:: TOPREAL3 semantic presentation begin begin theorem :: TOPREAL3:1 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Num 1) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ))) & (Bool (Num 2) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ))) & (Bool (Num 3) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) ($#k11_finseq_1 :::"*>"::: ) ))) ")" )) ; theorem :: TOPREAL3:2 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "p1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "p2")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ))) & (Bool (Set (Set "(" (Set (Var "p1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "p2")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ))) ")" )) ; theorem :: TOPREAL3:3 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "p1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ))) & (Bool (Set (Set "(" (Set (Var "p1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ))) ")" )) ; theorem :: TOPREAL3:4 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k4_real_1 :::"*"::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ))) & (Bool (Set (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "p")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k4_real_1 :::"*"::: ) (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ))) ")" ))) ; theorem :: TOPREAL3:5 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r1")) "," (Set (Var "s1")) "," (Set (Var "r2")) "," (Set (Var "s2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "r1")) "," (Set (Var "s1")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "r2")) "," (Set (Var "s2")) ($#k10_finseq_1 :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "p1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "r1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r2")) ")" ) "," (Set "(" (Set (Var "s1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s2")) ")" ) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p2"))) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "r1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r2")) ")" ) "," (Set "(" (Set (Var "s1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s2")) ")" ) ($#k10_finseq_1 :::"*>"::: ) )) ")" ))) ; theorem :: TOPREAL3:6 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) ; theorem :: TOPREAL3:7 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "u1")) "," (Set (Var "u2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "u1")) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Var "u2")) ($#r1_hidden :::"="::: ) (Set (Var "p2")))) "holds" (Bool (Set (Set "(" ($#k13_euclid :::"Pitag_dist"::: ) (Num 2) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "u1")) "," (Set (Var "u2")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k7_square_1 :::"sqrt"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))))) ; theorem :: TOPREAL3:8 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" )))) ; theorem :: TOPREAL3:9 (Bool "for" (Set (Var "r1")) "," (Set (Var "s1")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s1")))) "holds" (Bool "{" (Set (Var "p1")) where p1 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s1"))) ")" ) "}" ($#r1_hidden :::"="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set (Var "r1")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set (Var "s1")) ($#k19_euclid :::"]|"::: ) ) ")" ))) ; theorem :: TOPREAL3:10 (Bool "for" (Set (Var "r1")) "," (Set (Var "s1")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s1")))) "holds" (Bool "{" (Set (Var "p1")) where p1 "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) : (Bool "(" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s1"))) ")" ) "}" ($#r1_hidden :::"="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "r")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "s1")) "," (Set (Var "r")) ($#k19_euclid :::"]|"::: ) ) ")" ))) ; theorem :: TOPREAL3:11 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "," (Set (Var "r1")) "," (Set (Var "s1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set (Var "r1")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set (Var "s1")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "r"))))) ; theorem :: TOPREAL3:12 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r1")) "," (Set (Var "r")) "," (Set (Var "s1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "r")) ($#k19_euclid :::"]|"::: ) ) "," (Set ($#k19_euclid :::"|["::: ) (Set (Var "s1")) "," (Set (Var "r")) ($#k19_euclid :::"]|"::: ) ) ")" ))) "holds" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "r"))))) ; theorem :: TOPREAL3:13 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ))) ; theorem :: TOPREAL3:14 (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 (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) ")" ))) ; theorem :: TOPREAL3:15 (Bool "for" (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p")) "," (Set (Var "p1")) "," (Set (Var "q")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::">"::: ) (Set (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1)))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))) ; theorem :: TOPREAL3:16 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p3")) ($#k11_finseq_1 :::"*>"::: ) ))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p2")) "," (Set (Var "p3")) ")" ")" ))))) ; theorem :: TOPREAL3:17 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" ))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" ) "," (Set (Var "j")) ")" )))) ; theorem :: TOPREAL3:18 (Bool "for" (Set (Var "f")) "," (Set (Var "h")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k8_finseq_1 :::"^"::: ) (Set (Var "h")) ")" ) "," (Set (Var "j")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "j")) ")" )))) ; theorem :: TOPREAL3:19 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: TOPREAL3:20 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) ; theorem :: TOPREAL3:21 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p2")) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )))))) ; theorem :: TOPREAL3:22 (Bool "for" (Set (Var "p1")) "," (Set (Var "p2")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r1")) "," (Set (Var "s1")) "," (Set (Var "r2")) "," (Set (Var "s2")) "," (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Var "p1"))) & (Bool (Set (Var "p1")) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "s1")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Var "p2")) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "r2")) "," (Set (Var "s2")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "r2")) "," (Set (Var "s1")) ($#k19_euclid :::"]|"::: ) )) & (Bool (Set (Var "p2")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))))) ; theorem :: TOPREAL3:23 (Bool "for" (Set (Var "s")) "," (Set (Var "r1")) "," (Set (Var "r")) "," (Set (Var "s1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "s")) "," (Set (Var "r1")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "s")) "," (Set (Var "s1")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))) "holds" (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "s")) "," (Set "(" (Set "(" (Set (Var "r1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s1")) ")" ) ($#k6_real_1 :::"/"::: ) (Num 2) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )))) ; theorem :: TOPREAL3:24 (Bool "for" (Set (Var "r1")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "s1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "s")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "s1")) "," (Set (Var "s")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))) "holds" (Bool (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set (Var "r1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s1")) ")" ) ($#k6_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set (Var "s")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )))) ; theorem :: TOPREAL3:25 (Bool "for" (Set (Var "r1")) "," (Set (Var "r2")) "," (Set (Var "r")) "," (Set (Var "s1")) "," (Set (Var "s2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "r2")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "s1")) "," (Set (Var "s2")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Bool "not" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "s2")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )))) "holds" (Bool (Set ($#k19_euclid :::"|["::: ) (Set (Var "s1")) "," (Set (Var "r2")) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )))) ; theorem :: TOPREAL3:26 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Bool "not" (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set "(" ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "i")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ")" )) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) ")" )) "holds" (Bool "not" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))))))) ; theorem :: TOPREAL3:27 (Bool "for" (Set (Var "q")) "," (Set (Var "p2")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p2")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "p")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q")) "," (Set (Var "p2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p2")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: TOPREAL3:28 (Bool "for" (Set (Var "q")) "," (Set (Var "p2")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "p2")) ($#k17_euclid :::"`1"::: ) ))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p2")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p2")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "p")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q")) "," (Set (Var "p2")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p2")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: TOPREAL3:29 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: TOPREAL3:30 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: TOPREAL3:31 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: TOPREAL3:32 (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 (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: TOPREAL3:33 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::">"::: ) (Num 2)) & (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ))) ; theorem :: TOPREAL3:34 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ($#k11_finseq_1 :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "q"))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) ")" ))) ; theorem :: TOPREAL3:35 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ($#k11_finseq_1 :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "q"))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) ")" ))) ; theorem :: TOPREAL3:36 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ($#k11_finseq_1 :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "q"))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) ")" ))) ; theorem :: TOPREAL3:37 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set "(" (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ($#k11_finseq_1 :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "q"))) & (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) ")" ))) ; theorem :: TOPREAL3:38 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ")" ))))) ; theorem :: TOPREAL3:39 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">="::: ) (Num 2)) & (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")))))) "holds" (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set (Var "p")))))) ; theorem :: TOPREAL3:40 (Bool "for" (Set (Var "q")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "q")) ($#r1_hidden :::"<>"::: ) (Set (Var "p"))) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q")) "," (Set (Var "p")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool "not" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set (Var "f"))))))) ; theorem :: TOPREAL3:41 (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v4_topreal1 :::"being_S-Seq"::: ) ) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r2_hidden :::"in"::: ) (Set ($#k2_topreal1 :::"LSeg"::: ) "(" (Set (Var "f")) "," (Set (Var "m")) ")" )) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")))))) ; theorem :: TOPREAL3:42 (Bool "for" (Set (Var "p1")) "," (Set (Var "q")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "q")) ")" ))) & (Bool "(" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) "or" (Bool "(" (Bool (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ) & (Bool "(" (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) "or" (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) )) ")" )) "holds" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "q")) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "q")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: TOPREAL3:43 (Bool "for" (Set (Var "p1")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Bool "not" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p")) ")" ))) & (Bool (Set (Set (Var "p1")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: TOPREAL3:44 (Bool "for" (Set (Var "p1")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Num 2) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "p1")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" ))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "u")) "," (Set (Var "r")) ")" )) & (Bool (Bool "not" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p")) ")" ))) & (Bool (Set (Set (Var "p1")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k18_euclid :::"`2"::: ) ))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p")) "," (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) ")" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set ($#k19_euclid :::"|["::: ) (Set "(" (Set (Var "q")) ($#k17_euclid :::"`1"::: ) ")" ) "," (Set "(" (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ")" ) ($#k19_euclid :::"]|"::: ) ) "," (Set (Var "q")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set (Var "p1")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: TOPREAL3:45 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "n")))) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ))))) ; theorem :: TOPREAL3:46 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "n")))) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ))))) ;