:: JORDAN1C semantic presentation begin theorem :: JORDAN1C:1 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "i")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Num 1) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ")" ))))) ; theorem :: JORDAN1C:2 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ")" ))))) ; theorem :: JORDAN1C:3 (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "S")) "is" ($#v9_rltopsp1 :::"bounded"::: ) )) "holds" (Bool (Set (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "S"))) "is" ($#v5_xxreal_2 :::"real-bounded"::: ) )) ; theorem :: JORDAN1C:4 (Bool "for" (Set (Var "C1")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C2")) "," (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set (Set (Var "C1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "C2")))) & (Bool (Bool "not" (Set (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "C2"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "C2"))) "is" ($#v3_xxreal_2 :::"bounded_below"::: ) )) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_0 :::"min"::: ) "(" (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C1")) ")" ) "," (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C2")) ")" ) ")" )))) ; theorem :: JORDAN1C:5 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "X")) "is" ($#v9_rltopsp1 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "X"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "X")))) & (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "X"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "X")))) ")" ))) ; theorem :: JORDAN1C:6 (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")) ")" )))) ; theorem :: JORDAN1C:7 (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")) ")" )))) ; theorem :: JORDAN1C:8 (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")) ")" )))) ; theorem :: JORDAN1C:9 (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")) ")" )))) ; theorem :: JORDAN1C:10 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")))) & (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2) ")" ) ($#k1_int_1 :::"/]"::: ) ))) "holds" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "I"))))))) ; theorem :: JORDAN1C:11 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")))) & (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2) ")" ) ($#k1_int_1 :::"/]"::: ) ))) "holds" (Bool (Set (Set (Var "I")) ($#k3_real_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))))))) ; theorem :: JORDAN1C:12 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "J")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")))) & (Bool (Set (Var "J")) ($#r1_hidden :::"="::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2) ")" ) ($#k1_int_1 :::"/]"::: ) ))) "holds" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "J"))) & (Bool (Set (Set (Var "J")) ($#k3_real_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) ")" ))))) ; theorem :: JORDAN1C:13 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2) ")" ) ($#k1_int_1 :::"/]"::: ) ))) "holds" (Bool (Set (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "I")) ($#k5_real_1 :::"-"::: ) (Num 2) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )))))) ; theorem :: JORDAN1C:14 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2) ")" ) ($#k1_int_1 :::"/]"::: ) ))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "I")) ($#k5_real_1 :::"-"::: ) (Num 1) ")" ) ")" ))))))) ; theorem :: JORDAN1C:15 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "J")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "J")) ($#r1_hidden :::"="::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2) ")" ) ($#k1_int_1 :::"/]"::: ) ))) "holds" (Bool (Set (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "J")) ($#k5_real_1 :::"-"::: ) (Num 2) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )))))) ; theorem :: JORDAN1C:16 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "J")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Bool (Set (Var "J")) ($#r1_hidden :::"="::: ) (Set ($#k1_int_1 :::"[\"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Num 2) ")" ) ($#k1_int_1 :::"/]"::: ) ))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "C")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "C")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "J")) ($#k5_real_1 :::"-"::: ) (Num 1) ")" ) ")" ))))))) ; theorem :: JORDAN1C:17 (Bool "for" (Set (Var "C")) "being" ($#v4_pre_topc :::"closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")))) ")" )))) ; theorem :: JORDAN1C:18 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Num 1) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Set (Var "i")) ($#k3_real_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Set (Var "j")) ($#k3_real_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" )))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "r")))))))) ; theorem :: JORDAN1C:19 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")) ")" ))))) ; theorem :: JORDAN1C:20 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")) ")" ))))) ; theorem :: JORDAN1C:21 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")) ")" ))))) ; theorem :: JORDAN1C:22 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "C")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")) ")" ))))) ; theorem :: JORDAN1C:23 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C"))))) "holds" (Bool "ex" (Set (Var "n")) "," (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set (Var "j")) ")" )) & (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set (Var "C")))) ")" )))) ; theorem :: JORDAN1C:24 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "C")) "is" ($#v9_rltopsp1 :::"bounded"::: ) )) "holds" (Bool "not" (Bool (Set ($#k2_jordan2c :::"UBD"::: ) (Set (Var "C"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ;