:: TOPS_4 semantic presentation begin theorem :: TOPS_4:1 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "S")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "B")) "," (Set (Var "T")) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set (Var "A"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "A"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set (Var "B"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "B"))) "#)" )) & (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"U1"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) "#)" )) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) )) "holds" (Bool (Set (Var "g")) "is" ($#v1_t_0topsp :::"open"::: ) )))) ; theorem :: TOPS_4:2 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "holds" (Bool "(" (Bool (Set (Var "P")) "is" ($#v3_pre_topc :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k1_topreal9 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Var "P"))))) ")" ))) ; theorem :: TOPS_4:3 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "W")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "W")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "V")))) ")" )))) ")" ))) ; theorem :: TOPS_4:4 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "q")) "," (Set (Var "r")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "V")))))))) ")" )))) ; theorem :: TOPS_4:5 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M")) ")" ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M")) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "W")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "W")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ")" ))) ")" )))) ")" )))) ; theorem :: TOPS_4:6 (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M1")) ")" ) "," (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M2")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M1")) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M2")) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "q")) "," (Set (Var "s")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ")" ))))))) ")" ))) ; theorem :: TOPS_4:7 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k1_topreal9 :::"Ball"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) "," (Set (Var "r")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "V"))))))) ")" )))) ; theorem :: TOPS_4:8 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "W")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "W")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k1_topreal9 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ")" ))) ")" )))) ")" )))) ; theorem :: TOPS_4:9 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k1_topreal9 :::"Ball"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) "," (Set (Var "s")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k1_topreal9 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ")" )))))) ")" ))) ; theorem :: TOPS_4:10 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "V"))))))) ")" ))) ; theorem :: TOPS_4:11 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "V"))) & (Bool (Set (Var "V")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "p")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set (Var "p")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))) ")" )))) ")" ))) ; theorem :: TOPS_4:12 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "p")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set (Var "p")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))))) ")" )) ; theorem :: TOPS_4:13 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k1_topreal9 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ")" )))))) ")" ))) ; theorem :: TOPS_4:14 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_t_0topsp :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set ($#k1_topreal9 :::"Ball"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) "," (Set (Var "s")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "p")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set (Var "p")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))))) ")" ))) ; begin theorem :: TOPS_4:15 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M")) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))))) "holds" (Bool "ex" (Set (Var "W")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "W"))) ($#r1_tarski :::"c="::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "q")) "," (Set (Var "r")) ")" )) ")" ))))) ")" )))) ; theorem :: TOPS_4:16 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M")) ")" ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M")) (Bool "for" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "s")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "V")))))) ")" )))) ; theorem :: TOPS_4:17 (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M1")) ")" ) "," (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M2")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M1")) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M2")) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "p"))))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "s")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "q")) "," (Set (Var "r")) ")" )))))) ")" ))) ; theorem :: TOPS_4:18 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "W")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "W"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_topreal9 :::"Ball"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) "," (Set (Var "r")) ")" )) ")" )))) ")" )))) ; theorem :: TOPS_4:19 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k1_topreal9 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "s")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "V")))))) ")" )))) ; theorem :: TOPS_4:20 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k1_topreal9 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "s")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_topreal9 :::"Ball"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) "," (Set (Var "r")) ")" ))))) ")" ))) ; theorem :: TOPS_4:21 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "W")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "W"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ")" )))) ")" ))) ; theorem :: TOPS_4:22 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "p")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "p")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set (Var "V")))))) ")" ))) ; theorem :: TOPS_4:23 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "p")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "p")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))))) ")" )) ; theorem :: TOPS_4:24 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "," (Set ($#k3_topmetr :::"R^1"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k1_topreal9 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "s")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ))))) ")" ))) ; theorem :: TOPS_4:25 (Bool "for" (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "m")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "p")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "s")) ")" ) "," (Set "(" (Set (Var "p")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "s")) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k1_topreal9 :::"Ball"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) "," (Set (Var "r")) ")" ))))) ")" ))) ;