:: GOBRD14 semantic presentation begin theorem :: GOBRD14:1 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#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")) ($#r1_hidden :::"="::: ) (Set ($#k16_euclid :::"0.REAL"::: ) (Num 2))) & (Bool (Set (Var "P")) ($#r2_jordan2c :::"is_outside_component_of"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ")" ) ($#k3_subset_1 :::"`"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "P"))) ")" ))))) ; begin theorem :: GOBRD14:2 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))))) ; theorem :: GOBRD14:3 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_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 ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set "(" "(" (Num 1) "," (Num 2) ")" ($#k5_funct_4 :::"-->"::: ) "(" (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Num 1) ")" ")" ) ($#k17_euclid :::"`1"::: ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) "," (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ")" ) "," (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ")" ")" ))))) ; theorem :: GOBRD14:4 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#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 ($#k1_matrix_1 :::"width"::: ) (Set (Var "G"))))) "holds" (Bool (Set ($#k3_goboard5 :::"cell"::: ) "(" (Set (Var "G")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#v2_compts_1 :::"compact"::: ) ))) ; theorem :: GOBRD14:5 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G"))))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")) ")" ) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ")" ))))) ; theorem :: GOBRD14:6 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set (Var "j")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G")))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set (Var "G"))))) "holds" (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Set (Var "n")) ")" ) ")" ")" ) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ")" ))))) ; theorem :: GOBRD14:7 (Bool "for" (Set (Var "n")) "being" ($#m1_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) ")" ) "holds" (Bool (Num 3) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1))))) ; theorem :: GOBRD14:8 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#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 (Var "j")))) "holds" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1))) & (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool "ex" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "c"))) & (Bool (Set (Var "c")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1))) & (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "d"))) & (Bool (Set (Var "d")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "c")) "," (Set (Var "d")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ))) & (Bool (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "a")) "," (Set (Var "b")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "j")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" )) & (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "j")) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k7_nat_d :::"-'"::: ) (Num 2) ")" ) ")" ))) & (Bool (Set (Var "d")) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k2_nat_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k13_newton :::"|^"::: ) (Set "(" (Set (Var "j")) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ")" ) ($#k4_nat_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k7_nat_d :::"-'"::: ) (Num 2) ")" ) ")" ))) ")" ))))) ; theorem :: GOBRD14:9 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#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 ($#k4_tarski :::"["::: ) (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_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")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#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")) ")" ))))) ; theorem :: GOBRD14:10 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#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 ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "i")) ($#k2_nat_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")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#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")) ")" ))))) ; theorem :: GOBRD14:11 (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 "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "," (Set (Var "t")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "n"))) & (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k1_topreal6 :::"dist"::: ) "(" (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) "," (Set "(" (Set "(" ($#k1_jordan8 :::"Gauge"::: ) "(" (Set (Var "C")) "," (Set (Var "n")) ")" ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Num 2) "," (Num 1) ")" ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t"))) ")" )))) ; begin theorem :: GOBRD14:12 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "P")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) "or" (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) "or" (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) ")" ))) ; theorem :: GOBRD14:13 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "A1")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "A2")))) & (Bool (Set (Var "A1")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "A2"))) & (Bool "(" "for" (Set (Var "C1")) "," (Set (Var "C2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k1_pre_topc :::"|"::: ) (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) ")" ) "st" (Bool (Bool (Set (Var "C1")) ($#r1_hidden :::"="::: ) (Set (Var "A1"))) & (Bool (Set (Var "C2")) ($#r1_hidden :::"="::: ) (Set (Var "A2")))) "holds" (Bool "(" (Bool (Set (Var "C1")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) & (Bool (Set (Var "C2")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) ")" ) ")" ) & (Bool "not" (Bool "(" (Bool (Set (Var "A1")) ($#r1_hidden :::"="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) & (Bool (Set (Var "A2")) ($#r1_hidden :::"="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) ")" ))) "holds" (Bool "(" (Bool (Set (Var "A1")) ($#r1_hidden :::"="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) & (Bool (Set (Var "A2")) ($#r1_hidden :::"="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) ")" ))) ; theorem :: GOBRD14:14 (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 ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f"))) ($#r1_subset_1 :::"misses"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) ; theorem :: GOBRD14:15 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set (Set "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")) ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )))) ; theorem :: GOBRD14:16 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f"))))) & (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) ")" ) ")" ))) ; theorem :: GOBRD14:17 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) ")" ) ")" ))) ; theorem :: GOBRD14:18 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f"))))) ")" ) ")" ))) ; theorem :: GOBRD14:19 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")) ")" )))) ; theorem :: GOBRD14:20 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")) ")" )))) ; theorem :: GOBRD14:21 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) ; theorem :: GOBRD14:22 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) ; registrationlet "f" be ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::); cluster (Set ($#k3_topreal1 :::"L~"::: ) "f") -> ($#v1_jordan1 :::"Jordan"::: ) ; end; theorem :: GOBRD14:23 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: GOBRD14:24 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: GOBRD14:25 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: GOBRD14:26 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: GOBRD14:27 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: GOBRD14:28 (Bool "for" (Set (Var "C")) "being" ($#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) ")" ) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set (Var "C")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set "(" "(" (Num 1) "," (Num 2) ")" ($#k5_funct_4 :::"-->"::: ) "(" (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) "," (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) "," (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) "," (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set (Var "C")) ")" ) ")" ) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ")" ")" )))) ; theorem :: GOBRD14:29 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) ; theorem :: GOBRD14:30 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) ; theorem :: GOBRD14:31 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "g"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) ")" )))) ; theorem :: GOBRD14:32 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "g")) ")" )) "is" ($#v2_compts_1 :::"compact"::: ) )) ; theorem :: GOBRD14:33 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Bool "not" (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "g"))) "is" ($#v9_rltopsp1 :::"bounded"::: ) ))) ; theorem :: GOBRD14:34 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "g"))) ($#r2_jordan2c :::"is_outside_component_of"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))))) ; theorem :: GOBRD14:35 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "g"))) ($#r1_jordan2c :::"is_inside_component_of"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))))) ; theorem :: GOBRD14:36 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k2_jordan2c :::"UBD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "g"))))) ; theorem :: GOBRD14:37 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "g"))))) ; theorem :: GOBRD14:38 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r2_jordan2c :::"is_outside_component_of"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "g")))))) ; theorem :: GOBRD14:39 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_jordan2c :::"is_inside_component_of"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "P")) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "g")))))) ; theorem :: GOBRD14:40 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_jordan2c :::"is_inside_component_of"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k1_jordan2c :::"BDD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ))))) ; theorem :: GOBRD14:41 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "g")) ")" )))) ; theorem :: GOBRD14:42 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "g")) ")" )))) ; theorem :: GOBRD14:43 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "g")) ")" )))) ; theorem :: GOBRD14:44 (Bool "for" (Set (Var "g")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "g")) ")" )))) ;