:: JORDAN14 semantic presentation begin theorem :: JORDAN14:1 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) "or" (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) ")" )) ; theorem :: JORDAN14:2 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool (Set ($#k2_jordan2c :::"UBD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) "or" (Bool (Set ($#k2_jordan2c :::"UBD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) ")" )) ; theorem :: JORDAN14:3 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G")))) "holds" (Bool (Set ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) "," (Set (Var "G")) ")" ) "is" ($#v4_pre_topc :::"closed"::: ) )))) ; theorem :: JORDAN14:4 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ))) "iff" (Bool "(" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" )))) ; theorem :: JORDAN14:5 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) "is" ($#v2_connsp_1 :::"connected"::: ) )) ; registrationlet "f" be ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::); cluster (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" )) -> ($#v2_connsp_1 :::"connected"::: ) ; end; definitionlet "C" be ($#m1_subset_1 :::"Simple_closed_curve":::); let "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"SpanStart"::: "(" "C" "," "n" ")" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: JORDAN14:def 1 (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" "C" "," "n" ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_jordan1h :::"X-SpanStart"::: ) "(" "C" "," "n" ")" ")" ) "," (Set "(" ($#k3_jordan11 :::"Y-SpanStart"::: ) "(" "C" "," "n" ")" ")" ) ")" ); end; :: deftheorem defines :::"SpanStart"::: JORDAN14:def 1 : (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_jordan14 :::"SpanStart"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" ($#k3_jordan1h :::"X-SpanStart"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set "(" ($#k3_jordan11 :::"Y-SpanStart"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )))); theorem :: JORDAN14:6 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k1_jordan14 :::"SpanStart"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")))))) ; theorem :: JORDAN14:7 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool "(" (Bool (Set ($#k2_gobrd13 :::"right_cell"::: ) "(" (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "C"))) & (Bool (Set ($#k3_gobrd13 :::"left_cell"::: ) "(" (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "k")) "," (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "C"))) ")" ))) ; theorem :: JORDAN14:8 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "C")) ($#r1_subset_1 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; registrationlet "C" be ($#m1_subset_1 :::"Simple_closed_curve":::); let "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" "C" "," "n" ")" ")" ) ")" )) -> ($#v2_compts_1 :::"compact"::: ) ; end; theorem :: JORDAN14:9 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "C")) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:10 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "C")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:11 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:12 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set ($#k2_jordan2c :::"UBD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN14:13 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")))))) ; theorem :: JORDAN14:14 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k2_jordan2c :::"UBD"::: ) (Set (Var "C"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_jordan2c :::"UBD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN14:15 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")))))) ; theorem :: JORDAN14:16 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k2_jordan2c :::"UBD"::: ) (Set (Var "C"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:17 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k2_jordan2c :::"UBD"::: ) (Set (Var "C"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN14:18 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k2_jordan2c :::"UBD"::: ) (Set (Var "C"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:19 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Set (Var "P")) ($#r2_jordan2c :::"is_outside_component_of"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))))) ; theorem :: JORDAN14:20 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k2_jordan2c :::"UBD"::: ) (Set (Var "C"))) ($#r1_subset_1 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:21 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")))))) ; theorem :: JORDAN14:22 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (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 "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">"::: ) (Num 1)))) ; theorem :: JORDAN14:23 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:24 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (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 "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Var "j")) ($#r1_xxreal_0 :::">"::: ) (Num 1)))) ; theorem :: JORDAN14:25 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ))) "holds" (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:26 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k3_jordan11 :::"Y-SpanStart"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:27 (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set ($#k3_jordan1h :::"X-SpanStart"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "m")) ($#k7_nat_d :::"-'"::: ) (Set (Var "n")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" ($#k3_jordan1h :::"X-SpanStart"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2))))) ; theorem :: JORDAN14:28 (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "m")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))))) ; theorem :: JORDAN14:29 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) ) & (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 (Set "(" ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) "is" ($#v2_connsp_1 :::"connected"::: ) )))) ; theorem :: JORDAN14:30 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Set ($#k3_jordan11 :::"Y-SpanStart"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set "(" ($#k1_jordan11 :::"ApproxIndex"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" (Set "(" ($#k2_jordan11 :::"Y-InitStart"::: ) (Set (Var "C")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 2) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 2)))) "holds" (Bool (Set (Set "(" ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set "(" (Set "(" ($#k3_jordan1h :::"X-SpanStart"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set (Var "k")) ")" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN14:31 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#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 "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" )))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Num 2) ($#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) ")" ) ($#k7_real_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: JORDAN14:32 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ))))) ; theorem :: JORDAN14:33 (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_goboard1 :::"is_sequence_on"::: ) (Set (Var "G"))) & (Bool (Set (Var "f")) "is" ($#v1_topreal1 :::"special"::: ) )) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) ))))) ; theorem :: JORDAN14:34 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k2_goboard9 :::"LeftComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ))))) ; theorem :: JORDAN14:35 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" ))))) ; theorem :: JORDAN14:36 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_jordan1h :::"is_sufficiently_large_for"::: ) (Set (Var "C"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "m")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set "(" ($#k1_jordan13 :::"Span"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))) ;