,s2)] t32:=p2:less(r,s) t33:=lemma5(r,s,t32,t23):pos(s) ps1:=rpofp(s,t33):cut t34:=isp(real,[x:real]in(x,s2),s,pofrp(ps1),i,isprp1(s,t33)):in(pofrp(ps1),s2) t35:=nonemptyi(cut,sc2,ps1,t27(ps1,t34)):nonempty(cut,sc2) a@t36:=nonemptyapp(real,s2,p1b,nonempty(cut,sc2),[x:real][t:in(x,s2)]t35(x,t)):nonempty(cut,sc2) r0@[i:in"rp"(r0,sc1)][s0:cut][j:in"rp"(s0,sc2)] t37:=p2:less(pofrp(r0),pofrp(s0)) t38:=lessrpip(r0,s0,t37):less"rp"(r0,s0) 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 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)) 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)) stp:=pofrp(stc):real t41:=posi(stc):pos(stp) [s:real][l:less(s,stp)][p:pos(s)] ps2:=rpofp(s,p):cut t42:=lessrpip(ps2,stc,isless1(s,pofrp(ps2),stp,isprp1(s,p),l)):less"rp"(ps2,stc) t43:=t39:in"rp"(ps2,sc1) t44:=isp(real,[x:real]in(x,s1),pofrp(ps2),s,t26(ps2,t43),isprp2(s,p)):in(s,s1) l@[n:not(pos(s))][i:in(s,s2)] t45:=p2:less(r,s) t46:=n:con n@t47:=ore1(in(s,s1),in(s,s2),p0,[t:in(s,s2)]t46(t)):in(s,s1) l@t48:=th1"l.imp"(pos(s),in(s,s1),[t:pos(s)]t44(t),[t:not(pos(s))]t47(t)):in(s,s1) s@[m:more(s,stp)] t49:=lemma5(stp,s,lemma1(s,stp,m),t41):pos(s) ps3:=rpofp(s,t49):cut t50:=morerpip(ps3,stc,ismore1(s,pofrp(ps3),stp,isprp1(s,t49),m)):more"rp"(ps3,stc) t51:=t40:in"rp"(ps3,sc2) t52:=isp(real,[x:real]in(x,s2),pofrp(ps3),s,t28(ps3,t51),isprp2(s,t49)):in(s,s2) 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) t54:=somei(real,[x:real]prop3(x),stp,t53):some([x:real]prop3(x)) 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)) p2@[case2:some([x:real]and(neg(x),in(x,s2)))] sp1:=setof(real,[x:real]in(m0(x),s1)):set(real) sp2:=setof(real,[x:real]in(m0(x),s2)):set(real) [r:real][i:in(m0(r),s1)] t56:=estii(real,[x:real]in(m0(x),s1),r,i):in(r,sp1) r@[i:in(r,sp1)] t57:=estie(real,[x:real]in(m0(x),s1),r,i):in(m0(r),s1) r@[i:in(m0(r),s2)] t58:=estii(real,[x:real]in(m0(x),s2),r,i):in(r,sp2) r@[i:in(r,sp2)] t59:=estie(real,[x:real]in(m0(x),s2),r,i):in(m0(r),s2) 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),p0,[t:in(m0(r),s1)]t56(t),[t:in(m0(r),s2)]t58(t))):or(in(r,sp2),in(r,sp1)) [i:in(r,s2)] t61:=t58(m0(r),isp(real,[x:real]in(x,s2),r,m0(m0(r)),i,satz177a(r))):in(m0(r),sp2) t62:=nonemptyi(real,sp2,m0(r),t61):nonempty(real,sp2) case2@t63:=nonemptyapp(real,s2,p1b,nonempty(real,sp2),[x:real][t:in(x,s2)]t62(x,t)):nonempty(real,sp2) r@[i:in(r,s1)] t64:=t56(m0(r),isp(real,[x:real]in(x,s1),r,m0(m0(r)),i,satz177a(r))):in(m0(r),sp1) t65:=nonemptyi(real,sp1,m0(r),t64):nonempty(real,sp1) case2@t66:=nonemptyapp(real,s1,p1a,nonempty(real,sp1),[x:real][t:in(x,s1)]t65(x,t)):nonempty(real,sp1) r@[i:in(r,sp2)][s:real][j:in(s,sp1)] t67:=p2:less(m0(s),m0(r)) t68:=lemma1(s,r,satz183d(s,r,t67)):less(r,s) r@[a:and(neg(r),in(r,s2))] t69:=satz176c(r,ande1(neg(r),in(r,s2),a)):pos(m0(r)) 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) t71:=andi(pos(m0(r)),in(m0(r),sp2),t69,t58(m0(r),t70)):and(pos(m0(r)),in(m0(r),sp2)) t72:=somei(real,[x:real]and(pos(x),in(x,sp2)),m0(r),t71):some([x:real]and(pos(x),in(x,sp2))) 