:: JORDAN5D semantic presentation begin theorem :: JORDAN5D:1 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "h")) "," (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )))) ; theorem :: JORDAN5D:2 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 3) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i")))) "holds" (Bool (Set (Set (Var "i")) ($#k4_nat_d :::"mod"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: JORDAN5D:3 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "h"))))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "h")))) & (Bool (Set (Set (Var "h")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" )))) ; theorem :: JORDAN5D:4 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k28_seq_4 :::"Incr"::: ) (Set (Var "g")) ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k28_seq_4 :::"Incr"::: ) (Set (Var "g")) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k28_seq_4 :::"Incr"::: ) (Set (Var "g")) ")" ) ")" ))) ")" ))) ; theorem :: JORDAN5D:5 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "I")) "being" ($#m2_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 "h")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "I")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ")" ) "," (Set (Var "I")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) ")" ))) ; theorem :: JORDAN5D:6 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Num 1) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ))) ; theorem :: JORDAN5D:7 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m2_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 "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "ex" (Set (Var "k")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" )))) ; theorem :: JORDAN5D:8 (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 "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "ex" (Set (Var "k")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" )))) ; theorem :: JORDAN5D:9 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_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 "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) ")" )))) ; theorem :: JORDAN5D:10 (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 "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" )))) ; begin theorem :: JORDAN5D:11 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "being" ($#m2_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 "h"))))) "holds" (Bool "(" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) ")" ))) ; theorem :: JORDAN5D:12 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h"))))) "holds" (Bool "(" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) ")" ))) ; theorem :: JORDAN5D:13 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k14_pscomp_1 :::"W-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k14_pscomp_1 :::"W-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" )))))) ; theorem :: JORDAN5D:14 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k16_pscomp_1 :::"E-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k16_pscomp_1 :::"E-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" )))))) ; theorem :: JORDAN5D:15 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k15_pscomp_1 :::"N-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k15_pscomp_1 :::"N-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" )))))) ; theorem :: JORDAN5D:16 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k17_pscomp_1 :::"S-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k17_pscomp_1 :::"S-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" )))))) ; theorem :: JORDAN5D:17 (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")))) "}" )) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) ")" )))))) ; theorem :: JORDAN5D:18 (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")))) "}" )) "holds" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) ")" )))))) ; theorem :: JORDAN5D:19 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k14_pscomp_1 :::"W-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN5D:20 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k14_pscomp_1 :::"W-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN5D:21 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k16_pscomp_1 :::"E-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN5D:22 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k16_pscomp_1 :::"E-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN5D:23 (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")))) "}" )) "holds" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) ")" ))))) ; theorem :: JORDAN5D:24 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k17_pscomp_1 :::"S-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN5D:25 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k17_pscomp_1 :::"S-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN5D:26 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k15_pscomp_1 :::"N-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN5D:27 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" ) "}" )) "holds" (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k15_pscomp_1 :::"N-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN5D:28 (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")))) "}" )) "holds" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) ")" ))))) ; theorem :: JORDAN5D:29 (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")))) "}" )) "holds" (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) ")" ))))) ; theorem :: JORDAN5D:30 (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")))) "}" )) "holds" (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) ")" ))))) ; theorem :: JORDAN5D:31 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" )))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "I")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ))))) ; theorem :: JORDAN5D:32 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ")" ) "," (Set (Var "I")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ))))) ; theorem :: JORDAN5D:33 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" )))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Num 1) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ))))) ; theorem :: JORDAN5D:34 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "I")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ))))) ; theorem :: JORDAN5D:35 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_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 "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" )))) "holds" (Bool "ex" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" )))) ; theorem :: JORDAN5D:36 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" )))) "holds" (Bool "ex" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool "(" (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))) ")" )))) ; theorem :: JORDAN5D:37 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) ))) ; theorem :: JORDAN5D:38 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k18_euclid :::"`2"::: ) ))) ; theorem :: JORDAN5D:39 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ")" ) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) ))) ; theorem :: JORDAN5D:40 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ))) ; theorem :: JORDAN5D:41 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "I")) "," (Set (Var "i1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) "{" (Set (Var "j")) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "I")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Set (Var "j")) ")" )) ")" )) ")" ) "}" ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set ($#k6_seq_4 :::"min"::: ) (Set (Var "Y"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Set (Var "i1")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) ))))) ; theorem :: JORDAN5D:42 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "I")) "," (Set (Var "i1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) "{" (Set (Var "j")) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "h")))) & (Bool (Set (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "I")) ")" )) ")" )) ")" ) "}" ) & (Bool (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "I")) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set ($#k6_seq_4 :::"min"::: ) (Set (Var "Y"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "I")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) ))))) ; theorem :: JORDAN5D:43 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "I")) "," (Set (Var "i1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) "{" (Set (Var "j")) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "I")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "h")))) & (Bool (Set (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "I")) ")" )) ")" )) ")" ) "}" ) & (Bool (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "I")) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_2 :::"max"::: ) (Set (Var "Y"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "I")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) ))))) ; theorem :: JORDAN5D:44 (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "I")) "," (Set (Var "i1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "I"))) & (Bool (Set (Var "I")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) "{" (Set (Var "j")) where j "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "I")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Set (Var "j")) ")" )) ")" )) ")" ) "}" ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_2 :::"max"::: ) (Set (Var "Y"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "I")) "," (Set (Var "i1")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k18_euclid :::"`2"::: ) ))))) ; begin definitionlet "g" be ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::); func :::"i_s_w"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 1 (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Num 1) "," it ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," it ")" ) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"i_n_w"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 2 (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Num 1) "," it ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," it ")" ) ($#r1_hidden :::"="::: ) (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"i_s_e"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 3 (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ")" ) "," it ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ")" ) "," it ")" ) ($#r1_hidden :::"="::: ) (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"i_n_e"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 4 (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ")" ) "," it ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ")" ) "," it ")" ) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"i_w_s"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 5 (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) it "," (Num 1) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" it "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"i_e_s"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 6 (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) it "," (Num 1) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" it "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"i_w_n"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 7 (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) it "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" it "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"i_e_n"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 8 (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) it "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" it "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) "g" ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); end; :: deftheorem defines :::"i_s_w"::: JORDAN5D:def 1 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan5d :::"i_s_w"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Num 1) "," (Set (Var "b2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "b2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"i_n_w"::: JORDAN5D:def 2 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_jordan5d :::"i_n_w"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Num 1) "," (Set (Var "b2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "b2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"i_s_e"::: JORDAN5D:def 3 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_jordan5d :::"i_s_e"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ")" ) "," (Set (Var "b2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ")" ) "," (Set (Var "b2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"i_n_e"::: JORDAN5D:def 4 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_jordan5d :::"i_n_e"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ")" ) "," (Set (Var "b2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ")" ) "," (Set (Var "b2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"i_w_s"::: JORDAN5D:def 5 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_jordan5d :::"i_w_s"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "b2")) "," (Num 1) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "b2")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"i_e_s"::: JORDAN5D:def 6 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k6_jordan5d :::"i_e_s"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "b2")) "," (Num 1) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "b2")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"i_w_n"::: JORDAN5D:def 7 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k7_jordan5d :::"i_w_n"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "b2")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "b2")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"i_e_n"::: JORDAN5D:def 