From 0b0c46e907411bb03e06ecff169f722bf63761b9 Mon Sep 17 00:00:00 2001 From: Claudio Sacerdoti Coen Date: Tue, 23 Dec 2014 11:08:18 +0000 Subject: [PATCH] Additional contribs. --- helm/software/lambda-delta/.depend.opt | 181 + helm/software/lambda-delta/Make | 1 + helm/software/lambda-delta/Makefile | 71 + helm/software/lambda-delta/Makefile.common | 86 + helm/software/lambda-delta/README | 10 + helm/software/lambda-delta/automath/Make | 1 + helm/software/lambda-delta/automath/Omega.aut | 10 + helm/software/lambda-delta/automath/aut.ml | 25 + .../lambda-delta/automath/autLexer.mll | 75 + .../lambda-delta/automath/autOutput.ml | 100 + .../lambda-delta/automath/autOutput.mli | 20 + .../lambda-delta/automath/autParser.mly | 95 + .../lambda-delta/automath/autProcess.ml | 77 + .../lambda-delta/automath/autProcess.mli | 18 + .../lambda-delta/automath/grundlagen-orig.aut | 10713 ++++++++++++++++ .../lambda-delta/automath/grundlagen.aut | 10713 ++++++++++++++++ helm/software/lambda-delta/basic_ag/Make | 2 + helm/software/lambda-delta/basic_ag/bag.ml | 92 + .../lambda-delta/basic_ag/bagEnvironment.ml | 39 + .../lambda-delta/basic_ag/bagEnvironment.mli | 16 + .../lambda-delta/basic_ag/bagOutput.ml | 145 + .../lambda-delta/basic_ag/bagOutput.mli | 20 + .../lambda-delta/basic_ag/bagReduction.ml | 196 + .../lambda-delta/basic_ag/bagReduction.mli | 20 + .../lambda-delta/basic_ag/bagSubstitution.ml | 48 + .../lambda-delta/basic_ag/bagSubstitution.mli | 12 + .../software/lambda-delta/basic_ag/bagType.ml | 127 + .../lambda-delta/basic_ag/bagType.mli | 16 + .../lambda-delta/basic_ag/bagUntrusted.ml | 29 + .../lambda-delta/basic_ag/bagUntrusted.mli | 14 + helm/software/lambda-delta/basic_rg/Make | 2 + helm/software/lambda-delta/basic_rg/brg.ml | 82 + .../lambda-delta/basic_rg/brgEnvironment.ml | 35 + .../lambda-delta/basic_rg/brgEnvironment.mli | 14 + .../lambda-delta/basic_rg/brgOutput.ml | 258 + .../lambda-delta/basic_rg/brgOutput.mli | 25 + .../lambda-delta/basic_rg/brgReduction.ml | 210 + .../lambda-delta/basic_rg/brgReduction.mli | 26 + .../lambda-delta/basic_rg/brgSubstitution.ml | 46 + .../lambda-delta/basic_rg/brgSubstitution.mli | 15 + .../software/lambda-delta/basic_rg/brgType.ml | 131 + .../lambda-delta/basic_rg/brgType.mli | 16 + .../lambda-delta/basic_rg/brgUntrusted.ml | 36 + .../lambda-delta/basic_rg/brgUntrusted.mli | 14 + helm/software/lambda-delta/common/Make | 1 + helm/software/lambda-delta/common/entity.ml | 99 + .../software/lambda-delta/common/hierarchy.ml | 43 + .../lambda-delta/common/hierarchy.mli | 22 + helm/software/lambda-delta/common/library.ml | 125 + helm/software/lambda-delta/common/library.mli | 52 + helm/software/lambda-delta/common/output.ml | 75 + helm/software/lambda-delta/common/output.mli | 23 + helm/software/lambda-delta/complete_rg/Make | 1 + helm/software/lambda-delta/complete_rg/crg.ml | 78 + .../lambda-delta/complete_rg/crgAut.ml | 222 + .../lambda-delta/complete_rg/crgAut.mli | 17 + .../lambda-delta/complete_rg/crgBrg.ml | 64 + .../lambda-delta/complete_rg/crgBrg.mli | 12 + .../lambda-delta/complete_rg/crgOutput.ml | 148 + .../lambda-delta/complete_rg/crgOutput.mli | 14 + helm/software/lambda-delta/icons/basic-32.png | Bin 0 -> 587 bytes helm/software/lambda-delta/icons/crux-16.ico | Bin 0 -> 318 bytes helm/software/lambda-delta/icons/crux-32.png | Bin 0 -> 224 bytes .../software/lambda-delta/icons/helena-32.png | Bin 0 -> 549 bytes helm/software/lambda-delta/lib/Make | 1 + helm/software/lambda-delta/lib/cps.ml | 83 + helm/software/lambda-delta/lib/log.ml | 92 + helm/software/lambda-delta/lib/log.mli | 61 + helm/software/lambda-delta/lib/nUri.ml | 1 + helm/software/lambda-delta/lib/nUri.mli | 1 + helm/software/lambda-delta/lib/share.ml | 21 + helm/software/lambda-delta/lib/time.ml | 34 + helm/software/lambda-delta/toplevel/Make | 1 + helm/software/lambda-delta/toplevel/meta.ml | 25 + .../software/lambda-delta/toplevel/metaAut.ml | 216 + .../lambda-delta/toplevel/metaAut.mli | 18 + .../software/lambda-delta/toplevel/metaBag.ml | 67 + .../lambda-delta/toplevel/metaBag.mli | 12 + .../software/lambda-delta/toplevel/metaBrg.ml | 66 + .../lambda-delta/toplevel/metaBrg.mli | 12 + .../lambda-delta/toplevel/metaLibrary.ml | 36 + .../lambda-delta/toplevel/metaLibrary.mli | 18 + .../lambda-delta/toplevel/metaOutput.ml | 162 + .../lambda-delta/toplevel/metaOutput.mli | 20 + helm/software/lambda-delta/toplevel/top.ml | 278 + .../lambda-delta/xml/ld-html-entity.xsl | 34 + .../lambda-delta/xml/ld-html-library.xsl | 151 + .../lambda-delta/xml/ld-html-root.xsl | 67 + .../lambda-delta/xml/ld-html-term.xsl | 70 + helm/software/lambda-delta/xml/ld-html.xsl | 31 + helm/software/lambda-delta/xml/ld.dtd | 88 + .../formal_topology/overlap/apply_functor.ma | 122 + .../formal_topology/overlap/basic_pairs.ma | 241 + .../basic_pairs_to_basic_topologies.ma | 43 + .../overlap/basic_pairs_to_o-basic_pairs.ma | 148 + .../overlap/basic_topologies.ma | 193 + .../basic_topologies_to_o-basic_topologies.ma | 38 + .../formal_topology/overlap/categories.ma | 466 + .../overlap/concrete_spaces.ma | 109 + .../concrete_spaces_to_o-concrete_spaces.ma | 49 + .../overlap/cprop_connectives.ma | 192 + .../overlap/formal_topologies.ma | 96 + .../formal_topology/overlap/notation.ma | 20 + .../formal_topology/overlap/o-algebra.ma | 444 + .../formal_topology/overlap/o-basic_pairs.ma | 256 + .../o-basic_pairs_to_o-basic_topologies.ma | 119 + .../overlap/o-basic_topologies.ma | 188 + .../overlap/o-concrete_spaces.ma | 134 + .../overlap/o-formal_topologies.ma | 98 + .../formal_topology/overlap/o-saturations.ma | 43 + .../overlap/r-o-basic_pairs.ma | 253 + .../formal_topology/overlap/relations.ma | 293 + .../overlap/relations_to_o-algebra.ma | 264 + .../formal_topology/overlap/saturations.ma | 40 + .../overlap/saturations_to_o-saturations.ma | 36 + .../formal_topology/overlap/subsets.ma | 167 + .../ng_assembly/freescale/load_write.ma | 946 ++ .../ng_assembly/freescale/memory_abs.ma | 231 + .../ng_assembly/freescale/memory_bits.ma | 223 + .../ng_assembly/freescale/memory_func.ma | 87 + .../ng_assembly/freescale/memory_struct.ma | 624 + .../ng_assembly/freescale/memory_trees.ma | 227 + .../contribs/ng_assembly/freescale/model.ma | 357 + .../contribs/ng_assembly/freescale/multivm.ma | 1307 ++ .../ng_assembly/freescale/multivm_lemmas.ma | 57 + .../contribs/ng_assembly/freescale/opcode.ma | 180 + .../ng_assembly/freescale/opcode_base.ma | 330 + .../freescale/opcode_base_lemmas.ma | 101 + .../freescale/opcode_base_lemmas1.ma | 208 + .../freescale/opcode_base_lemmas_instrmode.ma | 425 + .../freescale/opcode_base_lemmas_opcode.ma | 200 + .../contribs/ng_assembly/freescale/status.ma | 1275 ++ .../ng_assembly/freescale/status_lemmas.ma | 1536 +++ .../ng_assembly/freescale/table_HC05.ma | 377 + .../ng_assembly/freescale/table_HC05_tests.ma | 94 + .../ng_assembly/freescale/table_HC08.ma | 474 + .../ng_assembly/freescale/table_HC08_tests.ma | 123 + .../ng_assembly/freescale/table_HCS08.ma | 486 + .../freescale/table_HCS08_tests.ma | 118 + .../ng_assembly/freescale/table_RS08.ma | 395 + .../ng_assembly/freescale/table_RS08_tests.ma | 78 + .../ng_assembly/freescale/translation.ma | 247 + .../freescale/translation_lemmas.ma | 88 + .../freescale_tests/medium_tests.ma | 887 ++ .../freescale_tests/medium_tests_tools.ma | 592 + .../freescale_tests/micro_tests1.ma | 143 + .../freescale_tests/micro_tests10.ma | 103 + .../freescale_tests/micro_tests2.ma | 87 + .../freescale_tests/micro_tests3.ma | 76 + .../freescale_tests/micro_tests4.ma | 152 + .../freescale_tests/micro_tests4bis.ma | 89 + .../freescale_tests/micro_tests5.ma | 93 + .../freescale_tests/micro_tests6.ma | 100 + .../freescale_tests/micro_tests7.ma | 78 + .../freescale_tests/micro_tests8.ma | 87 + .../freescale_tests/micro_tests9.ma | 104 + .../freescale_tests/micro_tests_tools.ma | 80 + 157 files changed, 44061 insertions(+) create mode 100644 helm/software/lambda-delta/.depend.opt create mode 100644 helm/software/lambda-delta/Make create mode 100644 helm/software/lambda-delta/Makefile create mode 100644 helm/software/lambda-delta/Makefile.common create mode 100644 helm/software/lambda-delta/README create mode 100644 helm/software/lambda-delta/automath/Make create mode 100644 helm/software/lambda-delta/automath/Omega.aut create mode 100644 helm/software/lambda-delta/automath/aut.ml create mode 100644 helm/software/lambda-delta/automath/autLexer.mll create mode 100644 helm/software/lambda-delta/automath/autOutput.ml create mode 100644 helm/software/lambda-delta/automath/autOutput.mli create mode 100644 helm/software/lambda-delta/automath/autParser.mly create mode 100644 helm/software/lambda-delta/automath/autProcess.ml create mode 100644 helm/software/lambda-delta/automath/autProcess.mli create mode 100644 helm/software/lambda-delta/automath/grundlagen-orig.aut create mode 100644 helm/software/lambda-delta/automath/grundlagen.aut create mode 100644 helm/software/lambda-delta/basic_ag/Make create mode 100644 helm/software/lambda-delta/basic_ag/bag.ml create mode 100644 helm/software/lambda-delta/basic_ag/bagEnvironment.ml create mode 100644 helm/software/lambda-delta/basic_ag/bagEnvironment.mli create mode 100644 helm/software/lambda-delta/basic_ag/bagOutput.ml create mode 100644 helm/software/lambda-delta/basic_ag/bagOutput.mli create mode 100644 helm/software/lambda-delta/basic_ag/bagReduction.ml create mode 100644 helm/software/lambda-delta/basic_ag/bagReduction.mli create mode 100644 helm/software/lambda-delta/basic_ag/bagSubstitution.ml create mode 100644 helm/software/lambda-delta/basic_ag/bagSubstitution.mli create mode 100644 helm/software/lambda-delta/basic_ag/bagType.ml create mode 100644 helm/software/lambda-delta/basic_ag/bagType.mli create mode 100644 helm/software/lambda-delta/basic_ag/bagUntrusted.ml create mode 100644 helm/software/lambda-delta/basic_ag/bagUntrusted.mli create mode 100644 helm/software/lambda-delta/basic_rg/Make create mode 100644 helm/software/lambda-delta/basic_rg/brg.ml create mode 100644 helm/software/lambda-delta/basic_rg/brgEnvironment.ml create mode 100644 helm/software/lambda-delta/basic_rg/brgEnvironment.mli create mode 100644 helm/software/lambda-delta/basic_rg/brgOutput.ml create mode 100644 helm/software/lambda-delta/basic_rg/brgOutput.mli create mode 100644 helm/software/lambda-delta/basic_rg/brgReduction.ml create mode 100644 helm/software/lambda-delta/basic_rg/brgReduction.mli create mode 100644 helm/software/lambda-delta/basic_rg/brgSubstitution.ml create mode 100644 helm/software/lambda-delta/basic_rg/brgSubstitution.mli create mode 100644 helm/software/lambda-delta/basic_rg/brgType.ml create mode 100644 helm/software/lambda-delta/basic_rg/brgType.mli create mode 100644 helm/software/lambda-delta/basic_rg/brgUntrusted.ml create mode 100644 helm/software/lambda-delta/basic_rg/brgUntrusted.mli create mode 100644 helm/software/lambda-delta/common/Make create mode 100644 helm/software/lambda-delta/common/entity.ml create mode 100644 helm/software/lambda-delta/common/hierarchy.ml create mode 100644 helm/software/lambda-delta/common/hierarchy.mli create mode 100644 helm/software/lambda-delta/common/library.ml create mode 100644 helm/software/lambda-delta/common/library.mli create mode 100644 helm/software/lambda-delta/common/output.ml create mode 100644 helm/software/lambda-delta/common/output.mli create mode 100644 helm/software/lambda-delta/complete_rg/Make create mode 100644 helm/software/lambda-delta/complete_rg/crg.ml create mode 100644 helm/software/lambda-delta/complete_rg/crgAut.ml create mode 100644 helm/software/lambda-delta/complete_rg/crgAut.mli create mode 100644 helm/software/lambda-delta/complete_rg/crgBrg.ml create mode 100644 helm/software/lambda-delta/complete_rg/crgBrg.mli create mode 100644 helm/software/lambda-delta/complete_rg/crgOutput.ml create mode 100644 helm/software/lambda-delta/complete_rg/crgOutput.mli create mode 100644 helm/software/lambda-delta/icons/basic-32.png create mode 100644 helm/software/lambda-delta/icons/crux-16.ico create mode 100644 helm/software/lambda-delta/icons/crux-32.png create mode 100644 helm/software/lambda-delta/icons/helena-32.png create mode 100644 helm/software/lambda-delta/lib/Make create mode 100644 helm/software/lambda-delta/lib/cps.ml create mode 100644 helm/software/lambda-delta/lib/log.ml create mode 100644 helm/software/lambda-delta/lib/log.mli create mode 120000 helm/software/lambda-delta/lib/nUri.ml create mode 120000 helm/software/lambda-delta/lib/nUri.mli