:: JORDAN1F semantic presentation begin theorem :: JORDAN1F:1 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "k")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ")" ))) ")" ))))) ; theorem :: JORDAN1F:2 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "k")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ")" ))) ")" ))))) ; theorem :: JORDAN1F:3 (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "k")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "n")) "," (Set (Var "i")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "i")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ")" ))) ")" ))))) ; theorem :: JORDAN1F:4 (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "j")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "k")) "," (Set (Var "i")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "n")) "," (Set (Var "i")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set (Var "i")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "i")) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ")" ))) ")" ))))) ; theorem :: JORDAN1F:5 (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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1F:6 (Bool "for" (Set (Var "C")) "being" ($#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) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1F:7 (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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1F:8 (Bool "for" (Set (Var "C")) "being" ($#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) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1F:9 (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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool "(" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))) ")" ) "or" (Bool "(" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_jordan6 :::"Lower_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))) ")" ) ")" ))) ; theorem :: JORDAN1F:10 (Bool "for" (Set (Var "C")) "being" ($#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 ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_jordan1e :::"Upper_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )))) ; theorem :: JORDAN1F:11 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool "ex" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" )) ")" )) & (Bool "(" "for" (Set (Var "i1")) "," (Set (Var "j1")) "," (Set (Var "i2")) "," (Set (Var "j2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i1")) "," (Set (Var "j1")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i2")) "," (Set (Var "j2")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ))) "holds" (Bool (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "i2")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "i1")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "j2")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "j1")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) "holds" (Bool (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f"))) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G")))))) ; theorem :: JORDAN1F:12 (Bool "for" (Set (Var "C")) "being" ($#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 ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k2_jordan1e :::"Lower_Seq"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" )))) ; theorem :: JORDAN1F:13 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_topreal2 :::"being_simple_closed_curve"::: ) ($#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 "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 "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Num 2))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" (Set "(" ($#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) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (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) ")" ")" ) ")" ) ")" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k8_jordan6 :::"Upper_Arc"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ")" ) ")" ) ")" )))) "holds" (Bool "ex" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) "," (Set (Var "j")) ")" )) ")" ))))) ;