8 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k8_jordan5d :::"i_e_n"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "b2")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ))) & (Bool (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "b2")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "g")) ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); theorem :: JORDAN5D:45 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_jordan5d :::"i_w_n"::: ) (Set (Var "h")))) & (Bool (Set ($#k7_jordan5d :::"i_w_n"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_jordan5d :::"i_e_n"::: ) (Set (Var "h")))) & (Bool (Set ($#k8_jordan5d :::"i_e_n"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k5_jordan5d :::"i_w_s"::: ) (Set (Var "h")))) & (Bool (Set ($#k5_jordan5d :::"i_w_s"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_jordan5d :::"i_e_s"::: ) (Set (Var "h")))) & (Bool (Set ($#k6_jordan5d :::"i_e_s"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) ")" )) ; theorem :: JORDAN5D:46 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k4_jordan5d :::"i_n_e"::: ) (Set (Var "h")))) & (Bool (Set ($#k4_jordan5d :::"i_n_e"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_jordan5d :::"i_s_e"::: ) (Set (Var "h")))) & (Bool (Set ($#k3_jordan5d :::"i_s_e"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_jordan5d :::"i_n_w"::: ) (Set (Var "h")))) & (Bool (Set ($#k2_jordan5d :::"i_n_w"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_jordan5d :::"i_s_w"::: ) (Set (Var "h")))) & (Bool (Set ($#k1_jordan5d :::"i_s_w"::: ) (Set (Var "h"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "h")) ")" ))) ")" )) ; definitionlet "g" be ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::); func :::"n_s_w"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 9 (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) it) & (Bool (Set it ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "g")) & (Bool (Set "g" ($#k1_funct_1 :::"."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"n_n_w"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 10 (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) it) & (Bool (Set it ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "g")) & (Bool (Set "g" ($#k1_funct_1 :::"."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"n_s_e"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 11 (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) it) & (Bool (Set it ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "g")) & (Bool (Set "g" ($#k1_funct_1 :::"."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"n_n_e"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 12 (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) it) & (Bool (Set it ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "g")) & (Bool (Set "g" ($#k1_funct_1 :::"."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"n_w_s"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 13 (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) it) & (Bool (Set it ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "g")) & (Bool (Set "g" ($#k1_funct_1 :::"."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"n_e_s"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 14 (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) it) & (Bool (Set it ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "g")) & (Bool (Set "g" ($#k1_funct_1 :::"."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"n_w_n"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 15 (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) it) & (Bool (Set it ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "g")) & (Bool (Set "g" ($#k1_funct_1 :::"."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); func :::"n_e_n"::: "g" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: JORDAN5D:def 16 (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) it) & (Bool (Set it ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "g")) & (Bool (Set "g" ($#k1_funct_1 :::"."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "g" ")" ))) ")" ); end; :: deftheorem defines :::"n_s_w"::: JORDAN5D:def 9 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k9_jordan5d :::"n_s_w"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b2"))) & (Bool (Set (Set (Var "b2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"n_n_w"::: JORDAN5D:def 10 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k10_jordan5d :::"n_n_w"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b2"))) & (Bool (Set (Set (Var "b2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"n_s_e"::: JORDAN5D:def 11 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k11_jordan5d :::"n_s_e"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b2"))) & (Bool (Set (Set (Var "b2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"n_n_e"::: JORDAN5D:def 12 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k12_jordan5d :::"n_n_e"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b2"))) & (Bool (Set (Set (Var "b2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"n_w_s"::: JORDAN5D:def 13 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k13_jordan5d :::"n_w_s"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b2"))) & (Bool (Set (Set (Var "b2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"n_e_s"::: JORDAN5D:def 14 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k14_jordan5d :::"n_e_s"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b2"))) & (Bool (Set (Set (Var "b2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"n_w_n"::: JORDAN5D:def 15 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k15_jordan5d :::"n_w_n"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b2"))) & (Bool (Set (Set (Var "b2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); :: deftheorem defines :::"n_e_n"::: JORDAN5D:def 16 : (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k16_jordan5d :::"n_e_n"::: ) (Set (Var "g")))) "iff" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b2"))) & (Bool (Set (Set (Var "b2")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))) ")" ) ")" ))); theorem :: JORDAN5D:47 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k15_jordan5d :::"n_w_n"::: ) (Set (Var "h"))) ($#r1_hidden :::"<>"::: ) (Set ($#k13_jordan5d :::"n_w_s"::: ) (Set (Var "h"))))) ; theorem :: JORDAN5D:48 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k9_jordan5d :::"n_s_w"::: ) (Set (Var "h"))) ($#r1_hidden :::"<>"::: ) (Set ($#k11_jordan5d :::"n_s_e"::: ) (Set (Var "h"))))) ; theorem :: JORDAN5D:49 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k16_jordan5d :::"n_e_n"::: ) (Set (Var "h"))) ($#r1_hidden :::"<>"::: ) (Set ($#k14_jordan5d :::"n_e_s"::: ) (Set (Var "h"))))) ; theorem :: JORDAN5D:50 (Bool "for" (Set (Var "h")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k10_jordan5d :::"n_n_w"::: ) (Set (Var "h"))) ($#r1_hidden :::"<>"::: ) (Set ($#k12_jordan5d :::"n_n_e"::: ) (Set (Var "h"))))) ;