create mode 100644 helm/software/lambda-delta/lib/share.ml create mode 100644 helm/software/lambda-delta/lib/time.ml create mode 100644 helm/software/lambda-delta/toplevel/Make create mode 100644 helm/software/lambda-delta/toplevel/meta.ml create mode 100644 helm/software/lambda-delta/toplevel/metaAut.ml create mode 100644 helm/software/lambda-delta/toplevel/metaAut.mli create mode 100644 helm/software/lambda-delta/toplevel/metaBag.ml create mode 100644 helm/software/lambda-delta/toplevel/metaBag.mli create mode 100644 helm/software/lambda-delta/toplevel/metaBrg.ml create mode 100644 helm/software/lambda-delta/toplevel/metaBrg.mli create mode 100644 helm/software/lambda-delta/toplevel/metaLibrary.ml create mode 100644 helm/software/lambda-delta/toplevel/metaLibrary.mli create mode 100644 helm/software/lambda-delta/toplevel/metaOutput.ml create mode 100644 helm/software/lambda-delta/toplevel/metaOutput.mli create mode 100644 helm/software/lambda-delta/toplevel/top.ml create mode 100644 helm/software/lambda-delta/xml/ld-html-entity.xsl create mode 100644 helm/software/lambda-delta/xml/ld-html-library.xsl create mode 100644 helm/software/lambda-delta/xml/ld-html-root.xsl create mode 100644 helm/software/lambda-delta/xml/ld-html-term.xsl create mode 100644 helm/software/lambda-delta/xml/ld-html.xsl create mode 100644 helm/software/lambda-delta/xml/ld.dtd create mode 100644 helm/software/matita/contribs/formal_topology/overlap/apply_functor.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_basic_topologies.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_o-basic_pairs.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/basic_topologies.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/basic_topologies_to_o-basic_topologies.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/categories.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/concrete_spaces.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/concrete_spaces_to_o-concrete_spaces.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/cprop_connectives.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/formal_topologies.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/notation.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/o-basic_topologies.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/o-formal_topologies.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/r-o-basic_pairs.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/relations.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/saturations.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/saturations_to_o-saturations.ma create mode 100644 helm/software/matita/contribs/formal_topology/overlap/subsets.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/load_write.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/memory_abs.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/memory_bits.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/memory_func.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/memory_struct.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/memory_trees.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/model.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/multivm.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/multivm_lemmas.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/opcode.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/opcode_base.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/opcode_base_lemmas.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/opcode_base_lemmas1.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/opcode_base_lemmas_instrmode.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/opcode_base_lemmas_opcode.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/status.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/status_lemmas.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/table_HC05.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/table_HC05_tests.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/table_HC08.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/table_HC08_tests.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/table_HCS08.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/table_HCS08_tests.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/table_RS08.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/table_RS08_tests.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/translation.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale/translation_lemmas.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/medium_tests.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/medium_tests_tools.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests1.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests10.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests2.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests3.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests4.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests4bis.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests5.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests6.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests7.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests8.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests9.ma create mode 100755 helm/software/matita/contribs/ng_assembly/freescale_tests/micro_tests_tools.ma diff --git a/helm/software/lambda-delta/.depend.opt b/helm/software/lambda-delta/.depend.opt new file mode 100644 index 000000000..f0d13c9a2 --- /dev/null +++ b/helm/software/lambda-delta/.depend.opt @@ -0,0 +1,181 @@ +lib/nUri.cmo: lib/nUri.cmi +lib/nUri.cmx: lib/nUri.cmi +lib/log.cmo: lib/cps.cmx lib/log.cmi +lib/log.cmx: lib/cps.cmx lib/log.cmi +lib/time.cmo: lib/log.cmi +lib/time.cmx: lib/log.cmx +automath/autProcess.cmi: automath/aut.cmx +automath/autProcess.cmo: automath/aut.cmx automath/autProcess.cmi +automath/autProcess.cmx: automath/aut.cmx automath/autProcess.cmi +automath/autOutput.cmi: automath/autProcess.cmi automath/aut.cmx +automath/autOutput.cmo: lib/log.cmi lib/cps.cmx automath/autProcess.cmi \ + automath/aut.cmx automath/autOutput.cmi +automath/autOutput.cmx: lib/log.cmx lib/cps.cmx