:: JGRAPH_1 semantic presentation begin theorem :: JGRAPH_1:1 (Bool "for" (Set (Var "G")) "being" ($#l1_graph_1 :::"Graph":::) (Bool "for" (Set (Var "IT")) "being" ($#v7_graph_1 :::"oriented"::: ) ($#m2_graph_1 :::"Chain"::: ) "of" (Set (Var "G")) (Bool "for" (Set (Var "vs")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "G"))) "st" (Bool (Bool (Set (Var "IT")) "is" ($#v1_graph_4 :::"Simple"::: ) ) & (Bool (Set (Var "vs")) ($#r3_graph_4 :::"is_oriented_vertex_seq_of"::: ) (Set (Var "IT")))) "holds" (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "vs")))) & (Bool (Set (Set (Var "vs")) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "vs")) ($#k1_funct_1 :::"."::: ) (Set (Var "m"))))) "holds" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "vs")))) ")" ))))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; func :::"PGraph"::: "X" -> ($#l1_graph_1 :::"MultiGraphStruct"::: ) equals :: JGRAPH_1:def 1 (Set ($#g1_graph_1 :::"MultiGraphStruct"::: ) "(#" "X" "," (Set ($#k2_zfmisc_1 :::"[:"::: ) "X" "," "X" ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" "X" "," "X" ")" ")" ) "," (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" "X" "," "X" ")" ")" ) "#)" ); end; :: deftheorem defines :::"PGraph"::: JGRAPH_1:def 1 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_jgraph_1 :::"PGraph"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#g1_graph_1 :::"MultiGraphStruct"::: ) "(#" (Set (Var "X")) "," (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "X")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "(" ($#k9_funct_3 :::"pr1"::: ) "(" (Set (Var "X")) "," (Set (Var "X")) ")" ")" ) "," (Set "(" ($#k10_funct_3 :::"pr2"::: ) "(" (Set (Var "X")) "," (Set (Var "X")) ")" ")" ) "#)" ))); theorem :: JGRAPH_1:2 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_jgraph_1 :::"PGraph"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "X")))) ; definitionlet "f" be ($#m1_hidden :::"FinSequence":::); func :::"PairF"::: "f" -> ($#m1_hidden :::"FinSequence":::) means :: JGRAPH_1:def 2 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) "f" ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) "f"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" "f" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"PairF"::: JGRAPH_1:def 2 : (Bool "for" (Set (Var "f")) "," (Set (Var "b2")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_jgraph_1 :::"PairF"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k4_tarski :::"]"::: ) )) ")" ) ")" ) ")" )); registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_jgraph_1 :::"PGraph"::: ) "X") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; theorem :: JGRAPH_1:3 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Var "f")) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_jgraph_1 :::"PGraph"::: ) (Set (Var "X")) ")" ))))) ; theorem :: JGRAPH_1:4 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "holds" (Bool (Set ($#k2_jgraph_1 :::"PairF"::: ) (Set (Var "f"))) "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set "(" ($#k1_jgraph_1 :::"PGraph"::: ) (Set (Var "X")) ")" ))))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "X")); :: original: :::"PairF"::: redefine func :::"PairF"::: "f" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set "(" ($#k1_jgraph_1 :::"PGraph"::: ) "X" ")" )); end; theorem :: JGRAPH_1:5 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_jgraph_1 :::"PairF"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k3_jgraph_1 :::"PairF"::: ) (Set (Var "f")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set "the" ($#u4_struct_0 :::"carrier'"::: ) "of" (Set "(" ($#k1_jgraph_1 :::"PGraph"::: ) (Set (Var "X")) ")" )))))) ; theorem :: JGRAPH_1:6 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "holds" (Bool (Set ($#k3_jgraph_1 :::"PairF"::: ) (Set (Var "f"))) "is" ($#v7_graph_1 :::"oriented"::: ) ($#m2_graph_1 :::"Chain"::: ) "of" (Set ($#k1_jgraph_1 :::"PGraph"::: ) (Set (Var "X")))))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "X")); :: original: :::"PairF"::: redefine func :::"PairF"::: "f" -> ($#v7_graph_1 :::"oriented"::: ) ($#m2_graph_1 :::"Chain"::: ) "of" (Set ($#k1_jgraph_1 :::"PGraph"::: ) "X"); end; theorem :: JGRAPH_1:7 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "f1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_jgraph_1 :::"PGraph"::: ) (Set (Var "X")) ")" )) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "f1")))) "holds" (Bool (Set (Var "f1")) ($#r3_graph_4 :::"is_oriented_vertex_seq_of"::: ) (Set ($#k4_jgraph_1 :::"PairF"::: ) (Set (Var "f"))))))) ; begin definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f", "g" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "X")); pred "g" :::"is_Shortcut_of"::: "f" means :: JGRAPH_1:def 3 (Bool "(" (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set "g" ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set "f" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) "f" ")" )) ($#r1_hidden :::"="::: ) (Set "g" ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) "g" ")" ))) & (Bool "ex" (Set (Var "fc")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_jgraph_1 :::"PairF"::: ) "f" ")" )(Bool "ex" (Set (Var "fvs")) "being" ($#m1_subset_1 :::"Subset":::) "of" "f"(Bool "ex" (Set (Var "sc")) "being" ($#v7_graph_1 :::"oriented"::: ) ($#v3_graph_2 :::"simple"::: ) ($#m2_graph_1 :::"Chain"::: ) "of" (Set ($#k1_jgraph_1 :::"PGraph"::: ) "X")(Bool "ex" (Set (Var "g1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_jgraph_1 :::"PGraph"::: ) "X" ")" )) "st" (Bool "(" (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "fc"))) ($#r1_hidden :::"="::: ) (Set (Var "sc"))) & (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "fvs"))) ($#r1_hidden :::"="::: ) "g") & (Bool (Set (Var "g1")) ($#r1_hidden :::"="::: ) "g") & (Bool (Set (Var "g1")) ($#r3_graph_4 :::"is_oriented_vertex_seq_of"::: ) (Set (Var "sc"))) ")" ))))) ")" ); end; :: deftheorem defines :::"is_Shortcut_of"::: JGRAPH_1:def 3 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_jgraph_1 :::"is_Shortcut_of"::: ) (Set (Var "f"))) "iff" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ))) & (Bool "ex" (Set (Var "fc")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_jgraph_1 :::"PairF"::: ) (Set (Var "f")) ")" )(Bool "ex" (Set (Var "fvs")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "f"))(Bool "ex" (Set (Var "sc")) "being" ($#v7_graph_1 :::"oriented"::: ) ($#v3_graph_2 :::"simple"::: ) ($#m2_graph_1 :::"Chain"::: ) "of" (Set ($#k1_jgraph_1 :::"PGraph"::: ) (Set (Var "X")))(Bool "ex" (Set (Var "g1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_jgraph_1 :::"PGraph"::: ) (Set (Var "X")) ")" )) "st" (Bool "(" (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "fc"))) ($#r1_hidden :::"="::: ) (Set (Var "sc"))) & (Bool (Set ($#k15_finseq_1 :::"Seq"::: ) (Set (Var "fvs"))) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "g1")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "g1")) ($#r3_graph_4 :::"is_oriented_vertex_seq_of"::: ) (Set (Var "sc"))) ")" ))))) ")" ) ")" ))); theorem :: JGRAPH_1:8 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "g")) ($#r1_jgraph_1 :::"is_Shortcut_of"::: ) (Set (Var "f")))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ))) ; theorem :: JGRAPH_1:9 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "st" (Bool (Set (Var "g")) ($#r1_jgraph_1 :::"is_Shortcut_of"::: ) (Set (Var "f")))))) ; theorem :: JGRAPH_1:10 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "g")) ($#r1_jgraph_1 :::"is_Shortcut_of"::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k4_jgraph_1 :::"PairF"::: ) (Set (Var "g")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k4_jgraph_1 :::"PairF"::: ) (Set (Var "f")) ")" ))))) ; theorem :: JGRAPH_1:11 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "g")) ($#r1_jgraph_1 :::"is_Shortcut_of"::: ) (Set (Var "f")))) "holds" (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "IT" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); attr "IT" is :::"nodic"::: means :: JGRAPH_1:def 4 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "not" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "i")) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "j")) ")" )) "or" (Bool "(" (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "i")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" "IT" ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool "(" (Bool (Set "IT" ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set "IT" ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) "or" (Bool (Set "IT" ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set "IT" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" ) "or" (Bool "(" (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "i")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" "IT" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool "(" (Bool (Set "IT" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set "IT" ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) "or" (Bool (Set "IT" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set "IT" ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" ) "or" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" "IT" "," (Set (Var "j")) ")" )) ")" )); end; :: deftheorem defines :::"nodic"::: JGRAPH_1:def 4 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "IT")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_jgraph_1 :::"nodic"::: ) ) "iff" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "not" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "i")) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "j")) ")" )) "or" (Bool "(" (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "i")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool "(" (Bool (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) "or" (Bool (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" ) "or" (Bool "(" (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "i")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool "(" (Bool (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) "or" (Bool (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "IT")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" ) ")" ) "or" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "IT")) "," (Set (Var "j")) ")" )) ")" )) ")" ))); theorem :: JGRAPH_1:12 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v1_goboard5 :::"s.c.c."::: ) )) ; theorem :: JGRAPH_1:13 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_goboard5 :::"s.c.c."::: ) ) & (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ))) "holds" (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) )) ; theorem :: JGRAPH_1:14 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jgraph_1 :::"nodic"::: ) ) & (Bool (Set ($#k4_jgraph_1 :::"PairF"::: ) (Set (Var "f"))) "is" ($#v1_graph_4 :::"Simple"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v1_goboard5 :::"s.c.c."::: ) )) ; theorem :: JGRAPH_1:15 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jgraph_1 :::"nodic"::: ) ) & (Bool (Set ($#k4_jgraph_1 :::"PairF"::: ) (Set (Var "f"))) "is" ($#v1_graph_4 :::"Simple"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) )) ; theorem :: JGRAPH_1:16 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "p2"))) "or" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p2")) "," (Set (Var "p3")) ")" ")" ))) ")" )) "or" (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p2")) "," (Set (Var "p3")) ")" )) "or" (Bool (Set (Var "p3")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" )) ")" ))) ; theorem :: JGRAPH_1:17 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_topreal1 :::"s.n.c."::: ) ) & (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Num 1) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Num 1) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 2) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) )) ; theorem :: JGRAPH_1:18 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set ($#k4_jgraph_1 :::"PairF"::: ) (Set (Var "f"))) "is" ($#v1_graph_4 :::"Simple"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Num 1)) ")" ))) ; theorem :: JGRAPH_1:19 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 1))) "holds" (Bool "(" (Bool (Set ($#k4_jgraph_1 :::"PairF"::: ) (Set (Var "f"))) "is" ($#v1_graph_4 :::"Simple"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) ")" ))) ; theorem :: JGRAPH_1:20 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_jgraph_1 :::"nodic"::: ) ) & (Bool (Set ($#k4_jgraph_1 :::"PairF"::: ) (Set (Var "f"))) "is" ($#v1_graph_4 :::"Simple"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) )) ; theorem :: JGRAPH_1:21 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "g")) ($#r1_jgraph_1 :::"is_Shortcut_of"::: ) (Set (Var "f"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))))) "holds" (Bool "ex" (Set (Var "k1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k1"))) & (Bool (Set (Set (Var "k1")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "k1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "k1")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) ")" )))) ; theorem :: JGRAPH_1:22 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "g")) ($#r1_jgraph_1 :::"is_Shortcut_of"::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) ; theorem :: JGRAPH_1:23 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "g")) ($#r1_jgraph_1 :::"is_Shortcut_of"::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: JGRAPH_1:24 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "g")) ($#r1_jgraph_1 :::"is_Shortcut_of"::: ) (Set (Var "f")))) "holds" (Bool (Set (Var "g")) "is" ($#v1_topreal1 :::"special"::: ) )) ; theorem :: JGRAPH_1:25 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Var "g")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")))) ")" ))) ; theorem :: JGRAPH_1:26 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")))) & (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2")))) & (Bool (Set (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f1"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f1")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) "," (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f2"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f1")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) "," (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f1")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f1"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f2")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) "," (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f2"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f2")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) "," (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f2")) ")" )))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f1"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f2"))))) ; begin theorem :: JGRAPH_1:27 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r2"))) & (Bool (Set (Var "r2")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "r1")) ($#k9_real_1 :::"-"::: ) (Set (Var "r2")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "b")) ($#k9_real_1 :::"-"::: ) (Set (Var "a"))))) ; theorem :: JGRAPH_1:28 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "x1")) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Var "x2")) ($#r1_hidden :::"="::: ) (Set (Var "p2")))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_metric_1 :::"dist"::: ) "(" (Set (Var "x1")) "," (Set (Var "x2")) ")" ))))) ; theorem :: JGRAPH_1:29 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#k5_square_1 :::"^2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" )))) ; theorem :: JGRAPH_1:30 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" )))) ; theorem :: JGRAPH_1:31 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ")" )))) ; theorem :: JGRAPH_1:32 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ")" ) ")" )))) ; theorem :: JGRAPH_1:33 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "p")) ($#k12_euclid :::".|"::: ) )) ")" )) ; theorem :: JGRAPH_1:34 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2")) ")" ) ($#k12_euclid :::".|"::: ) )) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2")) ")" ) ($#k12_euclid :::".