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))) 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)) [r:real][p:prop3(sp2,sp1,r)][s:real][l:less(s,m0(r))] t75:=ismore2(m0(m0(r)),r,m0(s),satz177(r),satz183c(s,m0(r),l)):more(m0(s),r) t76:=ande2(prop1(sp2,sp1,r),prop2(sp2,sp1,r),p):in(m0(s),sp1) t77:=isp(real,[x:real]in(x,s1),m0(m0(s)),s,t57(m0(s),t76),satz177(s)):in(s,s1) s@[m:more(s,m0(r))] t78:=isless2(m0(m0(r)),r,m0(s),satz177(r),satz183a(s,m0(r),m)):less(m0(s),r) t79:=ande1(prop1(sp2,sp1,r),prop2(sp2,sp1,r),p):in(m0(s),sp2) t80:=isp(real,[x:real]in(x,s2),m0(m0(s)),s,t59(m0(s),t79),satz177(s)):in(s,s2) 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)) t82:=somei(real,[x:real]prop3(x),m0(r),t81):some([x:real]prop3(x)) 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)) 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)] t84:=th4"l.some"(real,[x:real]and(neg(x),in(x,s2)),notcase2,r):not(and(neg(r),in(r,s2))) t85:=th3"l.and"(neg(r),in(r,s2),t84,satz169d(r,l)):not(in(r,s2)) t86:=ore1(in(r,s1),in(r,s2),p0,t85):in(r,s1) r@[m:more(r,0)] t87:=th4"l.some"(real,[x:real]and(pos(x),in(x,s1)),notcase1,r):not(and(pos(r),in(r,s1))) t88:=th3"l.and"(pos(r),in(r,s1),t87,satz169b(r,m)):not(in(r,s1)) t89:=ore2(in(r,s1),in(r,s2),p0,t88):in(r,s2) 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) t91:=somei(real,[x:real]prop3(x),0,t90):some([x:real]prop3(x)) 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)) 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)) t94:=onei(real,[x:real]prop3(x),t22,t93):one([x:real]prop3(x)) -5r205 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)))) 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)))) schnitt:=ind(real,[x:real]prop3".5r205"(x),satz205):real [r:real][l:less(r,schnitt)] satz205a:=ande1(prop1".5r205"(schnitt),prop2".5r205"(schnitt),oneax(real,[x:real]prop3".5r205"(x),satz205)):in(r,s1) r@[m:more(r,schnitt)] satz205b:=ande2(prop1".5r205"(schnitt),prop2".5r205"(schnitt),oneax(real,[x:real]prop3".5r205"(x),satz205)):in(r,s2) @[r0:cut][s0:cut] +iva dr:=pdofrp(r0):dif ds:=pdofrp(s0):dif t1:=lemmaivad1(r0,s0):eq"rp"(pd(dr,ds),pdofrp(pl"rp"(r0,s0))) -iva 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))) +*iva s0@t2:=lemmaivad2(r0,s0):eq"rp"(td(dr,ds),pdofrp(ts"rp"(r0,s0))) -iva 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))) [m:more(pofrp(r0),pofrp(s0))] +*iva m@t3:=moreex(pofrp(r0),pofrp(s0),dr,ds,innclass(dr),innclass(ds),m):mored(dr,ds) -iva m@lemmaiva3:=lemmaivad3(r0,s0,t3".iva"):more"rp"(r0,s0) s0@[m:more"rp"(r0,s0)] +*iva m@[l:less(pofrp(r0),pofrp(s0))] t4:=satz121(s0,r0,lemmaiva3(s0,r0,lemma2(pofrp(r0),pofrp(s0),l))):less"rp"(r0,s0) m@t5:=ec3e23(is"rp"(r0,s0),more"rp"(r0,s0),less"rp"(r0,s0),satz123b(r0,s0),m):not(less"rp"(r0,s0)) 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))) t7:=ec3e21(is"rp"(r0,s0),more"rp"(r0,s0),less"rp"(r0,s0),satz123b(r0,s0),m):nis"rp"(r0,s0) 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)) -iva 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)) @[x:nat][y:nat][m:more(rlofnt(x),rlofnt(y))] +*iva m@t9:=lemmaiva3(rpofnt(x),rpofnt(y),m):more"rp"(rpofnt(x),rpofnt(y)) t10:=satz154d(rtofn(x),rtofn(y),t9):more"rt"(rtofn(x),rtofn(y)) 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)) -iva m@lemmaiva5:=satz111a(x,y,t11".iva"):more"n"(x,y) y@[m:more"n"(x,y)] +*iva m@t12:=satz111d(x,y,m):moref(fr(x,1),fr(y,1)) 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)) t14:=satz154a(rtofn(x),rtofn(y),t13):more"rp"(rpofnt(x),rpofnt(y)) -iva m@lemmaiva6:=lemmaiva4(rpofnt(x),rpofnt(y),t14".iva"):more(rlofnt(x),rlofnt(y)) @[r:real] +int a0ir@[i:intrl(r)] t1:=intabsd(a0,intrlex(r,a0,a0ir,i)):intd(absd(a0)) t2:=intrlin(abs(r),absd(a0),aica,t1):intrl(abs(r)) -int [i:intrl(r)] intabs:=realapp1(r,intrl(abs(r)),[x:dif][t:inn(x,class(r))]t2".int"(r,x,t,i)):intrl(abs(r)) +*int i@t3:=intm0d(a0,intrlex(r,a0,a0ir,i)):intd(m0d(a0)) t4:=intrlin(m0(r),m0d(a0),micm0,t3):intrl(m0(r)) -int i@intm0:=realapp1(r,intrl(m0(r)),[x:dif][t:inn(x,class(r))]t4".int"(r,x,t,i)):intrl(m0(r)) +*int b1is@[i:intrl(r)][j:intrl(s)] t5:=intpd(a1,b1,intrlex(r,a1,a1ir,i),intrlex(s,b1,b1is,j)):intd(pd(a1,b1)) t6:=intrlin(pl(r,s),pd(a1,b1),picp,t5):intrl(pl(r,s)) -int i@[s:real][j:intrl(s)] 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)) intmn:=intpl(r,i,m0(s),intm0(s,j)):intrl(mn(r,s)) +*int j@t7:=inttd(a1,b1,intrlex(r,a1,a1ir,i),intrlex(s,b1,b1is,j)):intd(td(a1,b1)) t8:=intrlin(ts(r,s),td(a1,b1),tict,t7):intrl(ts(r,s)) -int 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)) r@[n:natrl(r)] +ivr24 t1:=satz24a(ntofrl(r,n)):lessis"n"(1,ntofrl(r,n)) 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)) -ivr24 satzr24:=satz167e(1rl,r,t2".ivr24"):lessis(1rl,r) r@[i:intrl(r)][s:real][j:intrl(s)][l:less(r,s)] +ivr25 t1:=satz182d(s,r,lemma2(r,s,l)):pos(mn(s,r)) t2:=intmn(s,j,r,i):intrl(mn(s,r)) t3:=posintnatrl(mn(s,r),t1,t2):natrl(mn(s,r)) t4:=satzr24(mn(s,r),t3):lessis(1rl,mn(s,r)) 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)) -ivr25 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) @[x:nat][y:nat] +ivr155 t1:=lemmaiva1(rpofnt(x),rpofnt(y)):is(pl(rlofnt(x),rlofnt(y)),pofrp(pl"rp"(rpofnt(x),rpofnt(y)))) 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)))) -ivr155 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))) 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))) +*ivr155 y@t3:=lemmaiva2(rpofnt(x),rpofnt(y)):is(ts(rlofnt(x),rlofnt(y)),pofrp(ts"rp"(rpofnt(x),rpofnt(y)))) 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)))) -ivr155 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))) 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))) @[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))] +7r161 [c0:dif][cit:inn(c0,class(t))][a0:dif][air:inn(a0,class(r))][b0:dif][bis:inn(b0,class(s))] 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)) 