automath/autProcess.cmx \ + automath/aut.cmx automath/autOutput.cmi +automath/autParser.cmi: automath/aut.cmx +automath/autParser.cmo: automath/aut.cmx automath/autParser.cmi +automath/autParser.cmx: automath/aut.cmx automath/autParser.cmi +automath/autLexer.cmo: lib/log.cmi automath/autParser.cmi +automath/autLexer.cmx: lib/log.cmx automath/autParser.cmx +common/hierarchy.cmo: lib/cps.cmx common/hierarchy.cmi +common/hierarchy.cmx: lib/cps.cmx common/hierarchy.cmi +common/output.cmo: lib/log.cmi common/output.cmi +common/output.cmx: lib/log.cmx common/output.cmi +common/entity.cmo: lib/nUri.cmi automath/aut.cmx +common/entity.cmx: lib/nUri.cmx automath/aut.cmx +common/library.cmi: common/hierarchy.cmi common/entity.cmx +common/library.cmo: lib/nUri.cmi common/hierarchy.cmi common/entity.cmx \ + lib/cps.cmx common/library.cmi +common/library.cmx: lib/nUri.cmx common/hierarchy.cmx common/entity.cmx \ + lib/cps.cmx common/library.cmi +basic_ag/bag.cmo: lib/log.cmi common/entity.cmx lib/cps.cmx +basic_ag/bag.cmx: lib/log.cmx common/entity.cmx lib/cps.cmx +basic_ag/bagOutput.cmi: lib/log.cmi basic_ag/bag.cmx +basic_ag/bagOutput.cmo: common/output.cmi lib/nUri.cmi lib/log.cmi \ + common/hierarchy.cmi common/entity.cmx basic_ag/bag.cmx \ + basic_ag/bagOutput.cmi +basic_ag/bagOutput.cmx: common/output.cmx lib/nUri.cmx lib/log.cmx \ + common/hierarchy.cmx common/entity.cmx basic_ag/bag.cmx \ + basic_ag/bagOutput.cmi +basic_ag/bagEnvironment.cmi: basic_ag/bag.cmx +basic_ag/bagEnvironment.cmo: lib/nUri.cmi lib/log.cmi common/entity.cmx \ + basic_ag/bag.cmx basic_ag/bagEnvironment.cmi +basic_ag/bagEnvironment.cmx: lib/nUri.cmx lib/log.cmx common/entity.cmx \ + basic_ag/bag.cmx basic_ag/bagEnvironment.cmi +basic_ag/bagSubstitution.cmi: basic_ag/bag.cmx +basic_ag/bagSubstitution.cmo: lib/share.cmx basic_ag/bag.cmx \ + basic_ag/bagSubstitution.cmi +basic_ag/bagSubstitution.cmx: lib/share.cmx basic_ag/bag.cmx \ + basic_ag/bagSubstitution.cmi +basic_ag/bagReduction.cmi: basic_ag/bag.cmx +basic_ag/bagReduction.cmo: lib/nUri.cmi lib/log.cmi common/entity.cmx \ + lib/cps.cmx basic_ag/bagSubstitution.cmi basic_ag/bagOutput.cmi \ + basic_ag/bagEnvironment.cmi basic_ag/bag.cmx basic_ag/bagReduction.cmi +basic_ag/bagReduction.cmx: lib/nUri.cmx lib/log.cmx common/entity.cmx \ + lib/cps.cmx basic_ag/bagSubstitution.cmx basic_ag/bagOutput.cmx \ + basic_ag/bagEnvironment.cmx basic_ag/bag.cmx basic_ag/bagReduction.cmi +basic_ag/bagType.cmi: common/hierarchy.cmi basic_ag/bag.cmx +basic_ag/bagType.cmo: lib/share.cmx lib/nUri.cmi lib/log.cmi \ + common/hierarchy.cmi common/entity.cmx lib/cps.cmx \ + basic_ag/bagReduction.cmi basic_ag/bagOutput.cmi \ + basic_ag/bagEnvironment.cmi basic_ag/bag.cmx basic_ag/bagType.cmi +basic_ag/bagType.cmx: lib/share.cmx lib/nUri.cmx lib/log.cmx \ + common/hierarchy.cmx common/entity.cmx lib/cps.cmx \ + basic_ag/bagReduction.cmx basic_ag/bagOutput.cmx \ + basic_ag/bagEnvironment.cmx basic_ag/bag.cmx basic_ag/bagType.cmi +basic_ag/bagUntrusted.cmi: common/hierarchy.cmi basic_ag/bag.cmx +basic_ag/bagUntrusted.cmo: lib/nUri.cmi lib/log.cmi common/entity.cmx \ + basic_ag/bagType.cmi basic_ag/bagEnvironment.cmi basic_ag/bag.cmx \ + basic_ag/bagUntrusted.cmi +basic_ag/bagUntrusted.cmx: lib/nUri.cmx lib/log.cmx common/entity.cmx \ + basic_ag/bagType.cmx basic_ag/bagEnvironment.cmx basic_ag/bag.cmx \ + basic_ag/bagUntrusted.cmi +basic_rg/brg.cmo: common/entity.cmx +basic_rg/brg.cmx: common/entity.cmx +basic_rg/brgOutput.cmi: lib/log.cmi common/library.cmi basic_rg/brg.cmx +basic_rg/brgOutput.cmo: common/output.cmi lib/nUri.cmi lib/log.cmi \ + common/library.cmi common/hierarchy.cmi common/entity.cmx lib/cps.cmx \ + basic_rg/brg.cmx basic_rg/brgOutput.cmi +basic_rg/brgOutput.cmx: common/output.cmx lib/nUri.cmx lib/log.cmx \ + common/library.cmx common/hierarchy.cmx common/entity.cmx lib/cps.cmx \ + basic_rg/brg.cmx basic_rg/brgOutput.cmi +basic_rg/brgEnvironment.cmi: basic_rg/brg.cmx +basic_rg/brgEnvironment.cmo: lib/nUri.cmi common/entity.cmx basic_rg/brg.cmx \ + basic_rg/brgEnvironment.cmi +basic_rg/brgEnvironment.cmx: lib/nUri.cmx common/entity.cmx basic_rg/brg.cmx \ + basic_rg/brgEnvironment.cmi +basic_rg/brgSubstitution.cmi: basic_rg/brg.cmx +basic_rg/brgSubstitution.cmo: basic_rg/brg.cmx basic_rg/brgSubstitution.cmi +basic_rg/brgSubstitution.cmx: basic_rg/brg.cmx basic_rg/brgSubstitution.cmi +basic_rg/brgReduction.cmi: lib/log.cmi common/entity.cmx basic_rg/brg.cmx +basic_rg/brgReduction.cmo: lib/share.cmx common/output.cmi lib/nUri.cmi \ + lib/log.cmi common/entity.cmx lib/cps.cmx basic_rg/brgOutput.cmi \ + basic_rg/brgEnvironment.cmi basic_rg/brg.cmx basic_rg/brgReduction.cmi +basic_rg/brgReduction.cmx: lib/share.cmx common/output.cmx lib/nUri.cmx \ + lib/log.cmx common/entity.cmx lib/cps.cmx basic_rg/brgOutput.cmx \ + basic_rg/brgEnvironment.cmx basic_rg/brg.cmx basic_rg/brgReduction.cmi +basic_rg/brgType.cmi: lib/log.cmi common/hierarchy.cmi common/entity.cmx \ + basic_rg/brgReduction.cmi basic_rg/brg.cmx +basic_rg/brgType.cmo: lib/share.cmx lib/nUri.cmi lib/log.cmi \ + common/hierarchy.cmi common/entity.cmx lib/cps.cmx \ + basic_rg/brgSubstitution.cmi basic_rg/brgReduction.cmi \ + basic_rg/brgOutput.cmi basic_rg/brgEnvironment.cmi basic_rg/brg.cmx \ + basic_rg/brgType.cmi +basic_rg/brgType.cmx: lib/share.cmx lib/nUri.cmx lib/log.cmx \ + common/hierarchy.cmx common/entity.cmx lib/cps.cmx \ + basic_rg/brgSubstitution.cmx basic_rg/brgReduction.cmx \ + basic_rg/brgOutput.cmx basic_rg/brgEnvironment.cmx basic_rg/brg.cmx \ + basic_rg/brgType.cmi +basic_rg/brgUntrusted.cmi: common/hierarchy.cmi basic_rg/brgType.cmi \ + basic_rg/brg.cmx +basic_rg/brgUntrusted.cmo: lib/nUri.cmi lib/log.cmi common/entity.cmx \ + basic_rg/brgType.cmi basic_rg/brgReduction.cmi \ + basic_rg/brgEnvironment.cmi basic_rg/brg.cmx basic_rg/brgUntrusted.cmi +basic_rg/brgUntrusted.cmx: lib/nUri.cmx lib/log.cmx common/entity.cmx \ + basic_rg/brgType.cmx basic_rg/brgReduction.cmx \ + basic_rg/brgEnvironment.cmx basic_rg/brg.cmx basic_rg/brgUntrusted.cmi +complete_rg/crg.cmo: common/entity.cmx +complete_rg/crg.cmx: common/entity.cmx +complete_rg/crgOutput.cmi: common/library.cmi complete_rg/crg.cmx +complete_rg/crgOutput.cmo: lib/nUri.cmi common/library.cmi \ + common/hierarchy.cmi