:: SPRECT_2 semantic presentation begin theorem :: SPRECT_2:1 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Set (Var "i")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))))) ; theorem :: SPRECT_2:2 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "j"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))))) ; theorem :: SPRECT_2:3 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" (Set (Var "k")) ($#k2_nat_1 :::"+"::: ) (Set (Var "i")) ")" ) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )))))) ; theorem :: SPRECT_2:4 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "j"))) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))))) ; theorem :: SPRECT_2:5 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_xxreal_0 :::">="::: ) (Num 1))))) ; theorem :: SPRECT_2:6 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j")))))) ; theorem :: SPRECT_2:7 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "not" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#v1_xboole_0 :::"empty"::: ) ))))) ; theorem :: SPRECT_2:8 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))))))) ; theorem :: SPRECT_2:9 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j"))))))) ; begin theorem :: SPRECT_2:10 (Bool "for" (Set (Var "X")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "X")))))) ; theorem :: SPRECT_2:11 (Bool "for" (Set (Var "X")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "X")))))) ; theorem :: SPRECT_2:12 (Bool "for" (Set (Var "X")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "X")))))) ; theorem :: SPRECT_2:13 (Bool "for" (Set (Var "X")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "X")))))) ; begin theorem :: SPRECT_2:14 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 1) ($#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 ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "k")) ")" ")" ) where k "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool "(" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) ")" ) "}" )))) ; theorem :: SPRECT_2:15 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k1_goboard1 :::"X_axis"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:16 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" ($#k2_goboard1 :::"Y_axis"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:17 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "a")) "," (Set (Var "c")) ")" )) & (Bool (Set (Set (Var "a")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "b")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "c")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "b")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Bool "not" (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b")))) & (Bool (Bool "not" (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "c"))))) "holds" (Bool "(" (Bool (Set (Set (Var "a")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "c")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k17_euclid :::"`1"::: ) )) ")" )) ; theorem :: SPRECT_2:18 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "a")) "," (Set (Var "c")) ")" )) & (Bool (Set (Set (Var "a")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "b")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "c")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "b")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Bool "not" (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b")))) & (Bool (Bool "not" (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "c"))))) "holds" (Bool "(" (Bool (Set (Set (Var "a")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "c")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k18_euclid :::"`2"::: ) )) ")" )) ; theorem :: SPRECT_2:19 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "a")) "," (Set (Var "c")) ")" )) & (Bool (Set (Set (Var "a")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "b")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "c")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "b")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Bool "not" (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b")))) & (Bool (Bool "not" (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "c"))))) "holds" (Bool "(" (Bool (Set (Set (Var "a")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "c")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k17_euclid :::"`1"::: ) )) ")" )) ; theorem :: SPRECT_2:20 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "a")) "," (Set (Var "c")) ")" )) & (Bool (Set (Set (Var "a")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "b")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "c")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "b")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Bool "not" (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b")))) & (Bool (Bool "not" (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "c"))))) "holds" (Bool "(" (Bool (Set (Set (Var "a")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "c")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k18_euclid :::"`2"::: ) )) ")" )) ; begin definitionlet "f", "g" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); pred "g" :::"is_in_the_area_of"::: "f" means :: SPRECT_2:def 1 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "g"))) "holds" (Bool "(" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" "g" ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" "g" ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" ))) & (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" "g" ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" "g" ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" ))) ")" )); end; :: deftheorem defines :::"is_in_the_area_of"::: SPRECT_2:def 1 : (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) "iff" (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool "(" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) ($#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 "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" )) ")" )); theorem :: SPRECT_2:21 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Var "f")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) ; theorem :: SPRECT_2:22 (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")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) "holds" (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "g")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:23 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:24 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#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 "h")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) "holds" (Bool (Set (Set (Var "g")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "h"))) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) ; theorem :: SPRECT_2:25 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) ; theorem :: SPRECT_2:26 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) ; theorem :: SPRECT_2:27 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) ; theorem :: SPRECT_2:28 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f")))) ; begin definitionlet "f", "g" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); pred "g" :::"is_a_h.c._for"::: "f" means :: SPRECT_2:def 2 (Bool "(" (Bool "g" ($#r1_sprect_2 :::"is_in_the_area_of"::: ) "f") & (Bool (Set (Set "(" "g" ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" ))) & (Bool (Set (Set "(" "g" ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) "g" ")" ) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" ))) ")" ); pred "g" :::"is_a_v.c._for"::: "f" means :: SPRECT_2:def 3 (Bool "(" (Bool "g" ($#r1_sprect_2 :::"is_in_the_area_of"::: ) "f") & (Bool (Set (Set "(" "g" ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" ))) & (Bool (Set (Set "(" "g" ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) "g" ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" ))) ")" ); end; :: deftheorem defines :::"is_a_h.c._for"::: SPRECT_2:def 2 : (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r2_sprect_2 :::"is_a_h.c._for"::: ) (Set (Var "f"))) "iff" (Bool "(" (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ) ")" )); :: deftheorem defines :::"is_a_v.c._for"::: SPRECT_2:def 3 : (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r3_sprect_2 :::"is_a_v.c._for"::: ) (Set (Var "f"))) "iff" (Bool "(" (Bool (Set (Var "g")) ($#r1_sprect_2 :::"is_in_the_area_of"::: ) (Set (Var "f"))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" ) ")" )); theorem :: SPRECT_2:29 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "g")) "," (Set (Var "h")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#v1_topreal1 :::"special"::: ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")))) & (Bool (Num 2) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")))) & (Bool (Set (Var "g")) ($#r2_sprect_2 :::"is_a_h.c._for"::: ) (Set (Var "f"))) & (Bool (Set (Var "h")) ($#r3_sprect_2 :::"is_a_v.c._for"::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "g"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "h")))))) ; begin definitionlet "f" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); attr "f" is :::"clockwise_oriented"::: means :: SPRECT_2:def 4 (Bool (Set (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" "f" "," (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" ) ")" ) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) "f" ")" ))); end; :: deftheorem defines :::"clockwise_oriented"::: SPRECT_2:def 4 : (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_sprect_2 :::"clockwise_oriented"::: ) ) "iff" (Bool (Set (Set "(" ($#k1_finseq_6 :::"Rotate"::: ) "(" (Set (Var "f")) "," (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" )); theorem :: SPRECT_2:30 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_sprect_2 :::"clockwise_oriented"::: ) ) "iff" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 2)) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) ")" )) ; registration cluster (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) ) -> ($#v2_compts_1 :::"compact"::: ) ; end; theorem :: SPRECT_2:31 (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) ; theorem :: SPRECT_2:32 (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ; theorem :: SPRECT_2:33 (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) ; theorem :: SPRECT_2:34 (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ; theorem :: SPRECT_2:35 (Bool (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Num 1) "," (Num 1) ($#k19_euclid :::"]|"::: ) ) ")" )) ; theorem :: SPRECT_2:36 (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set ($#k1_topreal1 :::"R^2-unit_square"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Num 1) ($#k19_euclid :::"]|"::: ) )) ; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#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) ")" ); cluster (Set ($#k1_sprect_1 :::"SpStSeq"::: ) "X") -> ($#v1_sprect_2 :::"clockwise_oriented"::: ) ; end; 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 :::"carrier"::: ) "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"::: ) ($#v1_finseq_6 :::"circular"::: ) ($#v1_topreal1 :::"special"::: ) ($#v2_topreal1 :::"unfolded"::: ) ($#v1_goboard5 :::"s.c.c."::: ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) for ($#m1_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )); end; theorem :: SPRECT_2:37 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "j"))) & (Bool "(" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ) "or" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ) ")" )) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; theorem :: SPRECT_2:38 (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 "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) & (Bool "(" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ) "or" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" ) ")" )) "holds" (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; theorem :: SPRECT_2:39 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:40 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:41 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:42 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:43 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:44 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:45 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:46 (Bool "for" (Set (Var "f")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:47 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_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 (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) "or" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#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")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "m")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: SPRECT_2:48 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_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 (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) "or" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#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")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "m")) ")" ")" ))))) ; theorem :: SPRECT_2:49 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_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 (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) "or" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" )) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) "," (Set (Var "i")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "n")) "," (Set (Var "m")) ")" ")" ))))) ; theorem :: SPRECT_2:50 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_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 (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) & (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) "or" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))) ")" )) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) "," (Set (Var "i")) ")" ")" )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "m")) "," (Set (Var "n")) ")" ")" ))))) ; theorem :: SPRECT_2:51 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) ))) ; theorem :: SPRECT_2:52 (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 ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) ; theorem :: SPRECT_2:53 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ))) ; theorem :: SPRECT_2:54 (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 ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) ; theorem :: SPRECT_2:55 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) ))) ; theorem :: SPRECT_2:56 (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 ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) ; theorem :: SPRECT_2:57 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) ))) ; theorem :: SPRECT_2:58 (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 ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) ; theorem :: SPRECT_2:59 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ")" ) ($#r1_subset_1 :::"misses"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ")" ))) ; theorem :: SPRECT_2:60 (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 "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1))) & (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k17_euclid :::"`1"::: ) )) "or" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set (Var "f"))) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; theorem :: SPRECT_2:61 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k17_euclid :::"`1"::: ) )) "or" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#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")) ")" ) ")" ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "p")) ($#k12_finseq_1 :::"*>"::: ) )) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; begin theorem :: SPRECT_2:62 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k12_pscomp_1 :::"NE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; theorem :: SPRECT_2:63 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k12_pscomp_1 :::"NE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; theorem :: SPRECT_2:64 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k13_pscomp_1 :::"SE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; theorem :: SPRECT_2:65 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k12_pscomp_1 :::"NE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) )) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; theorem :: SPRECT_2:66 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k11_pscomp_1 :::"NW-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; theorem :: SPRECT_2:67 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k10_pscomp_1 :::"SW-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k12_finseq_1 :::"*>"::: ) ) ($#k8_finseq_1 :::"^"::: ) (Set "(" ($#k3_finseq_6 :::"mid"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) "," (Set (Var "j")) ")" ")" )) "is" ($#m2_finseq_1 :::"S-Sequence_in_R2":::)))) ; registrationlet "f" be ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::); cluster (Set ($#k3_topreal1 :::"L~"::: ) "f") -> ($#v1_topreal2 :::"being_simple_closed_curve"::: ) ; end; begin theorem :: SPRECT_2:68 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))))) ; theorem :: SPRECT_2:69 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (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 (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">"::: ) (Num 1))) ; theorem :: SPRECT_2:70 (Bool "for" (Set (Var "z")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Set (Var "z")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ))) & (Bool (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )))) "holds" (Bool (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))))) ; theorem :: SPRECT_2:71 (Bool "for" (Set (Var "z")) "being" ($#~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 "z")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )))) "holds" (Bool (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))))) ; theorem :: SPRECT_2:72 (Bool "for" (Set (Var "z")) "being" ($#~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 "z")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ))) & (Bool (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )))) "holds" (Bool (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))))) ; theorem :: SPRECT_2:73 (Bool "for" (Set (Var "z")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Set (Var "z")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )))) "holds" (Bool (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))))) ; theorem :: SPRECT_2:74 (Bool "for" (Set (Var "z")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Set (Var "z")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ))) & (Bool (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )))) "holds" (Bool (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))))) ; theorem :: SPRECT_2:75 (Bool "for" (Set (Var "z")) "being" ($#~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 "z")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ))) & (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )))) "holds" (Bool (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))))) ; theorem :: SPRECT_2:76 (Bool "for" (Set (Var "z")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#v1_sprect_2 :::"clockwise_oriented"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (Set (Var "z")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" )))) "holds" (Bool (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "z")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "z"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "z"))))) ; theorem :: SPRECT_2:77 (Bool "for" (Set (Var "f")) "being" ($#~v3_funct_1 "non" ($#v3_funct_1 :::"constant"::: ) ) ($#v2_goboard5 :::"standard"::: ) ($#m2_finseq_1 :::"special_circular_sequence":::) "st" (Bool (Bool (Set (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 (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" ) ")" ) ($#k4_finseq_4 :::".."::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) ;