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)) 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) 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) t5:=isin(r,s,a0,b0,air,bis,satzd161b(c0,a0,b0,t1,t2,t3,t4)):is(r,s) -7r161 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) t@[n:not(neg(t))] +*7r161 n@[c0:dif][cit:inn(c0,class(t))] t6:=th3"l.imp"(negd(c0),neg(t),n,[u:negd(c0)]negin(t,c0,cit,u)):not(negd(c0)) [a0:dif][a:and(not(negd(a0)),eq"rp"(td(a0,a0),c0))] ar:=realof(a0):real 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)) 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) t9:=andi(not(neg(ar)),is(ts(ar,ar),t),t7,t8):and(not(neg(ar)),is(ts(ar,ar),t)) 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))) 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))) -7r161 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))) 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))) sqrt:=ind(real,[u:real]and(not(neg(u)),is(ts(u,u),t)),satzr161):real +*7r161 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)) -7r161 n@thsqrt1a:=ande1(not(neg(sqrt)),is(ts(sqrt,sqrt),t),t12".7r161"):not(neg(sqrt)) thsqrt1b:=ande2(not(neg(sqrt)),is(ts(sqrt,sqrt),t),t12".7r161"):is(ts(sqrt,sqrt),t) [x:real][o:not(neg(x))][i:is(ts(x,x),t)] thsqrt2:=satzr161b(x,sqrt,andi(not(neg(x)),is(ts(x,x),t),o,i),t12".7r161"):is(x,sqrt(t,n)) o@[i:is(t,ts(x,x))] thsqrt3:=symis(real,x,sqrt(t,n),thsqrt2(symis(real,t,ts(x,x),i))):is(sqrt(t,n),x) @[r:real][s:real][n:not(neg(r))][o:not(neg(s))][i:is(r,s)] 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)) r@[n:not(neg(r))][i:is(r,0)] 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) n@[o:nis(r,0)] +sqrt 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) -sqrt 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)) @[r:real][s:real][t:real][n:nis(t,0)] +v0 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)) -v0 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)) +*v0 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)) -v0 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)) r@[n:nis(r,0)] lemma8:=satz204b(r,r,n,ov(r,r,n),1rl,satz204c(r,r,n),satz195(r)):is(ov(r,r,n),1rl) 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) r@[i:is(r,m0(r))] +*v0 i@[p:pos(m0(r))] t3:=pnotn(m0(r),p):con i@[n:neg(m0(r))] t4:=nnotp(m0(r),n):con -v0 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) 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) 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))) @[x:real][ix:intrl(x)][y:real][iy:intrl(y)][ly:lessis(y,x)] +shift t1:=satz190a(x,x,1rl,0,moreisi2(x,x,refis(real,x)),satz169a(1rl,natpos(1rl,natrl1))):more(pl(x,1rl),pl(x,0)) t2:=ismore2(pl(x,0),x,pl(x,1rl),pl02(x,0,refis(real,0)),t1):more(pl(x,1rl),x) t3:=satz172d(pl(x,1rl),x,y,t2,satz168b(y,x,ly)):more(pl(x,1rl),y) t4:=satz182d(pl(x,1rl),y,t3):pos(mn(pl(x,1rl),y)) t5:=intmn(pl(x,1rl),intpl(x,ix,1rl,intrl1),y,iy):intrl(mn(pl(x,1rl),y)) t6:=posintnatrl(mn(pl(x,1rl),y),t4,t5):natrl(mn(pl(x,1rl),y)) -shift