common/entity.cmx complete_rg/crg.cmx lib/cps.cmx \ + complete_rg/crgOutput.cmi +complete_rg/crgOutput.cmx: lib/nUri.cmx common/library.cmx \ + common/hierarchy.cmx common/entity.cmx complete_rg/crg.cmx lib/cps.cmx \ + complete_rg/crgOutput.cmi +complete_rg/crgAut.cmi: common/entity.cmx complete_rg/crg.cmx \ + automath/aut.cmx +complete_rg/crgAut.cmo: lib/nUri.cmi common/entity.cmx complete_rg/crg.cmx \ + lib/cps.cmx automath/aut.cmx complete_rg/crgAut.cmi +complete_rg/crgAut.cmx: lib/nUri.cmx common/entity.cmx complete_rg/crg.cmx \ + lib/cps.cmx automath/aut.cmx complete_rg/crgAut.cmi +complete_rg/crgBrg.cmi: complete_rg/crg.cmx basic_rg/brg.cmx +complete_rg/crgBrg.cmo: common/entity.cmx complete_rg/crg.cmx lib/cps.cmx \ + basic_rg/brg.cmx complete_rg/crgBrg.cmi +complete_rg/crgBrg.cmx: common/entity.cmx complete_rg/crg.cmx lib/cps.cmx \ + basic_rg/brg.cmx complete_rg/crgBrg.cmi +toplevel/meta.cmo: common/entity.cmx +toplevel/meta.cmx: common/entity.cmx +toplevel/metaOutput.cmi: toplevel/meta.cmx +toplevel/metaOutput.cmo: lib/nUri.cmi toplevel/meta.cmx lib/log.cmi \ + common/entity.cmx lib/cps.cmx toplevel/metaOutput.cmi +toplevel/metaOutput.cmx: lib/nUri.cmx toplevel/meta.cmx lib/log.cmx \ + common/entity.cmx lib/cps.cmx toplevel/metaOutput.cmi +toplevel/metaLibrary.cmi: toplevel/meta.cmx +toplevel/metaLibrary.cmo: toplevel/metaOutput.cmi toplevel/metaLibrary.cmi +toplevel/metaLibrary.cmx: toplevel/metaOutput.cmx toplevel/metaLibrary.cmi +toplevel/metaAut.cmi: toplevel/meta.cmx automath/aut.cmx +toplevel/metaAut.cmo: lib/nUri.cmi toplevel/meta.cmx common/entity.cmx \ + lib/cps.cmx automath/aut.cmx toplevel/metaAut.cmi +toplevel/metaAut.cmx: lib/nUri.cmx toplevel/meta.cmx common/entity.cmx \ + lib/cps.cmx automath/aut.cmx toplevel/metaAut.cmi +toplevel/metaBag.cmi: toplevel/meta.cmx basic_ag/bag.cmx +toplevel/metaBag.cmo: toplevel/meta.cmx lib/cps.cmx basic_ag/bag.cmx \ + toplevel/metaBag.cmi +toplevel/metaBag.cmx: toplevel/meta.cmx lib/cps.cmx basic_ag/bag.cmx \ + toplevel/metaBag.cmi +toplevel/metaBrg.cmi: toplevel/meta.cmx basic_rg/brg.cmx +toplevel/metaBrg.cmo: toplevel/meta.cmx common/entity.cmx lib/cps.cmx \ + basic_rg/brg.cmx toplevel/metaBrg.cmi +toplevel/metaBrg.cmx: toplevel/meta.cmx common/entity.cmx lib/cps.cmx \ + basic_rg/brg.cmx toplevel/metaBrg.cmi +toplevel/top.cmo: lib/time.cmx common/output.cmi lib/nUri.cmi \ + toplevel/metaOutput.cmi toplevel/metaLibrary.cmi toplevel/metaBrg.cmi \ + toplevel/metaBag.cmi toplevel/metaAut.cmi toplevel/meta.cmx lib/log.cmi \ + common/library.cmi common/hierarchy.cmi common/entity.cmx \ + complete_rg/crgOutput.cmi complete_rg/crgBrg.cmi complete_rg/crgAut.cmi \ + complete_rg/crg.cmx lib/cps.cmx basic_rg/brgUntrusted.cmi \ + basic_rg/brgReduction.cmi basic_rg/brgOutput.cmi basic_rg/brg.cmx \ + basic_ag/bagUntrusted.cmi basic_ag/bagType.cmi basic_ag/bagOutput.cmi \ + basic_ag/bag.cmx automath/autProcess.cmi automath/autParser.cmi \ + automath/autOutput.cmi automath/autLexer.cmx +toplevel/top.cmx: lib/time.cmx common/output.cmx lib/nUri.cmx \ + toplevel/metaOutput.cmx toplevel/metaLibrary.cmx toplevel/metaBrg.cmx \ + toplevel/metaBag.cmx toplevel/metaAut.cmx toplevel/meta.cmx lib/log.cmx \ + common/library.cmx common/hierarchy.cmx common/entity.cmx \ + complete_rg/crgOutput.cmx complete_rg/crgBrg.cmx complete_rg/crgAut.cmx \ + complete_rg/crg.cmx lib/cps.cmx basic_rg/brgUntrusted.cmx \ + basic_rg/brgReduction.cmx basic_rg/brgOutput.cmx basic_rg/brg.cmx \ + basic_ag/bagUntrusted.cmx basic_ag/bagType.cmx basic_ag/bagOutput.cmx \ + basic_ag/bag.cmx automath/autProcess.cmx automath/autParser.cmx \ + automath/autOutput.cmx automath/autLexer.cmx diff --git a/helm/software/lambda-delta/Make b/helm/software/lambda-delta/Make new file mode 100644 index 000000000..5730e32e6 --- /dev/null +++ b/helm/software/lambda-delta/Make @@ -0,0 +1 @@ +lib automath common basic_ag basic_rg complete_rg toplevel diff --git a/helm/software/lambda-delta/Makefile b/helm/software/lambda-delta/Makefile new file mode 100644 index 000000000..9903f4fc4 --- /dev/null +++ b/helm/software/lambda-delta/Makefile @@ -0,0 +1,71 @@ +MAIN = helena + +REQUIRES = unix str + +KEEP = README automath/*.aut + +CLEAN = etc/log.txt + +TAGS = test test-si test-si-fast hal xml-si-drg xml-si-old profile + +XMLS = xml/brg/grundlagen/l/not.ld.xml xml/brg/grundlagen/l/et.ld.xml \ + xml/brg/grundlagen/l/e/st/eq/landau/n/rt/rp/r/c/8283/t234.ld.xml \ + xml/brg/grundlagen/l/e/pairis1.ld.xml + +include Makefile.common + +INPUT = automath/grundlagen.aut + +INPUT-ORIG = automath/grundlagen-orig.aut + +test: $(MAIN).opt + @echo " HELENA -o -p -r $(INPUT)" + $(H)./$(MAIN).opt -o -p -r -S 3 $(O) $(INPUT) > etc/log.txt + +test-si: $(MAIN).opt + @echo " HELENA -o -p -r -u $(INPUT)" + $(H)./$(MAIN).opt -o -p -r -u -S 3 $(O) $(INPUT) > etc/log.txt + +test-si-fast: $(MAIN).opt + @echo " HELENA -o -r -u $(INPUT)" + $(H)./$(MAIN).opt -o -r -u -S 1 $(O) $(INPUT) > etc/log.txt + +profile: $(MAIN).opt + @echo " HELENA -r -u $(INPUT) (30 TIMES)" + $(H)for T in `seq 30`; do ./$(MAIN).opt -r -u -S 1 $(O) automath/grundlagen.aut >> etc/log.txt; done + $(H)grep "at exit" etc/log.txt | sort | uniq | less + +hal: $(MAIN).opt + @echo " HELENA -o -x -m $(INPUT)" + $(H)./$(MAIN).opt -o -x -m -s 1 -S 1 $(INPUT) > etc/log.txt + +xml-si-old: $(MAIN).opt + @echo " HELENA -o -u -x -s 2 $(INPUT)" + $(H)./$(MAIN).opt -o -u -x -s 2 -S 1 $(INPUT) > etc/log.txt + +xml-si-drg: $(MAIN).opt + @echo " HELENA -u -x -s 1 $(INPUT)" + $(H)./$(MAIN).opt -u -x -s 1 -S 1 $(INPUT) > etc/log.txt + +%.ld: BASEURL = --stringparam baseurl $(LDDLURL) + +%.ld: + @echo " XSLT $@" + $(H)mkdir -p $(LDDLDIR)/$(@D) + $(H)$(XSLT) -o $(LDDLDIR)/$@.html $(BASEURL) xml/ld-html.xsl xml/$@.xml + +etc/make-html.sh xml/index.txt index: + @echo " GENERATE INDEXES" + $(H)find xml -name "*.ld.xml" | sed s/.xml//g | sed s/xml/ld:/g > xml/index.txt + $(H)sed "s/^/make --no-print-directory /" xml/index.txt | sed s.ld:/.. > etc/make-html.sh + +html: etc/make-html.sh + @echo " MAKE */*.ld" + $(H). $< + +test-html: + @$(MAKE) --no-print-directory $(XMLS:xml/%.xml=%) + +lddl: index + @echo " GENERATE lddl.tar.bz2" + $(H)tar -cjf $(DOWNDIR)/lddl.tar.bz2 -X etc/exclude.txt xml diff --git a/helm/software/lambda-delta/Makefile.common