:: SPRECT_3 semantic presentation begin theorem :: SPRECT_3:1 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "holds" (Bool (Set (Set "(" (Set (Var "g")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "g")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))))) ; theorem :: SPRECT_3:2 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_matrix_1 :::"Indices"::: ) (Set "(" "(" (Set (Var "a")) "," (Set (Var "b")) ")" ($#k3_matrix_2 :::"]["::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_enumset1 :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Num 1) "," (Num 1) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Num 1) "," (Num 2) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Num 2) "," (Num 1) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Num 2) "," (Num 2) ($#k4_tarski :::"]"::: ) ) ($#k2_enumset1 :::"}"::: ) ))) ; begin theorem :: SPRECT_3:3 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_xboole_0 :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "q")) ")" )))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q")))))) ; theorem :: SPRECT_3:4 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "q")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p")) ")" )))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q")))))) ; theorem :: SPRECT_3:5 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set (Var "p")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "q")) ")" )))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q"))))) ; theorem :: SPRECT_3:6 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" )) & (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ))) "holds" (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Var "r"))))) ; begin theorem :: SPRECT_3:7 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ))) "holds" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) )))) ; theorem :: SPRECT_3:8 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "B")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) )) ; theorem :: SPRECT_3:9 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "r")) "," (Set (Var "s")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "r")) "," (Set (Var "s")) ")" ))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k18_euclid :::"`2"::: ) ))) ; theorem :: SPRECT_3:10 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q")) "," (Set (Var "r")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) )) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q")) "," (Set (Var "r")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: SPRECT_3:11 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ) "is" ($#v1_sppol_1 :::"horizontal"::: ) ) & (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "s")) "," (Set (Var "r")) ")" ) "is" ($#v2_sppol_1 :::"vertical"::: ) ) & (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "s")) "," (Set (Var "r")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ))) ; begin theorem :: SPRECT_3:12 (Bool "for" (Set (Var "j")) "," (Set (Var "k")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "k")) ")" ")" ) ($#k18_euclid :::"`2"::: ) )))) ; theorem :: SPRECT_3:13 (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "," (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m2_finseq_1 :::"Go-board":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set (Var "G")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "G"))))) "holds" (Bool (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "G")) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "k")) "," (Set (Var "j")) ")" ")" ) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: SPRECT_3:14 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "C")) ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "C")) ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set (Var "C")) ")" )))) ; theorem :: SPRECT_3:15 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "C"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "C")) ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "C")) ")" ) ")" ))) ; registrationlet "C" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); cluster (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) "C" ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) "C" ")" ) ")" ) -> ($#v1_sppol_1 :::"horizontal"::: ) ; end; theorem :: SPRECT_3:16 (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"<>"::: ) (Set (Var "p"))) & (Bool "(" (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "or" (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) & (Bool (Set (Var "g")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set ($#k4_matrix_2 :::"<*"::: ) (Set (Var "p")) ($#k4_matrix_2 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g"))) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ))) ; theorem :: SPRECT_3:17 (Bool "for" (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"S-Sequence_in_R2":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Num 1) "," (Set (Var "j")) ")" ")" )))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "p")) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j"))) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))))) ; theorem :: SPRECT_3:18 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "h")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "h")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))))) ; theorem :: SPRECT_3:19 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j")))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set (Var "j")) ")" ")" ) ")" ))))) ; theorem :: SPRECT_3:20 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set "(" (Set (Var "j")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "j")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ")" ))))) ; theorem :: SPRECT_3:21 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "g")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool "(" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k17_euclid :::"`1"::: ) )) "or" (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) & (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "g"))) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) )) ; theorem :: SPRECT_3:22 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k3_jordan3 :::"R_Cut"::: ) "(" (Set (Var "f")) "," (Set (Var "p")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1))))) ; theorem :: SPRECT_3:23 (Bool "for" (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"S-Sequence_in_R2":::) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) ")" )) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")) ")" ) "," (Set (Var "p")) ")" ))) "holds" (Bool ($#r1_jordan5c :::"LE"::: ) (Set (Var "q")) "," (Set (Var "p")) "," (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) "," (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )))))) ; begin theorem :: SPRECT_3:24 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f"))) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))) "is" ($#v3_pre_topc :::"open"::: ) ) ")" )) ; registrationlet "f" be ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::); cluster (Set ($#k3_topreal1 :::"L~"::: ) "f") -> ($#~v1_sppol_1 "non" ($#v1_sppol_1 :::"horizontal"::: ) ) ($#~v2_sppol_1 "non" ($#v2_sppol_1 :::"vertical"::: ) ) ; cluster (Set ($#k2_goboard9 :::"LeftComp"::: ) "f") -> ($#v2_topreal4 :::"being_Region"::: ) ; cluster (Set ($#k3_goboard9 :::"RightComp"::: ) "f") -> ($#v2_topreal4 :::"being_Region"::: ) ; end; theorem :: SPRECT_3:25 (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 ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))) ($#r1_subset_1 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_3:26 (Bool "for" (Set (Var "f")) "being" ($#~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_topreal1 :::"L~"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_3:27 (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 ($#k7_jordan5d :::"i_w_n"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k8_jordan5d :::"i_e_n"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_3:28 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "ex" (Set (Var "i")) "being" ($#m2_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 "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" )) ")" ))) ; theorem :: SPRECT_3:29 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) "," (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ")" )) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set "(" (Set "(" ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" )) ")" ))) ; theorem :: SPRECT_3:30 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ))) & (Bool (Bool "not" (Set (Var "i")) ($#r1_hidden :::"="::: ) (Num 1)))) "holds" (Bool (Set (Var "j")) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))))) ; theorem :: SPRECT_3:31 (Bool "for" (Set (Var "f")) "being" ($#~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 (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2) ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" )))) ; begin theorem :: SPRECT_3:32 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" "not" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) "or" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) "or" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) "or" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) "or" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ))) ; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set ($#k5_numbers :::"NAT"::: ) )) bbbadV5_RELAT_1((Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) bbbadV1_FINSEQ_6((Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ($#v1_topreal1 :::"special"::: ) ($#v2_topreal1 :::"unfolded"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#v1_sprect_1 :::"rectangular"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )); end; theorem :: SPRECT_3:33 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"S-Sequence_in_R2":::) "st" (Bool (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r2_subset_1 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")))))) ; theorem :: SPRECT_3:34 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_sprect_1 :::"SpStSeq"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "f")))) ; theorem :: SPRECT_3:35 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ) ")" ) "}" )) ; theorem :: SPRECT_3:36 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 4) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ($#k6_matrix_2 :::"]["::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 3) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2) ")" ) ")" ))) ; theorem :: SPRECT_3:37 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool "(" (Bool (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "p")) where p "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" "not" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) "or" "not" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "or" "not" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ) "}" ) & (Bool (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "q")) where q "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ) "}" ) ")" )) ; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) bbbadV1_RELAT_1() bbbadV4_RELAT_1((Set ($#k5_numbers :::"NAT"::: ) )) bbbadV5_RELAT_1((Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ($#v1_funct_1 :::"Function-like"::: ) ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) bbbadV1_FINSET_1() ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) bbbadV1_FINSEQ_6((Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ($#v1_topreal1 :::"special"::: ) ($#v2_topreal1 :::"unfolded"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_1 :::"rectangular"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )); end; registration cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_FINSEQ_6((Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ))) ($#v1_topreal1 :::"special"::: ) ($#v2_topreal1 :::"unfolded"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#v1_sprect_1 :::"rectangular"::: ) -> ($#v1_sprect_1 :::"rectangular"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )); end; theorem :: SPRECT_3:38 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"S-Sequence_in_R2":::) "st" (Bool (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ) ")" ) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k11_pscomp_1 :::"NW-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))))) ; theorem :: SPRECT_3:39 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"S-Sequence_in_R2":::) "st" (Bool (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ) ")" ) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k13_pscomp_1 :::"SE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))))) ; theorem :: SPRECT_3:40 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "(" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) "or" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) "or" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) "or" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" )) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))))) ; theorem :: SPRECT_3:41 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f"))))) ; begin theorem :: SPRECT_3:42 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_finseq_4 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) "iff" (Bool "(" (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set (Var "p")) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set (Var "q")) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) ")" ) ")" ))) ; theorem :: SPRECT_3:43 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set (Var "p")) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) "or" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) "or" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) "or" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" )) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: SPRECT_3:44 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k1_xboole_0 :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1)) & (Bool (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_finseq_4 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set "(" (Set "(" (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "q")) ")" ) ")" ) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))))) ; theorem :: SPRECT_3:45 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_3:46 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set (Var "p")) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_3:47 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_finseq_4 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: SPRECT_3:48 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))) & (Bool (Bool "not" (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))) & (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set (Var "q")) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: SPRECT_3:49 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) ($#k3_matrix_1 :::"*"::: ) "(" (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_3:50 (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_finseq_4 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "g")))) "holds" (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set (Var "p")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "q")) ")" ) ")" ) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "g"))))) ; theorem :: SPRECT_3:51 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "g"))) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) ; theorem :: SPRECT_3:52 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set (Var "p")) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool (Set (Var "g")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k3_jordan3 :::"R_Cut"::: ) "(" (Set (Var "g")) "," (Set (Var "p")) ")" ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_3:53 (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 "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) "iff" (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set ($#k1_sprect_1 :::"SpStSeq"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ))) ; theorem :: SPRECT_3:54 (Bool "for" (Set (Var "f")) "being" ($#v1_sprect_1 :::"rectangular"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "g")) "being" ($#m2_finseq_1 :::"S-Sequence_in_R2":::) "st" (Bool (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k2_goboard9 :::"LeftComp"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k3_goboard9 :::"RightComp"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set (Var "g")) "," (Set "(" ($#k2_jordan5c :::"Last_Point"::: ) "(" (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "g")) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ) ")" ) "," (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ")" ) ")" ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_3:55 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_matrix_1 :::"width"::: ) (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k3_goboard5 :::"cell"::: ) "(" (Set "(" ($#k2_goboard2 :::"GoB"::: ) (Set (Var "f")) ")" ) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_sprect_1 :::"SpStSeq"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ))))) ; theorem :: SPRECT_3:56 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool (Set ($#k4_matrix_2 :::"<*"::: ) (Set (Var "p")) ($#k4_matrix_2 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool (Set (Var "g")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k2_jordan3 :::"L_Cut"::: ) "(" (Set (Var "g")) "," (Set (Var "p")) ")" ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))))) ;