:: JORDAN19 semantic presentation begin definitionlet "C" be ($#m1_subset_1 :::"Simple_closed_curve":::); func :::"Upper_Appr"::: "C" -> ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) means :: JORDAN19:def 1 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" "C" "," (Set (Var "i")) ")" ")" ) ")" )))); func :::"Lower_Appr"::: "C" -> ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) means :: JORDAN19:def 2 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" "C" "," (Set (Var "i")) ")" ")" ) ")" )))); end; :: deftheorem defines :::"Upper_Appr"::: JORDAN19:def 1 : (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan19 :::"Upper_Appr"::: ) (Set (Var "C")))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" )))) ")" ))); :: deftheorem defines :::"Lower_Appr"::: JORDAN19:def 2 : (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_jordan19 :::"Lower_Appr"::: ) (Set (Var "C")))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" )))) ")" ))); definitionlet "C" be ($#m1_subset_1 :::"Simple_closed_curve":::); func :::"North_Arc"::: "C" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JORDAN19:def 3 (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set "(" ($#k1_jordan19 :::"Upper_Appr"::: ) "C" ")" )); func :::"South_Arc"::: "C" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JORDAN19:def 4 (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set "(" ($#k2_jordan19 :::"Lower_Appr"::: ) "C" ")" )); end; :: deftheorem defines :::"North_Arc"::: JORDAN19:def 3 : (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k3_jordan19 :::"North_Arc"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set "(" ($#k1_jordan19 :::"Upper_Appr"::: ) (Set (Var "C")) ")" )))); :: deftheorem defines :::"South_Arc"::: JORDAN19:def 4 : (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k4_jordan19 :::"South_Arc"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set "(" ($#k2_jordan19 :::"Lower_Appr"::: ) (Set (Var "C")) ")" )))); theorem :: JORDAN19:1 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set "(" (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "m"))))) ; theorem :: JORDAN19:2 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "E")) "being" ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "m")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ))))) ; theorem :: JORDAN19:3 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (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 ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))))) ; theorem :: JORDAN19:4 (Bool "for" (Set (Var "C")) "being" ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "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 ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))))) ; theorem :: JORDAN19:5 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Num 1) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Num 1) ")" ")" ) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Num 1) ")" ")" ) ")" ) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )))))) ; theorem :: JORDAN19:6 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )) & (Bool (Bool "not" (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ))))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "k")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ))))))) ; theorem :: JORDAN19:7 (Bool "for" (Set (Var "M")) "being" ($#v8_metric_1 :::"symmetric"::: ) ($#v9_metric_1 :::"triangle"::: ) ($#l1_metric_1 :::"MetrStruct"::: ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "x")) "," (Set (Var "r")) ")" )) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "x")) "," (Set (Var "r")) ")" ))) "holds" (Bool (Set ($#k4_metric_1 :::"dist"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r"))))))) ; theorem :: JORDAN19:8 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN19:9 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN19:10 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")))))) ; theorem :: JORDAN19:11 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#v2_connsp_1 :::"connected"::: ) ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")))))) ; theorem :: JORDAN19:12 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:13 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "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 ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:14 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:15 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "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 ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:16 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:17 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "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 ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:18 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:19 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "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 ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ) ($#r2_subset_1 :::"meets"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:20 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" )) ($#r2_subset_1 :::"meets"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:21 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" )) ($#r2_subset_1 :::"meets"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:22 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" )) ($#r2_subset_1 :::"meets"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:23 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" )) ($#r2_subset_1 :::"meets"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:24 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" )) ($#r2_subset_1 :::"meets"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:25 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "k")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "k")) ")" ")" ) ")" ")" )) ($#r2_subset_1 :::"meets"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:26 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_jordan19 :::"North_Arc"::: ) (Set (Var "C"))))) "holds" (Bool "not" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k4_jordan19 :::"South_Arc"::: ) (Set (Var "C"))))))) ; theorem :: JORDAN19:27 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_jordan19 :::"North_Arc"::: ) (Set (Var "C"))))) "holds" (Bool "not" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k4_jordan19 :::"South_Arc"::: ) (Set (Var "C"))))))) ;