b/helm/software/lambda-delta/Makefile.common new file mode 100644 index 000000000..ca219ca66 --- /dev/null +++ b/helm/software/lambda-delta/Makefile.common @@ -0,0 +1,86 @@ +H=@ + +LDDLURL = http://helm.cs.unibo.it/lambda-delta/static/lddl +LDDLDIR = /projects/helm/public_html/lambda-delta/static/lddl +DOWNDIR = /projects/helm/public_html/lambda-delta/download + +DIRECTORIES = $(shell cat Make) + +INCLUDES = $(DIRECTORIES:%=-I %) + +OCAMLDEP = ocamlfind ocamldep -native $(INCLUDES) +OCAMLOPT = ocamlfind opt -linkpkg -package "$(REQUIRES)" $(INCLUDES) +OCAMLLEX = ocamllex.opt +OCAMLYACC = ocamlyacc -v +XMLLINT = xmllint --noout +XSLT = xsltproc +TAR = tar -czf etc/$(MAIN:%=%.tgz) + +define DIR_TEMPLATE + MODULES += $$(addprefix $(1)/,$$(shell cat $(1)/Make)) +endef + +define MOD_TEMPLATE + SOURCES += $$(if $$(wildcard $(1).ml[yi]),$(1).mli $(1).ml,$(1).ml) + CMXS += $(1).cmx + CLEAN += $(1).cmi $(1).cmx $(1).o + CLEAN += $$(if $$(wildcard $(1).ml[ly]),$(1).ml,) + CLEAN += $$(if $$(wildcard $(1).mly),$(1).mli $(1).output,) + KEEP += $$(if $$(wildcard $(1).mly),$(1).mly,\ + $$(if $$(wildcard $(1).mll),$(1).mll,\ + $$(if $$(wildcard $(1).mli),$(1).mli $(1).ml,$(1).ml)\ + )\ + ) +endef + +define INCLUDE_TEMPLATE + ifeq ($(MAKECMDGOALS), $(1)) + include .depend.opt + endif +endef + +$(foreach DIR, $(DIRECTORIES), $(eval $(call DIR_TEMPLATE, $(DIR)))) +$(foreach MOD, $(MODULES), $(eval $(call MOD_TEMPLATE, $(MOD)))) + +OBJECTS = $(patsubst %.ml,%.cmx,$(SOURCES:%.mli=%.cmi)) +CLEAN += $(MAIN).opt + +all opt: .depend.opt + @$(MAKE) --no-print-directory $(MAIN).opt + +$(MAIN).opt: $(OBJECTS) + @echo " OCAMLOPT -o $(MAIN).opt" + $(H)$(OCAMLOPT) -o $(MAIN).opt $(CMXS) + +.depend.opt: $(SOURCES) + @echo " OCAMLDEP -native" + $(H)$(OCAMLDEP) $^ > .depend.opt + +clean: + @echo " CLEAN . $(DIRECTORIES)" + $(H)find -name "*~" | xargs $(RM) $(CLEAN) + +lint: $(XMLS) + @echo XMLLINT --valid + $(H)$(XMLLINT) --valid $^ + +tgz: clean + @echo " TAR -czf $(MAIN:%=%.tgz) . $(DIRECTORIES)" + $(H)find -name "Make*" | xargs $(TAR) $(KEEP) + +%.ml %.mli: %.mly + @echo " OCAMLYACC $<" + $(H)$(OCAMLYACC) $< +%.ml: %.mll + @echo " OCAMLLEX $<" + $(H)$(OCAMLLEX) $< +%.cmi: %.mli + @echo " OCAMLOPT $<" + $(H)$(OCAMLOPT) -c $< +%.cmx: %.ml + @echo " OCAMLOPT $<" + $(H)$(OCAMLOPT) -c $< + +TAGS += $(MAIN).opt + +$(foreach TAG, $(TAGS), $(eval $(call INCLUDE_TEMPLATE, $(TAG)))) diff --git a/helm/software/lambda-delta/README b/helm/software/lambda-delta/README new file mode 100644 index 000000000..8a0f0d674 --- /dev/null +++ b/helm/software/lambda-delta/README @@ -0,0 +1,10 @@ +Helena 0.8.1 M + +* type "make" or "make opt" to compile the native executable + +* type "make test-si" to parse the grundlagen + it generates a log.txt with the grundlagen contents statistics + +* type "make clean" to remove the products of compilation + +* type "make tgz" to make a tar.gz of the source files diff --git a/helm/software/lambda-delta/automath/Make b/helm/software/lambda-delta/automath/Make new file mode 100644 index 000000000..29d237864 --- /dev/null +++ b/helm/software/lambda-delta/automath/Make @@ -0,0 +1 @@ +aut autProcess autOutput autParser autLexer diff --git a/helm/software/lambda-delta/automath/Omega.aut b/helm/software/lambda-delta/automath/Omega.aut new file mode 100644 index 000000000..2466a606e --- /dev/null +++ b/helm/software/lambda-delta/automath/Omega.aut @@ -0,0 +1,10 @@ +# The lambda-term \Omega +# This book is not accepted in AUT-QE because [y:'type'] is not allowed +# This book is accepted in lambda-delta with sort inclusion but Omega is not +# valid if sort inclusion is allowed on the term backbone only +# This book is valid in lambda-delta with unrestricted sort inclusion + ++l +@ Delta := [x:[y:'type']'type']x : [x:[y:'type']'type']'type' + Omega := Delta : 'type' +-l diff --git a/helm/software/lambda-delta/automath/aut.ml b/helm/software/lambda-delta/automath/aut.ml new file mode 100644 index 000000000..513bd4a47 --- /dev/null +++ b/helm/software/lambda-delta/automath/aut.ml @@ -0,0 +1,25 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department, University of Bologna, Italy. + ||I|| + ||T|| HELM is free software; you can redistribute it and/or + ||A|| modify it under the terms of the GNU General Public License + \ / version 2 or (at your option) any later version. + \ / This software is distributed as is, NO WARRANTY. + V_______________________________________________________________ *) + +type id = string (* identifier *) + +type qid = id * bool * id list (* qualified identifier: name, local?, path *) + +type term = Sort of bool (* sorts: true = TYPE, false = PROP *) + | GRef of qid * term list (* reference: name, arguments *) + | Appl of term * term (* application: argument, function *) + | Abst of id * term * term (* abstraction: name, domain, scope *) + +type entity = Section of (bool * id) option (* section: Some true = open, Some false = reopen, None = close last *) + | Context of qid option (* context: Some = last node, None = root *) + | Block of id * term (* block opener: name, domain *) + | Decl of id * term (* declaration: name, domain *) + | Def of id * term * bool * term (* definition: name, domain, transparent?, body *) diff --git a/helm/software/lambda-delta/automath/autLexer.mll b/helm/software/lambda-delta/automath/autLexer.mll new file mode 100644 index 000000000..006c056b9 --- /dev/null +++ b/helm/software/lambda-delta/automath/autLexer.mll @@ -0,0 +1,75 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department, University of Bologna, Italy. + ||I|| + ||T|| HELM is free software; you can redistribute it and/or + ||A|| modify it under the terms of the GNU General Public License + \ / version 2 or (at your option) any later version. + \ / This software is distributed as is, NO WARRANTY. + V_______________________________________________________________ *) + +{ + module L = Log + module P = AutParser + + let debug = false + let out s = if debug then L.warn s else () +} + +let LC = ['#' '%'] +let OC = "{" +let CC = "}" +let SPC = [' ' '\t' '\n']+ +let NL = "\n" +let ID = ['0'-'9' 'A'-'Z' 'a'-'z' '_' '\'' '`']+ + +rule line_comment = parse + | NL { () } + | OC { block_comment lexbuf; line_comment lexbuf } + | _ { line_comment