:: JORDAN1A semantic presentation begin definitionlet "f" be ($#m1_hidden :::"FinSequence":::); func :::"Center"::: "f" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) equals :: JORDAN1A:def 1 (Set (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) "f" ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)); end; :: deftheorem defines :::"Center"::: JORDAN1A:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "holds" (Bool (Set ($#k1_jordan1a :::"Center"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k3_nat_d :::"div"::: ) (Num 2) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)))); theorem :: JORDAN1A:1 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) "is" ($#v1_abian :::"odd"::: ) )) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set (Var "f")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1)))) ; theorem :: JORDAN1A:2 (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) "is" ($#v1_abian :::"even"::: ) )) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set (Var "f")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Num 2)))) ; begin registration cluster ($#~v1_xboole_0 "non" ($#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"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))); end; theorem :: JORDAN1A:3 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "D"))))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "D")))))) ; theorem :: JORDAN1A:4 (Bool "for" (Set (Var "D")) "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 "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "D"))))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "D")))))) ; theorem :: JORDAN1A:5 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "D"))))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "D")))))) ; theorem :: JORDAN1A:6 (Bool "for" (Set (Var "D")) "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 "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "D"))))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "D")))))) ; theorem :: JORDAN1A:7 (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "D"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "D")))) ; theorem :: JORDAN1A:8 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k6_jordan6 :::"Vertical_Line"::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" )))) ; theorem :: JORDAN1A:9 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k6_jordan6 :::"Vertical_Line"::: ) (Set (Var "r"))))) ; theorem :: JORDAN1A:10 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k6_jordan6 :::"Vertical_Line"::: ) (Set (Var "s"))))) "holds" (Bool (Set (Var "A")) "is" ($#v2_sppol_1 :::"vertical"::: ) ))) ; theorem :: JORDAN1A:11 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) "," (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_funct_2 :::"."::: ) (Set (Var "q")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set (Var "r")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" )))) ; theorem :: JORDAN1A:12 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) "," (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_funct_2 :::"."::: ) (Set (Var "q")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" )))) ; theorem :: JORDAN1A:13 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k6_jordan6 :::"Vertical_Line"::: ) (Set (Var "s")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k6_jordan6 :::"Vertical_Line"::: ) (Set (Var "s"))))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k6_jordan6 :::"Vertical_Line"::: ) (Set (Var "s")))))) ; theorem :: JORDAN1A:14 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "B")))) "holds" (Bool (Set (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "A"))) ($#r1_xboole_0 :::"meets"::: ) (Set (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "B"))))) ; theorem :: JORDAN1A:15 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "B"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k6_jordan6 :::"Vertical_Line"::: ) (Set (Var "s")))) & (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set ($#k6_jordan6 :::"Vertical_Line"::: ) (Set (Var "s"))))) "holds" (Bool (Set (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "A"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "B")))))) ; begin theorem :: JORDAN1A:16 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")) ")" ) ")" ")" ) ")" )))) ; theorem :: JORDAN1A:17 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")) ")" ) "," (Set (Var "j")) ")" ")" ) ")" )))) ; theorem :: JORDAN1A:18 (Bool "for" (Set (Var "j1")) "," (Set (Var "j2")) "," (Set (Var "i1")) "," (Set (Var "i2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Var "j1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Var "j2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: JORDAN1A:19 (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "," (Set (Var "j1")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i2"))) & (Bool (Set (Var "i2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Var "j1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j2"))) & (Bool (Set (Var "j2")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i2")) "," (Set (Var "j2")) ")" ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN1A:20 (Bool "for" (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "t")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))))) ; theorem :: JORDAN1A:21 (Bool "for" (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "t")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))))) ; theorem :: JORDAN1A:22 (Bool "for" (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "t")) "," (Num 1) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))))) ; theorem :: JORDAN1A:23 (Bool "for" (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")) ")" ) "," (Set (Var "t")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))))) ; theorem :: JORDAN1A:24 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool "not" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#v1_xboole_0 :::"empty"::: ) )))) ; theorem :: JORDAN1A:25 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#v2_connsp_1 :::"connected"::: ) ))) ; theorem :: JORDAN1A:26 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G"))))) "holds" (Bool "not" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) "is" ($#v9_rltopsp1 :::"bounded"::: ) )))) ; theorem :: JORDAN1A:27 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G"))))) "holds" (Bool "not" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")) ")" ) ")" ) "is" ($#v9_rltopsp1 :::"bounded"::: ) )))) ; begin theorem :: JORDAN1A:28 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 3))))) ; theorem :: JORDAN1A:29 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "i")) ")" ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "j")) ")" ")" ))))) ; theorem :: JORDAN1A:30 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "i")) ")" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "j")) ")" ")" ))))) ; theorem :: JORDAN1A:31 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (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 "D")) "," (Set (Var "m")) ")" ")" )))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2))) & (Bool (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ))) ")" ))) ; theorem :: JORDAN1A:32 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" )))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2))) & (Bool (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ))) ")" ))) ; theorem :: JORDAN1A:33 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (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 "D")) "," (Set (Var "m")) ")" ")" ))) & (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 "D")) "," (Set (Var "m")) ")" ")" )))) "holds" (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2))) & (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "j")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2)))) "holds" (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i1")) "," (Set (Var "j1")) ")" ))))) ; theorem :: JORDAN1A:34 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" )))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2))) & (Bool (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ))) ")" ))) ; theorem :: JORDAN1A:35 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" )))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2))) & (Bool (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ))) ")" ))) ; theorem :: JORDAN1A:36 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" ))) & (Bool "(" (Bool "(" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "i")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" ) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: JORDAN1A:37 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "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 "D")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" ))) & (Bool "(" (Bool "(" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "D")) "," (Set (Var "m")) ")" ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN1A:38 (Bool "for" (Set (Var "i")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Num 1) ")" ")" )))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Num 1) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "i")) ")" ")" ) ($#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")) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Num 2))))) ; theorem :: JORDAN1A:39 (Bool "for" (Set (Var "i")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Num 1) ")" ")" )))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Num 1) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Num 1) ")" ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Num 2))))) ; theorem :: JORDAN1A:40 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "," (Set (Var "m")) "being" ($#m2_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) ")" ) "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 "E")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN1A:41 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "i")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: JORDAN1A:42 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "," (Set (Var "m")) "being" ($#m2_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) ")" ) "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 "E")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "i")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: JORDAN1A:43 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "j")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "j")) "," (Num 1) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN1A:44 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m2_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) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ")" ) ($#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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) ")" ")" ) ")" )))) ; theorem :: JORDAN1A:45 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "j")) "being" ($#m2_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) ")" ) "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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "j")) ")" ")" ) ")" )))) ; theorem :: JORDAN1A:46 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "j")) "being" ($#m2_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) ")" ) "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 "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k1_jordan1a :::"Center"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) ")" ) ")" ")" ) ")" )))) ; theorem :: JORDAN1A:47 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_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) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" )))) "holds" (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i1"))) & (Bool (Set (Var "i1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2))) & (Bool (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "j")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j1"))) & (Bool (Set (Var "j1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "j")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2)))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i1")) "," (Set (Var "j1")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) "," (Set (Var "i")) "," (Set (Var "j")) ")" ))))) ; theorem :: JORDAN1A:48 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Num 3) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" )))) "holds" (Bool "for" (Set (Var "i1")) "," (Set (Var "j1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "i")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2))) & (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "j")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2)))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) "," (Set "(" (Set (Var "i1")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set (Var "j1")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "m")) ")" ")" ) "," (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ))))) ; theorem :: JORDAN1A:49 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set ($#k6_numbers :::"0"::: ) ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k2_jordan2c :::"UBD"::: ) (Set (Var "C")))))) ; theorem :: JORDAN1A:50 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "E")) "," (Set (Var "C")) "being" ($#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 "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "E")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k2_jordan2c :::"UBD"::: ) (Set (Var "E")))))) ; begin theorem :: JORDAN1A:51 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k4_topreal1 :::"north_halfline"::: ) (Set (Var "p"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))))) ; theorem :: JORDAN1A:52 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k5_topreal1 :::"east_halfline"::: ) (Set (Var "p"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))))) ; theorem :: JORDAN1A:53 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k6_topreal1 :::"south_halfline"::: ) (Set (Var "p"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))))) ; theorem :: JORDAN1A:54 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k7_topreal1 :::"west_halfline"::: ) (Set (Var "p"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))))) ; theorem :: JORDAN1A:55 