shiftl:=ntofrl(mn(pl(x,1rl),y),t6".shift"):nat [n:1to(shiftl)] +*shift n@n1:=inn"n"(shiftl,n):nat t7:=1top(shiftl,n):lessis"n"(n1,shiftl) n2:=rlofnt(n1):real t8:=natintrl(n2,natrli(n1)):intrl(n2) -shift n@shiftr:=mn(pl(n2".shift",y),1rl):real intshiftr:=intmn(pl(n2".shift",y),intpl(n2".shift",t8".shift",y,iy),1rl,intrl1):intrl(shiftr) [m:1to(shiftl)][i:is(shiftr(n),shiftr(m))] +*shift 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) 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)) 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)) t11a:=isntirl(n1(n),n1(m),t10a):is"n"(n1(n),n1(m)) -shift i@iseshiftr:=isinne(shiftl,n,m,t11a".shift"):is"e"(1to(shiftl),n,m) +*shift n@[m:more(shiftr,x)] t9:=satz188d(shiftr,x,1rl,m):more(pl(shiftr,1rl),pl(x,1rl)) t10:=ismore1(pl(shiftr,1rl),pl(n2,y),pl(x,1rl),plmn(pl(n2,y),1rl),t9):more(pl(n2,y),pl(x,1rl)) t11:=satz188d(pl(n2,y),pl(x,1rl),m0(y),t10):more(mn(pl(n2,y),y),mn(pl(x,1rl),y)) t12:=ismore1(mn(pl(n2,y),y),n2,mn(pl(x,1rl),y),mnpl(n2,y),t11):more(n2,mn(pl(x,1rl),y)) 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)) t14:=lemmaiva5(ntofrl(n2,natrli(n1)),shiftl,t13):more"n"(ntofrl(n2,natrli(n1)),shiftl) t15:=ismore1"n"(ntofrl(n2,natrli(n1)),n1,shiftl,isntrl2(n1),t14):more"n"(n1,shiftl) 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)) -shift n@shiftrls:=satz167e(shiftr,x,t16".shift"):lessis(shiftr,x) +*shift n@[m:more(y,shiftr)] t17:=satz188d(y,shiftr,1rl,m):more(pl(y,1rl),pl(shiftr,1rl)) 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)) t19:=satz188a(1rl,n2,y,t18):more(1rl,n2) t20:=lemmaiva5(1,n1,t19):more"n"(1,n1) 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)) -shift n@lsshiftr:=satz167e(y,shiftr,t21".shift"):lessis(y,shiftr) ly@[u:real][a:and3(intrl(u),lessis(y,u),lessis(u,x))] +*shift a@t22:=and3e1(intrl(u),lessis(y,u),lessis(u,x),a):intrl(u) t23:=and3e2(intrl(u),lessis(y,u),lessis(u,x),a):lessis(y,u) t24:=and3e3(intrl(u),lessis(y,u),lessis(u,x),a):lessis(u,x) [l:less(u,x)] 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)) a@[i:is(u,x)] 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)) 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)) ul:=shiftl(u,t22,y,iy,t23):nat 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)) 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)) t30:=satz10e(ul,shiftl,t29):lessis"n"(ul,shiftl) -shift a@shiftl1:=outn(shiftl,ul".shift",t30".shift"):1to(shiftl) +*shift a@t31:=isinoutn(shiftl,ul,t30):is"n"(ul,n1(shiftl1)) 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)) 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) -shift 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)) shiftinv2:=symis(real,u,shiftr(shiftl1),shiftinv1):is(shiftr(shiftl1),u) ly@[alpha:'type'] seq:=[t:real][u:intrl(t)][v:lessis(y,t)][w:lessis(t,x)]alpha:'type' [s:seq] 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,s,