lexbuf } + | eof { () } +and block_comment = parse + | CC { () } + | OC { block_comment lexbuf; block_comment lexbuf } + | LC { line_comment lexbuf; block_comment lexbuf } + | _ { block_comment lexbuf } +and token = parse + | SPC { token lexbuf } + | LC { line_comment lexbuf; token lexbuf } + | OC { block_comment lexbuf; token lexbuf } + | "_E" { out "E"; P.E } + | "'_E'" { out "E"; P.E } + | "---" { out "EB"; P.EB } + | "'eb'" { out "EB"; P.EB } + | "EB" { out "EB"; P.EB } + | "--" { out "EXIT"; P.EXIT } + | "PN" { out "PN"; P.PN } + | "'pn'" { out "PN"; P.PN } + | "PRIM" { out "PN"; P.PN } + | "'prim'" { out "PN"; P.PN } + | "???" { out "PN"; P.PN } + | "PROP" { out "PROP"; P.PROP } + | "'prop'" { out "PROP"; P.PROP } + | "TYPE" { out "TYPE"; P.TYPE } + | "'type'" { out "TYPE"; P.TYPE } + | ID { out "ID"; P.IDENT (Lexing.lexeme lexbuf) } + | ":=" { out "DEF"; P.DEF } + | "(" { out "OP"; P.OP } + | ")" { out "CP"; P.CP } + | "[" { out "OB"; P.OB } + | "]" { out "CB"; P.CB } + | "<" { out "OA"; P.OA } + | ">" { out "CA"; P.CA } + | "@" { out "AT"; P.AT } + | "~" { out "TD"; P.TD } + | "\"" { out "QT"; P.QT } + | ":" { out "CN"; P.CN } + | "," { out "CM"; P.CM } + | ";" { out "SC"; P.SC } + | "." { out "FS"; P.FS } + | "+" { out "PLUS"; P.PLUS } + | "-" { out "MINUS"; P.MINUS } + | "*" { out "TIMES"; P.TIMES } + | "=" { out "DEF"; P.DEF } + | eof { out "EOF"; P.EOF } diff --git a/helm/software/lambda-delta/automath/autOutput.ml b/helm/software/lambda-delta/automath/autOutput.ml new file mode 100644 index 000000000..668de7462 --- /dev/null +++ b/helm/software/lambda-delta/automath/autOutput.ml @@ -0,0 +1,100 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department, University of Bologna, Italy. + ||I|| + ||T|| HELM is free software; you can redistribute it and/or + ||A|| modify it under the terms of the GNU General Public License + \ / version 2 or (at your option) any later version. + \ / This software is distributed as is, NO WARRANTY. + V_______________________________________________________________ *) + +module P = Printf +module C = Cps +module L = Log +module A = Aut +module R = AutProcess + +type counters = { + sections: int; + contexts: int; + blocks: int; + decls: int; + defs: int; + sorts: int; + grefs: int; + appls: int; + absts: int; + pars: int; + xnodes: int +} + +let initial_counters = { + sections = 0; contexts = 0; blocks = 0; decls = 0; defs = 0; + sorts = 0; grefs = 0; appls = 0; absts = 0; pars = 0; xnodes = 0 +} + +let rec count_term f c = function + | A.Sort _ -> + f {c with sorts = succ c.sorts; xnodes = succ c.xnodes} + | A.GRef (_, ts) -> + let c = {c with grefs = succ c.grefs} in + let c = {c with pars = c.pars + List.length ts} in + let c = {c with xnodes = succ c.xnodes + List.length ts} in + C.list_fold_left f count_term c ts + | A.Appl (v, t) -> + let c = {c with appls = succ c.appls; xnodes = succ c.xnodes} in + let f c = count_term f c t in + count_term f c v + | A.Abst (_, w, t) -> + let c = {c with absts = succ c.absts; xnodes = succ c.xnodes} in + let f c = count_term f c t in + count_term f c w + +let count_entity f c = function + | A.Section _ -> + f {c with sections = succ c.sections} + | A.Context _ -> + f {c with contexts = succ c.contexts} + | A.Block (_, w) -> + let c = {c with blocks = succ c.blocks; xnodes = succ c.xnodes} in + count_term f c w + | A.Decl (_, w) -> + let c = {c with decls = succ c.decls; xnodes = succ c.xnodes} in + count_term f c w + | A.Def (_, w, _, t) -> + let c = {c with defs = succ c.defs; xnodes = succ c.xnodes} in + let f c = count_term f c t in + count_term f c w + +let print_counters f c = + let terms = c.sorts + c.grefs + c.appls + c.absts in + let entities = c.sections + c.contexts + c.blocks + c.decls + c.defs in + L.warn (P.sprintf " Automath representation summary"); + L.warn (P.sprintf " Total book entities: %7u" entities); + L.warn (P.sprintf " Section entities: %7u" c.sections); + L.warn (P.sprintf " Context entities: %7u" c.contexts); + L.warn (P.sprintf " Block entities: %7u" c.blocks); + L.warn (P.sprintf " Declaration entities: %7u" c.decls); + L.warn (P.sprintf " Definition entities: %7u" c.defs); + L.warn (P.sprintf " Total Parameter items: %7u" c.pars); + L.warn (P.sprintf " Application items: %7u" c.pars); + L.warn (P.sprintf " Total term items: %7u" terms); + L.warn (P.sprintf " Sort items: %7u" c.sorts); + L.warn (P.sprintf " Reference items: %7u" c.grefs); + L.warn (P.sprintf " Application items: %7u" c.appls); + L.warn (P.sprintf " Abstraction items: %7u" c.absts); + L.warn (P.sprintf " Global Int. Complexity: unknown"); + L.warn (P.sprintf " + Abbreviation nodes: %7u" c.xnodes); + f () + +let print_process_counters f c = + let f iao iar iac iag = + L.warn (P.sprintf " Automath process summary"); + L.warn (P.sprintf " Implicit after opening: %7u" iao); + L.warn (P.sprintf " Implicit after reopening: %7u" iar); + L.warn (P.sprintf " Implicit after closing: %7u" iac); + L.warn (P.sprintf " Implicit after global: %7u" iag); + f () + in + R.get_counters f c diff --git a/helm/software/lambda-delta/automath/autOutput.mli b/helm/software/lambda-delta/automath/autOutput.mli new file mode 100644 index 000000000..611c1bf6b --- /dev/null +++ b/helm/software/lambda-delta/automath/autOutput.mli @@ -0,0 +1,20 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department, University of Bologna, Italy. + ||I|| + ||T|| HELM is free software; you can redistribute it and/or + ||A|| modify it under the terms of the GNU General Public License + \ / version 2 or (at your option) any later version. + \ / This software is distributed as is, NO WARRANTY. + V_______________________________________________________________ *) + +type counters + +val initial_counters: counters + +val count_entity: (counters -> 'a) -> counters -> Aut.entity -> 'a + +val print_counters: (unit -> 'a) -> counters -> 'a + +val print_process_counters: (unit -> 'a) -> AutProcess.status -> 'a diff --git a/helm/software/lambda-delta/automath/autParser.mly b/helm/software/lambda-delta/automath/autParser.mly new file mode 100644 index 000000000..c772837b3 --- /dev/null +++ b/helm/software/lambda-delta/automath/autParser.mly @@ -0,0 +1,95 @@ +/* Copyright (C) 2000, HELM Team. + * + * This file is part