|"::: ) )) ")" )) ; theorem :: JGRAPH_1:35 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p2")) ")" ))) ")" )))) ; theorem :: JGRAPH_1:36 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ))) "holds" (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p1")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2")) ")" ) ($#k12_euclid :::".|"::: ) )) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "p1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2")) ")" ) ($#k12_euclid :::".|"::: ) )) ")" ))) ; begin theorem :: JGRAPH_1:37 (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M")) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "P")) "is" ($#v2_compts_1 :::"compact"::: ) ) & (Bool (Set (Var "Q")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "Q")) "is" ($#v2_compts_1 :::"compact"::: ) )) "holds" (Bool (Set ($#k7_weierstr :::"min_dist_min"::: ) "(" (Set (Var "P")) "," (Set (Var "Q")) ")" ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: JGRAPH_1:38 (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M")) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "P")) "is" ($#v2_compts_1 :::"compact"::: ) ) & (Bool (Set (Var "Q")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "Q")) "is" ($#v2_compts_1 :::"compact"::: ) )) "holds" (Bool "(" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "Q"))) "iff" (Bool (Set ($#k7_weierstr :::"min_dist_min"::: ) "(" (Set (Var "P")) "," (Set (Var "Q")) ")" ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: JGRAPH_1:39 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) "," (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Var "c")) "," (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) ")" )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "g")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "g"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) "," (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "g"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Var "c")) "," (Set (Var "d"))) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) ")" ) ")" )))) ; theorem :: JGRAPH_1:40 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) "," (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Var "c")) "," (Set (Var "d"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) ")" )) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "g")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "g"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) "," (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "g"))) ($#r1_goboard4 :::"lies_between"::: ) (Set (Var "c")) "," (Set (Var "d"))) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) ")" )) ")" ) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) ")" ) ")" )))) ; theorem :: JGRAPH_1:41 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) )) ")" )) ; theorem :: JGRAPH_1:42 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" )) ; theorem :: JGRAPH_1:43 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ))) ; theorem :: JGRAPH_1:44 (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "q1")) "," (Set (Var "q2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "p1")) "," (Set (Var "p2"))) & (Bool (Set (Var "Q")) ($#r1_topreal1 :::"is_an_arc_of"::: ) (Set (Var "q1")) "," (Set (Var "q2"))) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) ")" ) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "Q")))) "holds" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) ")" ) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool "(" (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ) & (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "Q")))) "holds" (Bool "(" (Bool (Set (Set (Var "q1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "q2")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )) "holds" (Bool (Set (Var "P")) ($#r2_subset_1 :::"meets"::: ) (Set (Var "Q"))))) ; theorem :: JGRAPH_1:45 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "(" (Set (Var "Y")) ($#k1_pre_topc :::"|"::: ) (Set (Var "P")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "f1"))) & (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool (Set (Var "f1")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))) ; theorem :: JGRAPH_1:46 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) (Bool "for" (Set (Var "P")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_compts_1 :::"compact"::: ) ) & (Bool (Set (Var "Y")) "is" ($#v8_pre_topc :::"T_2"::: ) ) & (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "(" (Set (Var "Y")) ($#k1_pre_topc :::"|"::: ) (Set (Var "P")) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_funct_2 :::"="::: ) (Set (Var "f1"))) & (Bool (Set (Var "f1")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) ) ")" ))))) ; theorem :: JGRAPH_1:47 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "O")) "," (Set (Var "I")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "st" (Bool (Bool (Set (Var "O")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Num 1)) & (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "a"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "b"))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "O")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "c"))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "I")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "d"))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k5_topmetr :::"I[01]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) ")" ) ")" )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r2_subset_1 :::"meets"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "g"))))))) ;