:: TOPREAL8 semantic presentation begin theorem :: TOPREAL8:1 (Bool "for" (Set (Var "A")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k2_tarski :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ))) ; registration cluster ($#v1_zfmisc_1 :::"trivial"::: ) bbbadV1_RELAT_1() ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; begin registration cluster bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set ($#k5_numbers :::"NAT"::: ) )) ($#v1_funct_1 :::"Function-like"::: ) ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: TOPREAL8:2 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) ; theorem :: TOPREAL8:3 (Bool "for" (Set (Var "D")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v1_finseq_6 :::"circular"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 2)))) ; theorem :: TOPREAL8:4 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")))) "or" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ; theorem :: TOPREAL8:5 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k6_finseq_4 :::"|--"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "g"))))))) ; theorem :: TOPREAL8:6 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_funct_1 :::"one-to-one"::: ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))))) ; theorem :: TOPREAL8:7 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k3_graph_2 :::"^'"::: ) (Set (Var "g")) ")" )))) ; theorem :: TOPREAL8:8 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k4_finseq_4 :::".."::: ) (Set "(" (Set (Var "f")) ($#k3_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ))))) ; theorem :: TOPREAL8:9 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"FinSequence":::) (Bool "for" (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "f")) ($#k3_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ))))) ; theorem :: TOPREAL8:10 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k3_graph_2 :::"^'"::: ) (Set (Var "g")) ")" )))) ; theorem :: TOPREAL8:11 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "g")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)))) "holds" (Bool (Set (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g"))) "is" ($#v1_finseq_6 :::"circular"::: ) )))) ; theorem :: TOPREAL8:12 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_finseq_1 :::"Matrix":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "g")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1))) & (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "M"))) & (Bool (Set (Var "g")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "M")))) "holds" (Bool (Set (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g"))) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "M"))))))) ; theorem :: TOPREAL8:13 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool (Set "(" (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ($#k2_graph_2 :::"-cut"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "k"))))))) ; theorem :: TOPREAL8:14 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set "(" (Num 1) "," (Set (Var "k")) ")" ($#k2_graph_2 :::"-cut"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "k"))))))) ; theorem :: TOPREAL8:15 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "p")) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g")) ")" ) ($#k5_finseq_4 :::"-|"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set "(" (Num 1) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ($#k2_graph_2 :::"-cut"::: ) (Set (Var "f")))))))) ; theorem :: TOPREAL8:16 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ) ($#k2_finseq_5 :::":-"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "g"))))) ; theorem :: TOPREAL8:17 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ) ($#k1_finseq_5 :::"-:"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "f"))))) ; theorem :: TOPREAL8:18 (Bool "for" (Set (Var "D")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "g")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (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 ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1))) ")" )) "holds" (Bool (Set ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set "(" (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "f"))))))) ; begin theorem :: TOPREAL8:19 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Num 2) ")" )))) ; theorem :: TOPREAL8:20 (Bool "for" (Set (Var "f")) "being" ($#v1_goboard5 :::"s.c.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "n"))) "is" ($#v3_topreal1 :::"s.n.c."::: ) ))) ; theorem :: TOPREAL8:21 (Bool "for" (Set (Var "f")) "being" ($#v1_goboard5 :::"s.c.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n"))) "is" ($#v3_topreal1 :::"s.n.c."::: ) ))) ; theorem :: TOPREAL8:22 (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 4))) "holds" (Bool (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "n"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: TOPREAL8:23 (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 4))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "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 (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")))))) ; theorem :: TOPREAL8:24 (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 4))) "holds" (Bool (Set (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Set (Var "n"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; theorem :: TOPREAL8:25 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_topreal1 :::"special"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set "(" (Set (Var "m")) "," (Set (Var "n")) ")" ($#k2_graph_2 :::"-cut"::: ) (Set (Var "f"))) "is" ($#v1_topreal1 :::"special"::: ) ))) ; theorem :: TOPREAL8:26 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_topreal1 :::"special"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "g")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#v1_topreal1 :::"special"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)))) "holds" (Bool (Set (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g"))) "is" ($#v1_topreal1 :::"special"::: ) ))) ; theorem :: TOPREAL8:27 (Bool "for" (Set (Var "f")) "being" ($#v1_finseq_6 :::"circular"::: ) ($#v2_topreal1 :::"unfolded"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 4))) "holds" (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f")) ($#k2_rfinseq :::"/^"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2) ")" ) ($#k2_tarski :::"}"::: ) ))) ; theorem :: TOPREAL8:28 (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) ")" )))) ; theorem :: TOPREAL8:29 (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "j")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "g")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )))) ; theorem :: TOPREAL8:30 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "g")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "g")) "," (Num 1) ")" )))) ; theorem :: TOPREAL8:31 (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "g")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "j")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "g")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))))) ; theorem :: TOPREAL8:32 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_topreal1 :::"unfolded"::: ) ($#v3_topreal1 :::"s.n.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#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 "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k2_tarski :::"}"::: ) )))) ; theorem :: TOPREAL8:33 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#v2_funct_1 :::"one-to-one"::: ) ($#v2_topreal1 :::"unfolded"::: ) ($#v3_topreal1 :::"s.n.c."::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k2_tarski :::"}"::: ) )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g"))) "is" ($#v1_goboard5 :::"s.c.c."::: ) )) ; theorem :: TOPREAL8:34 (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" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "g")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1))) & (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "g")) "," (Num 1) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g"))) "is" ($#v2_topreal1 :::"unfolded"::: ) )) ; theorem :: TOPREAL8:35 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "f")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Bool "not" (Set (Var "g")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f")) ($#k4_graph_2 :::"^'"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )))) ; theorem :: TOPREAL8:36 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" )) ")" ) & (Bool (Bool "not" (Set (Var "f")) "is" ($#v3_funct_1 :::"constant"::: ) )) & (Bool (Set (Var "f")) "is" ($#v1_finseq_6 :::"circular"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_goboard5 :::"s.c.c."::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 4))) "holds" (Bool "ex" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set (Var "g")) "is" ($#v2_topreal1 :::"unfolded"::: ) ) & (Bool (Set (Var "g")) "is" ($#v1_goboard5 :::"s.c.c."::: ) ) & (Bool (Set (Var "g")) "is" ($#v1_topreal1 :::"special"::: ) ) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) ")" )))) ;