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "ex" (Set (Var "k")) "," (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "t")) ")" )) ")" )))) ; theorem :: JORDAN1A:56 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "ex" (Set (Var "k")) "," (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "t")) "," (Num 1) ")" )) ")" )))) ; theorem :: JORDAN1A:57 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "ex" (Set (Var "k")) "," (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "t")) ")" )) ")" )))) ; theorem :: JORDAN1A:58 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "," (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "t")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ))) "holds" (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1A:59 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "," (Set (Var "t")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "t")) ")" ))) "holds" (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1A:60 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "," (Set (Var "t")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "t")) "," (Num 1) ")" ))) "holds" (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1A:61 (Bool "for" (Set (Var "k")) "," (Set (Var "n")) "," (Set (Var "t")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "t")) ")" ))) "holds" (Bool (Set (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r2_hidden :::"in"::: ) (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1A:62 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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_hidden :::"="::: ) (Set (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ))))) ; theorem :: JORDAN1A:63 (Bool "for" (Set (Var "n")) "being" ($#m2_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")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ))))) ; theorem :: JORDAN1A:64 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k13_complex1 :::"/"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ))))) ; theorem :: JORDAN1A:65 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ")" ) ")" ))))) ; theorem :: JORDAN1A:66 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 (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 "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ) ")" ) ")" ))))) ; theorem :: JORDAN1A:67 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1A:68 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1A:69 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ")" ))))) ; theorem :: JORDAN1A:70 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN1A:71 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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) ")" ) "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")) ")" ")" )))) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) "," (Set (Var "i")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: JORDAN1A:72 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: JORDAN1A:73 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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) ")" ) "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")) ")" ")" )))) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "i")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: JORDAN1A:74 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k4_topreal1 :::"north_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ))))) ; theorem :: JORDAN1A:75 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k5_topreal1 :::"east_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ))))) ; theorem :: JORDAN1A:76 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k6_topreal1 :::"south_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "x")) ($#k18_euclid :::"`2"::: ) ))))) ; theorem :: JORDAN1A:77 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k7_topreal1 :::"west_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "x")) ($#k17_euclid :::"`1"::: ) ))))) ; theorem :: JORDAN1A:78 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "C")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k4_topreal1 :::"north_halfline"::: ) (Set (Var "x")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) ")" ))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) )))) ; theorem :: JORDAN1A:79 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "C")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k5_topreal1 :::"east_halfline"::: ) (Set (Var "x")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) ")" ))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) )))) ; theorem :: JORDAN1A:80 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "C")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k6_topreal1 :::"south_halfline"::: ) (Set (Var "x")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) ")" ))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) )))) ; theorem :: JORDAN1A:81 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "C")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k7_topreal1 :::"west_halfline"::: ) (Set (Var "x")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) ")" ))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) )))) ; theorem :: JORDAN1A:82 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "C")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k4_topreal1 :::"north_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))))) ; theorem :: JORDAN1A:83 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "C")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k5_topreal1 :::"east_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))))) ; theorem :: JORDAN1A:84 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "C")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k6_topreal1 :::"south_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))))) ; theorem :: JORDAN1A:85 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "C")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k7_topreal1 :::"west_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))))) ; theorem :: JORDAN1A:86 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "C"))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Set (Set "(" ($#k4_topreal1 :::"north_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )))))) ; theorem :: JORDAN1A:87 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "C"))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Set (Set "(" ($#k5_topreal1 :::"east_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )))))) ; theorem :: JORDAN1A:88 (Bool "for" (Set (Var "n")) "being" ($#m2_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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "C"))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Set (Set "(" ($#k6_topreal1 :::"south_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )))))) ; theorem :: JORDAN1A:89 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#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 "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "C"))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Set (Set "(" ($#k7_topreal1 :::"west_halfline"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )))))) ;