of HELM, an Hypertextual, Electronic + * Library of Mathematics, developed at the Computer Science + * Department, University of Bologna, Italy. + * + * HELM is free software; you can redistribute it and/or + * modify it under the terms of the GNU General Public License + * as published by the Free Software Foundation; either version 2 + * of the License, or (at your option) any later version. + * + * HELM is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with HELM; if not, write to the Free Software + * Foundation, Inc., 59 Temple Place - Suite 330, Boston, + * MA 02111-1307, USA. + * + * For details, see the HELM World-Wide-Web page, + * http://cs.unibo.it/helm/. + */ + +%{ + module A = Aut +%} + %token NUM + %token IDENT + %token EOF MINUS PLUS TIMES AT FS CN CM SC QT TD OP CP OB CB OA CA + %token TYPE PROP DEF EB E PN EXIT + + %start book + %type book +%% + path: MINUS {} | FS {} ; + oftype: CN {} | CM {} ; + star: TIMES {} | AT {} ; + sc: E {} | SC {} | CN {} ; + eof: SC {} | EOF {} ; + + expand: + | { true } + | TD { false } + ; + local: + | { false } + | path { true } + ; + + idents: + | IDENT { [$1] } + | IDENT path idents { $1 :: $3 } + ; + qid: + | IDENT { ($1, true, []) } + | IDENT QT QT { ($1, true, []) } + | IDENT QT local idents QT { ($1, $3, $4) } + ; + term: + | TYPE { A.Sort true } + | PROP { A.Sort false } + | qid { A.GRef ($1, []) } + | qid OP CP { A.GRef ($1, []) } + | qid OP terms CP { A.GRef ($1, $3) } + | OA term CA term { A.Appl ($2, $4) } + | OB IDENT oftype term CB term { A.Abst ($2, $4, $6) } + ; + terms: + | term { [$1] } + | term CM terms { $1 :: $3 } + ; + entity: + | PLUS IDENT { A.Section (Some (true, $2)) } + | PLUS TIMES IDENT { A.Section (Some (false, $3)) } + | MINUS IDENT { A.Section None } + | EXIT { A.Section None } + | star { A.Context None } + | qid star { A.Context (Some $1) } + | IDENT DEF EB sc term { A.Block ($1, $5) } + | IDENT sc term DEF EB { A.Block ($1, $3) } + | OB IDENT oftype term CB { A.Block ($2, $4) } + | IDENT DEF PN sc term { A.Decl ($1, $5) } + | IDENT sc term DEF PN { A.Decl ($1, $3) } + | IDENT DEF expand term sc term { A.Def ($1, $6, $3, $4) } + | IDENT sc term DEF expand term { A.Def ($1, $3, $5, $6) } + ; + entities: + | { [] } + | entity entities { $1 :: $2 } + ; + book: + | entities eof { $1 } + ; diff --git a/helm/software/lambda-delta/automath/autProcess.ml b/helm/software/lambda-delta/automath/autProcess.ml new file mode 100644 index 000000000..0009e021e --- /dev/null +++ b/helm/software/lambda-delta/automath/autProcess.ml @@ -0,0 +1,77 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department, University of Bologna, Italy. + ||I|| + ||T|| HELM is free software; you can redistribute it and/or + ||A|| modify it under the terms of the GNU General Public License + \ / version 2 or (at your option) any later version. + \ / This software is distributed as is, NO WARRANTY. + V_______________________________________________________________ *) + +module A = Aut + +type status = { + opening : bool; (* just opened section *) + reopening: bool; (* just reopened section *) + closing : bool; (* just closed section *) + explicit : bool; (* just found explicit context *) + block : bool; (* just found block opener *) + iao : int; (* implicit context after opening section *) + iar : int; (* implicit context after reopening section *) + iac : int; (* implicit context after closing section *) + iag : int (* implicit context after global statement *) +} + +(* internal functions *******************************************************) + +let orc_reset f st = + f {st with opening = false; reopening = false; closing = false} + +let orc_count f st = + let st = if st.opening then {st with iao = succ st.iao} else st in + let st = if st.reopening then {st with iar = succ st.iar} else st in + let st = if st.closing then {st with iac = succ st.iac} else st in + f st + +let exp_count f st = + let st = + if st.explicit || st.block then st else {st with iag = succ st.iag} + in + f st + +let proc_section f st = function + | Some (true, _) -> f {st with opening = true} + | Some (false, _) -> f {st with reopening = true} + | None -> f {st with closing = true} + +let proc_context f st = + orc_reset f {st with explicit = true} + +let proc_block f st = + orc_count (orc_reset f) {st with explicit = false; block = true} + +let proc_global f st = + let f st = + orc_count (orc_reset f) {st with explicit = false; block = false} + in + exp_count f st + +let proc_entity f st entity = match entity with + | A.Section section -> proc_section f st section entity + | A.Context _ -> proc_context f st entity + | A.Block _ -> proc_block f st entity + | A.Decl _ -> proc_global f st entity + | A.Def _ -> proc_global f st entity + +(* interface functions ******************************************************) + +let initial_status = { + opening = false; reopening = false; closing = false; + explicit = false; block = false; + iao = 0; iar = 0; iac = 0; iag = 0 +} + +let process_entity = proc_entity + +let get_counters f st = f st.iao st.iar st.iac st.iag diff --git a/helm/software/lambda-delta/automath/autProcess.mli b/helm/software/lambda-delta/automath/autProcess.mli new file mode 100644 index 000000000..8f10b653e --- /dev/null +++ b/helm/software/lambda-delta/automath/autProcess.mli @@ -0,0 +1,18 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department, University of Bologna, Italy. + ||I|| + ||T|| HELM is free software; you can redistribute it and/or + ||A|| modify it under the terms of the GNU General Public License + \ / version 2 or (at your option) any later version. + \ / This software is distributed as is, NO WARRANTY. + V_______________________________________________________________ *) + +type status + +val initial_status: status + +val process_entity: (status -> Aut.entity -> 'a) -> status -> Aut.entity -> 'a + +val get_counters: (int -> int -> int -> int -> 'a) -> status -> 'a diff --git a/helm/software/lambda-delta/automath/grundlagen-orig.aut b/helm/software/lambda-delta/automath/grundlagen-orig.aut new file mode 100644 index 000000000..4a856c245 --- /dev/null +++ b/helm/software/lambda-delta/automath/grundlagen-orig.aut @@ -0,0 +1,10713 @@ +# 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,