:: FRECHET semantic presentation begin theorem :: FRECHET:1 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "S"))) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T"))))) ; theorem :: FRECHET:2 (Bool "for" (Set (Var "T1")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "T2")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T1")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T2"))))) "holds" (Bool (Set (Var "S")) "is" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T2")))))) ; theorem :: FRECHET:3 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Basis":::) "of" (Set (Var "x")) "holds" (Bool (Set (Var "B")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); cluster ($#v1_tops_2 :::"open"::: ) "x" ($#v1_yellow_8 :::"-quasi_basis"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T"))))); end; theorem :: FRECHET:4 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set (Var "B"))) "is" ($#v3_pre_topc :::"open"::: ) ))) ; theorem :: FRECHET:5 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "T"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool "(" "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ) & (Bool "(" "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tops_2 :::"closed"::: ) )) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" )) "holds" (Bool (Set (Var "T")) "is" ($#l1_pre_topc :::"TopSpace":::))) ; theorem :: FRECHET:6 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set (Var "S")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set (Var "A"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ) ")" )) "holds" (Bool (Set (Var "S")) "is" ($#l1_pre_topc :::"TopSpace":::))))) ; theorem :: FRECHET:7 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k8_metric_1 :::"RealSpace"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "x")) "," (Set (Var "r")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x")) ($#k5_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set (Var "x")) ($#k3_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) )))) ; theorem :: FRECHET:8 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" ))) ")" )) ; theorem :: FRECHET:9 (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "S")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ")" ) "," (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 4) ")" ) ")" ) ($#k2_rcomp_1 :::".["::: ) )) ")" )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "S"))) "is" ($#v4_pre_topc :::"closed"::: ) )) ; theorem :: FRECHET:10 (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k5_numbers :::"NAT"::: ) ))) "holds" (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) )) ; definitionlet "M" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Const "M")) ")" ); func :::"Balls"::: "x" -> ($#m1_subset_1 :::"Subset-Family":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) "M" ")" ) means :: FRECHET:def 1 (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" "M" "st" (Bool "(" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) "x") & (Bool it ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "y")) "," (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) "}" ) ")" )); end; :: deftheorem defines :::"Balls"::: FRECHET:def 1 : (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M")) ")" ) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_frechet :::"Balls"::: ) (Set (Var "x")))) "iff" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "y")) "," (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ")" ")" ) where n "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) : (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) "}" ) ")" )) ")" )))); registrationlet "M" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Const "M")) ")" ); cluster (Set ($#k1_frechet :::"Balls"::: ) "x") -> ($#v1_tops_2 :::"open"::: ) "x" ($#v1_yellow_8 :::"-quasi_basis"::: ) ; end; registrationlet "M" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Const "M")) ")" ); cluster (Set ($#k1_frechet :::"Balls"::: ) "x") -> ($#v4_card_3 :::"countable"::: ) ; end; theorem :: FRECHET:11 (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M")) ")" ) (Bool "for" (Set (Var "x9")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "x9")))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k1_frechet :::"Balls"::: ) (Set (Var "x")) ")" ) "st" (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "x9")) "," (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))))))) ; theorem :: FRECHET:12 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k6_subset_1 :::"\"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) ")" ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")) ")" )))) ; theorem :: FRECHET:13 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "A")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "B"))) ($#r1_hidden :::"="::: ) (Set (Var "B")))) ; theorem :: FRECHET:14 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "A")) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "B")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "x")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "B"))))) ; theorem :: FRECHET:15 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "A")) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "B")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "x")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k6_subset_1 :::"\"::: ) (Set (Var "B")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: FRECHET:16 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "A")) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "B")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "C")) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "C")) ($#k6_subset_1 :::"\"::: ) (Set (Var "B")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: FRECHET:17 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "A")) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "B")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "B")))) ; theorem :: FRECHET:18 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "A")) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "B")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "C")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "C")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "B"))))) ; theorem :: FRECHET:19 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "A")) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set (Var "B")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_relat_1 :::"""::: ) (Set "(" (Set (Var "C")) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "C")) ($#k6_subset_1 :::"\"::: ) (Set (Var "B"))))) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; attr "T" is :::"first-countable"::: means :: FRECHET:def 2 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Basis":::) "of" (Set (Var "x")) "st" (Bool (Set (Var "B")) "is" ($#v4_card_3 :::"countable"::: ) ))); end; :: deftheorem defines :::"first-countable"::: FRECHET:def 2 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_frechet :::"first-countable"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Basis":::) "of" (Set (Var "x")) "st" (Bool (Set (Var "B")) "is" ($#v4_card_3 :::"countable"::: ) ))) ")" )); theorem :: FRECHET:20 (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) "holds" (Bool (Set ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) (Set (Var "M"))) "is" ($#v1_frechet :::"first-countable"::: ) )) ; theorem :: FRECHET:21 (Bool (Set ($#k3_topmetr :::"R^1"::: ) ) "is" ($#v1_frechet :::"first-countable"::: ) ) ; registration cluster (Set ($#k3_topmetr :::"R^1"::: ) ) -> ($#v1_frechet :::"first-countable"::: ) ; end; definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "S" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "T")); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); pred "S" :::"is_convergent_to"::: "x" means :: FRECHET:def 3 (Bool "for" (Set (Var "U1")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set (Var "U1")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool "x" ($#r2_hidden :::"in"::: ) (Set (Var "U1")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set "S" ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set (Var "U1")))))); end; :: deftheorem defines :::"is_convergent_to"::: FRECHET:def 3 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "S")) ($#r1_frechet :::"is_convergent_to"::: ) (Set (Var "x"))) "iff" (Bool "for" (Set (Var "U1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "U1")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "U1")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set (Var "S")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set (Var "U1")))))) ")" )))); theorem :: FRECHET:22 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "S")) ($#r2_funct_2 :::"="::: ) (Set (Set ($#k5_numbers :::"NAT"::: ) ) ($#k8_funcop_1 :::"-->"::: ) (Set (Var "x"))))) "holds" (Bool (Set (Var "S")) ($#r1_frechet :::"is_convergent_to"::: ) (Set (Var "x")))))) ; definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "S" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "T")); attr "S" is :::"convergent"::: means :: FRECHET:def 4 (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" "st" (Bool "S" ($#r1_frechet :::"is_convergent_to"::: ) (Set (Var "x")))); end; :: deftheorem defines :::"convergent"::: FRECHET:def 4 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v2_frechet :::"convergent"::: ) ) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Set (Var "S")) ($#r1_frechet :::"is_convergent_to"::: ) (Set (Var "x")))) ")" ))); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "S" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "T")); func :::"Lim"::: "S" -> ($#m1_subset_1 :::"Subset":::) "of" "T" means :: FRECHET:def 5 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "S" ($#r1_frechet :::"is_convergent_to"::: ) (Set (Var "x"))) ")" )); end; :: deftheorem defines :::"Lim"::: FRECHET:def 5 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_frechet :::"Lim"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool (Set (Var "S")) ($#r1_frechet :::"is_convergent_to"::: ) (Set (Var "x"))) ")" )) ")" )))); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; attr "T" is :::"Frechet"::: means :: FRECHET:def 6 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))))) "holds" (Bool "ex" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" "T" "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_frechet :::"Lim"::: ) (Set (Var "S")))) ")" )))); end; :: deftheorem defines :::"Frechet"::: FRECHET:def 6 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v3_frechet :::"Frechet"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))))) "holds" (Bool "ex" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_frechet :::"Lim"::: ) (Set (Var "S")))) ")" )))) ")" )); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; attr "T" is :::"sequential"::: means :: FRECHET:def 7 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" "T" "st" (Bool (Bool (Set (Var "S")) "is" ($#v2_frechet :::"convergent"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k2_frechet :::"Lim"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) ")" )); end; :: deftheorem defines :::"sequential"::: FRECHET:def 7 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v4_frechet :::"sequential"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v2_frechet :::"convergent"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k2_frechet :::"Lim"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) ")" )) ")" )); theorem :: FRECHET:23 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v1_frechet :::"first-countable"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v3_frechet :::"Frechet"::: ) )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_frechet :::"first-countable"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_frechet :::"Frechet"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: FRECHET:24 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k2_frechet :::"Lim"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A")))))) ; theorem :: FRECHET:25 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v2_frechet :::"convergent"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k2_frechet :::"Lim"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" )) "holds" (Bool (Set (Var "T")) "is" ($#v4_frechet :::"sequential"::: ) )) ; theorem :: FRECHET:26 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v3_frechet :::"Frechet"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v4_frechet :::"sequential"::: ) )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_frechet :::"Frechet"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_frechet :::"sequential"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; begin definitionfunc :::"REAL?"::: -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#l1_pre_topc :::"TopSpace":::) means :: FRECHET:def 8 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k1_numbers :::"REAL"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#k1_tarski :::"}"::: ) ))) & (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "," it "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_partfun1 :::"id"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set ($#k5_numbers :::"NAT"::: ) ) ($#k7_funcop_1 :::"-->"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" it "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set (Var "A"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ) ")" ) ")" )) ")" ); end; :: deftheorem defines :::"REAL?"::: FRECHET:def 8 : (Bool "for" (Set (Var "b1")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k3_frechet :::"REAL?"::: ) )) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b1"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k1_numbers :::"REAL"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#k1_tarski :::"}"::: ) ))) & (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "," (Set (Var "b1")) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_partfun1 :::"id"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k1_funct_4 :::"+*"::: ) (Set "(" (Set ($#k5_numbers :::"NAT"::: ) ) ($#k7_funcop_1 :::"-->"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "b1")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set (Var "A"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ) ")" ) ")" )) ")" ) ")" )); theorem :: FRECHET:27 (Bool (Set ($#k1_numbers :::"REAL"::: ) ) "is" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k3_frechet :::"REAL?"::: ) )) ; theorem :: FRECHET:28 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_frechet :::"REAL?"::: ) ) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set ($#k1_numbers :::"REAL"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) "iff" (Bool "ex" (Set (Var "O")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool "(" (Bool (Set (Var "O")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set ($#k5_numbers :::"NAT"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "O"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "O")) ($#k7_subset_1 :::"\"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#k1_tarski :::"}"::: ) ))) ")" )) ")" )) ; theorem :: FRECHET:29 (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_frechet :::"REAL?"::: ) )) & (Bool (Bool "not" (Set ($#k1_numbers :::"REAL"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "iff" (Bool "(" (Bool (Set (Var "A")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) )) & (Bool (Set (Set ($#k5_numbers :::"NAT"::: ) ) ($#k8_subset_1 :::"/\"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) ")" )) ; theorem :: FRECHET:30 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_frechet :::"REAL?"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "(" (Bool (Set (Set ($#k5_numbers :::"NAT"::: ) ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) "iff" (Bool "(" (Bool (Bool "not" (Set ($#k1_numbers :::"REAL"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ) ")" ))) ; theorem :: FRECHET:31 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_frechet :::"REAL?"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) )) ; theorem :: FRECHET:32 (Bool (Bool "not" (Set ($#k3_frechet :::"REAL?"::: ) ) "is" ($#v1_frechet :::"first-countable"::: ) )) ; theorem :: FRECHET:33 (Bool (Set ($#k3_frechet :::"REAL?"::: ) ) "is" ($#v3_frechet :::"Frechet"::: ) ) ; theorem :: FRECHET:34 (Bool "ex" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool "(" (Bool (Set (Var "T")) "is" ($#v3_frechet :::"Frechet"::: ) ) & (Bool (Bool "not" (Set (Var "T")) "is" ($#v1_frechet :::"first-countable"::: ) )) ")" )) ; begin theorem :: FRECHET:35 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_partfun1 :::"tolerates"::: ) (Set (Var "g")))) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_4 :::"+*"::: ) (Set (Var "g")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "g")) ")" )))) ; theorem :: FRECHET:36 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ))) ;