:: JORDAN22 semantic presentation begin theorem :: JORDAN22:1 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_jordan19 :::"Upper_Appr"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ))))) ; theorem :: JORDAN22:2 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k2_jordan19 :::"Lower_Appr"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ")" ) ")" ))))) ; registrationlet "C" be ($#m1_subset_1 :::"Simple_closed_curve":::); cluster (Set ($#k8_jordan6 :::"Upper_Arc"::: ) "C") -> ($#v2_connsp_1 :::"connected"::: ) ; cluster (Set ($#k9_jordan6 :::"Lower_Arc"::: ) "C") -> ($#v2_connsp_1 :::"connected"::: ) ; end; theorem :: JORDAN22:3 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_jordan19 :::"Upper_Appr"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) "is" ($#v2_compts_1 :::"compact"::: ) ) & (Bool (Set (Set "(" ($#k1_jordan19 :::"Upper_Appr"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) "is" ($#v2_connsp_1 :::"connected"::: ) ) ")" ))) ; theorem :: JORDAN22:4 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k2_jordan19 :::"Lower_Appr"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) "is" ($#v2_compts_1 :::"compact"::: ) ) & (Bool (Set (Set "(" ($#k2_jordan19 :::"Lower_Appr"::: ) (Set (Var "C")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) "is" ($#v2_connsp_1 :::"connected"::: ) ) ")" ))) ; registrationlet "C" be ($#m1_subset_1 :::"Simple_closed_curve":::); cluster (Set ($#k3_jordan19 :::"North_Arc"::: ) "C") -> ($#v2_compts_1 :::"compact"::: ) ; cluster (Set ($#k4_jordan19 :::"South_Arc"::: ) "C") -> ($#v2_compts_1 :::"compact"::: ) ; end; theorem :: JORDAN22:5 (Bool "for" (Set (Var "R")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_domain_1 :::"["::: ) (Num 1) "," (Num 1) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "R")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN22:6 (Bool "for" (Set (Var "R")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_domain_1 :::"["::: ) (Num 1) "," (Num 2) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "R")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN22:7 (Bool "for" (Set (Var "R")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_domain_1 :::"["::: ) (Num 2) "," (Num 1) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "R")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN22:8 (Bool "for" (Set (Var "i")) "," (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#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) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k"))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" )))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" )))) ; theorem :: JORDAN22:9 (Bool "for" (Set (Var "m")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#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) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ")" )))) ; theorem :: JORDAN22:10 (Bool "for" (Set (Var "i")) "," (Set (Var "m")) "," (Set (Var "k")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k"))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" )))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" )))) ; theorem :: JORDAN22:11 (Bool "for" (Set (Var "m")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Num 1) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "k")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Num 1) ")" ")" ) ")" )))) ; theorem :: JORDAN22:12 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "," (Set (Var "t")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Num 1) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t"))) ")" ))))) ; theorem :: JORDAN22:13 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ")" ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k6_jordan6 :::"Vertical_Line"::: ) (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN22:14 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ")" ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k6_jordan6 :::"Vertical_Line"::: ) (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ) ")" ))))) ; theorem :: JORDAN22:15 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )))) ; theorem :: JORDAN22:16 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ")" ")" ) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) )))) ; theorem :: JORDAN22:17 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_jordan21 :::"UMP"::: ) (Set (Var "C")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN22:18 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k2_jordan21 :::"LMP"::: ) (Set (Var "C")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set "(" ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN22:19 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "i")) ")" )) ")" )))) ; theorem :: JORDAN22:20 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "i")) ")" )) ")" )))) ; theorem :: JORDAN22:21 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ))))) ; theorem :: JORDAN22:22 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ))))) ; theorem :: JORDAN22:23 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k1_jordan21 :::"UMP"::: ) (Set (Var "C")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN22:24 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k2_jordan21 :::"LMP"::: ) (Set (Var "C")) ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN22:25 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set (Set "(" ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN22:26 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set (Set "(" ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN22:27 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set (Set "(" ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k1_jordan21 :::"UMP"::: ) (Set "(" ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN22:28 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set (Set "(" ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k2_jordan21 :::"LMP"::: ) (Set "(" ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ")" ) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) )))) ; begin theorem :: JORDAN22:29 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_jordan19 :::"North_Arc"::: ) (Set (Var "C"))))) ; theorem :: JORDAN22:30 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_jordan19 :::"North_Arc"::: ) (Set (Var "C"))))) ; theorem :: JORDAN22:31 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_jordan19 :::"South_Arc"::: ) (Set (Var "C"))))) ; theorem :: JORDAN22:32 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_jordan19 :::"South_Arc"::: ) (Set (Var "C"))))) ; theorem :: JORDAN22:33 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k1_jordan21 :::"UMP"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k3_jordan19 :::"North_Arc"::: ) (Set (Var "C"))))) ; theorem :: JORDAN22:34 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k2_jordan21 :::"LMP"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_jordan19 :::"South_Arc"::: ) (Set (Var "C"))))) ; theorem :: JORDAN22:35 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k3_jordan19 :::"North_Arc"::: ) (Set (Var "C"))) ($#r1_tarski :::"c="::: ) (Set (Var "C")))) ; theorem :: JORDAN22:36 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k4_jordan19 :::"South_Arc"::: ) (Set (Var "C"))) ($#r1_tarski :::"c="::: ) (Set (Var "C")))) ; theorem :: JORDAN22:37 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool "(" (Bool "(" (Bool (Set ($#k2_jordan21 :::"LMP"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set (Var "C")))) & (Bool (Set ($#k1_jordan21 :::"UMP"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set (Var "C")))) ")" ) "or" (Bool "(" (Bool (Set ($#k1_jordan21 :::"UMP"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set (Var "C")))) & (Bool (Set ($#k2_jordan21 :::"LMP"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set (Var "C")))) ")" ) ")" )) ; theorem :: JORDAN22:38 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_jordan19 :::"North_Arc"::: ) (Set (Var "C")) ")" )))) ; theorem :: JORDAN22:39 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_jordan19 :::"North_Arc"::: ) (Set (Var "C")) ")" )))) ; theorem :: JORDAN22:40 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k4_jordan19 :::"South_Arc"::: ) (Set (Var "C")) ")" )))) ; theorem :: JORDAN22:41 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C"))) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k4_jordan19 :::"South_Arc"::: ) (Set (Var "C")) ")" )))) ;