:: TOPGEN_4 semantic presentation begin definitionlet "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; func :::"TotFam"::: "T" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "T" equals :: TOPGEN_4:def 1 (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T")); end; :: deftheorem defines :::"TotFam"::: TOPGEN_4:def 1 : (Bool "for" (Set (Var "T")) "being" ($#l1_struct_0 :::"1-sorted"::: ) "holds" (Bool (Set ($#k1_topgen_4 :::"TotFam"::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T")))))); theorem :: TOPGEN_4:1 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v4_card_3 :::"countable"::: ) ) "iff" (Bool (Set ($#k7_setfam_1 :::"COMPLEMENT"::: ) (Set (Var "F"))) "is" ($#v4_card_3 :::"countable"::: ) ) ")" ))) ; registration cluster (Set ($#k3_numbers :::"RAT"::: ) ) -> ($#v4_card_3 :::"countable"::: ) ; end; scheme :: TOPGEN_4:sch 1 FraenCoun11{ P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "{" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "n")) ($#k6_domain_1 :::"}"::: ) ) where n "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k3_numbers :::"RAT"::: ) ) : (Bool P1[(Set (Var "n"))]) "}" "is" ($#v4_card_3 :::"countable"::: ) ) proof end; theorem :: TOPGEN_4:2 (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")) "holds" (Bool (Set ($#k2_topgen_1 :::"Der"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) : (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ")" ))) "}" ))) ; registration cluster ($#v8_struct_0 :::"finite"::: ) -> ($#v5_waybel23 :::"second-countable"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; registration cluster (Set ($#k1_numbers :::"REAL"::: ) ) -> ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) ; end; registration cluster ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) -> ($#~v1_finset_1 "non" ($#v1_finset_1 :::"finite"::: ) ) for ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_finset_1 "non" ($#v1_finset_1 :::"finite"::: ) ) -> ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) for ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v4_card_3 "non" ($#v4_card_3 :::"countable"::: ) ) for ($#m1_hidden :::"set"::: ) ; end; theorem :: TOPGEN_4:3 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool (Set ($#k2_topgen_1 :::"Der"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" ))) ; theorem :: TOPGEN_4:4 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Basis":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "V")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "V")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "V")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool "ex" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set (Var "B"))) & (Bool (Set (Var "W")) ($#r1_tarski :::"c="::: ) (Set (Var "V"))) & (Bool (Set (Var "W")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))))) ; begin theorem :: TOPGEN_4:5 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k4_topgen_1 :::"density"::: ) (Set (Var "T"))) ($#r1_ordinal1 :::"c="::: ) (Set ($#k2_waybel23 :::"weight"::: ) (Set (Var "T"))))) ; theorem :: TOPGEN_4:6 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v7_topgen_1 :::"separable"::: ) ) "iff" (Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) ) & (Bool (Set (Var "A")) "is" ($#v4_card_3 :::"countable"::: ) ) ")" )) ")" )) ; theorem :: TOPGEN_4:7 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v5_waybel23 :::"second-countable"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v7_topgen_1 :::"separable"::: ) )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v5_waybel23 :::"second-countable"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_topgen_1 :::"separable"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: TOPGEN_4:8 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) )) "holds" (Bool (Set ($#k2_tops_1 :::"Fr"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_tops_1 :::"Fr"::: ) (Set (Var "A")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k2_tops_1 :::"Fr"::: ) (Set (Var "B")) ")" ))))) ; theorem :: TOPGEN_4:9 (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 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_pcomps_1 :::"locally_finite"::: ) )) "holds" (Bool (Set ($#k2_tops_1 :::"Fr"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_topgen_1 :::"Fr"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TOPGEN_4:10 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v7_topgen_1 :::"separable"::: ) ) "iff" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" )) ($#r1_ordinal1 :::"c="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )) ")" )) ; theorem :: TOPGEN_4:11 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v7_topgen_1 :::"separable"::: ) ) "iff" (Bool (Set (Var "T")) "is" ($#v1_orders_4 :::"countable"::: ) ) ")" )) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "F" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); attr "F" is :::"all-open-containing"::: means :: TOPGEN_4:def 2 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) "F")); end; :: deftheorem defines :::"all-open-containing"::: TOPGEN_4:def 2 : (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 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v1_topgen_4 :::"all-open-containing"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) ")" ))); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "F" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); attr "F" is :::"all-closed-containing"::: means :: TOPGEN_4:def 3 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) "F")); end; :: deftheorem defines :::"all-closed-containing"::: TOPGEN_4:def 3 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_topgen_4 :::"all-closed-containing"::: ) ) "iff" (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 (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) ")" ))); definitionlet "T" be ($#m1_hidden :::"set"::: ) ; let "F" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); attr "F" is :::"closed_for_countable_unions"::: means :: TOPGEN_4:def 4 (Bool "for" (Set (Var "G")) "being" ($#v4_card_3 :::"countable"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" "T" "st" (Bool (Bool (Set (Var "G")) ($#r1_tarski :::"c="::: ) "F")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "G"))) ($#r2_hidden :::"in"::: ) "F")); end; :: deftheorem defines :::"closed_for_countable_unions"::: TOPGEN_4:def 4 : (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ) "iff" (Bool "for" (Set (Var "G")) "being" ($#v4_card_3 :::"countable"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "G")) ($#r1_tarski :::"c="::: ) (Set (Var "F")))) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "G"))) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) ")" ))); registrationlet "T" be ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_prob_1 :::"compl-closed"::: ) ($#v4_prob_1 :::"sigma-multiplicative"::: ) -> ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "T" ")" )); end; theorem :: TOPGEN_4:12 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) )) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "F"))))) ; registrationlet "T" be ($#m1_hidden :::"set"::: ) ; cluster ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "T" ")" )); end; theorem :: TOPGEN_4:13 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#m1_subset_1 :::"SigmaField":::) "of" (Set (Var "T"))) "iff" (Bool "(" (Bool (Set (Var "F")) "is" ($#v1_prob_1 :::"compl-closed"::: ) ) & (Bool (Set (Var "F")) "is" ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ) ")" ) ")" ))) ; definitionlet "T" be ($#m1_hidden :::"set"::: ) ; let "F" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); attr "F" is :::"closed_for_countable_meets"::: means :: TOPGEN_4:def 5 (Bool "for" (Set (Var "G")) "being" ($#v4_card_3 :::"countable"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" "T" "st" (Bool (Bool (Set (Var "G")) ($#r1_tarski :::"c="::: ) "F")) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "G"))) ($#r2_hidden :::"in"::: ) "F")); end; :: deftheorem defines :::"closed_for_countable_meets"::: TOPGEN_4:def 5 : (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v4_topgen_4 :::"closed_for_countable_meets"::: ) ) "iff" (Bool "for" (Set (Var "G")) "being" ($#v4_card_3 :::"countable"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "G")) ($#r1_tarski :::"c="::: ) (Set (Var "F")))) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "G"))) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) ")" ))); theorem :: TOPGEN_4:14 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_topgen_4 :::"all-closed-containing"::: ) ) & (Bool (Set (Var "F")) "is" ($#v1_prob_1 :::"compl-closed"::: ) ) "iff" (Bool "(" (Bool (Set (Var "F")) "is" ($#v1_topgen_4 :::"all-open-containing"::: ) ) & (Bool (Set (Var "F")) "is" ($#v1_prob_1 :::"compl-closed"::: ) ) ")" ) ")" ))) ; theorem :: TOPGEN_4:15 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_prob_1 :::"compl-closed"::: ) )) "holds" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k7_setfam_1 :::"COMPLEMENT"::: ) (Set (Var "F")))))) ; theorem :: TOPGEN_4:16 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G"))) & (Bool (Set (Var "G")) "is" ($#v1_prob_1 :::"compl-closed"::: ) )) "holds" (Bool (Set ($#k7_setfam_1 :::"COMPLEMENT"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set (Var "G"))))) ; theorem :: TOPGEN_4:17 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v4_topgen_4 :::"closed_for_countable_meets"::: ) ) & (Bool (Set (Var "F")) "is" ($#v1_prob_1 :::"compl-closed"::: ) ) "iff" (Bool "(" (Bool (Set (Var "F")) "is" ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ) & (Bool (Set (Var "F")) "is" ($#v1_prob_1 :::"compl-closed"::: ) ) ")" ) ")" ))) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_prob_1 :::"compl-closed"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) -> ($#v2_topgen_4 :::"all-closed-containing"::: ) ($#v4_topgen_4 :::"closed_for_countable_meets"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); cluster ($#v1_prob_1 :::"compl-closed"::: ) ($#v2_topgen_4 :::"all-closed-containing"::: ) ($#v4_topgen_4 :::"closed_for_countable_meets"::: ) -> ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); end; begin registrationlet "T" be ($#m1_hidden :::"set"::: ) ; let "F" be ($#v4_card_3 :::"countable"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); cluster (Set ($#k7_setfam_1 :::"COMPLEMENT"::: ) "F") -> ($#v4_card_3 :::"countable"::: ) ; end; registrationlet "T" be ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v1_tops_2 :::"open"::: ) ($#v2_tops_2 :::"closed"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); end; registrationlet "T" be ($#l1_pre_topc :::"TopSpace":::); cluster ($#v4_card_3 :::"countable"::: ) ($#v1_tops_2 :::"open"::: ) ($#v2_tops_2 :::"closed"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); end; theorem :: TOPGEN_4:18 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) "is" ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")))) ; registration cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v4_card_3 :::"countable"::: ) for ($#m1_hidden :::"set"::: ) ; end; begin theorem :: TOPGEN_4:19 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "A")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) "iff" (Bool (Set (Var "F")) "is" ($#v1_tops_2 :::"open"::: ) ) ")" )))) ; theorem :: TOPGEN_4:20 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "A")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool (Set (Var "F")) "is" ($#v2_tops_2 :::"closed"::: ) ) ")" )))) ; definitionlet "T" be ($#m1_hidden :::"set"::: ) ; let "F", "G" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); :: original: :::"INTERSECTION"::: redefine func :::"INTERSECTION"::: "(" "F" "," "G" ")" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "T"; :: original: :::"UNION"::: redefine func :::"UNION"::: "(" "F" "," "G" ")" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "T"; end; theorem :: TOPGEN_4:21 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tops_2 :::"closed"::: ) ) & (Bool (Set (Var "G")) "is" ($#v2_tops_2 :::"closed"::: ) )) "holds" (Bool (Set ($#k2_topgen_4 :::"INTERSECTION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ) "is" ($#v2_tops_2 :::"closed"::: ) ))) ; theorem :: TOPGEN_4:22 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tops_2 :::"closed"::: ) ) & (Bool (Set (Var "G")) "is" ($#v2_tops_2 :::"closed"::: ) )) "holds" (Bool (Set ($#k3_topgen_4 :::"UNION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ) "is" ($#v2_tops_2 :::"closed"::: ) ))) ; theorem :: TOPGEN_4:23 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tops_2 :::"open"::: ) ) & (Bool (Set (Var "G")) "is" ($#v1_tops_2 :::"open"::: ) )) "holds" (Bool (Set ($#k2_topgen_4 :::"INTERSECTION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ) "is" ($#v1_tops_2 :::"open"::: ) ))) ; theorem :: TOPGEN_4:24 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tops_2 :::"open"::: ) ) & (Bool (Set (Var "G")) "is" ($#v1_tops_2 :::"open"::: ) )) "holds" (Bool (Set ($#k3_topgen_4 :::"UNION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ) "is" ($#v1_tops_2 :::"open"::: ) ))) ; theorem :: TOPGEN_4:25 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k2_topgen_4 :::"INTERSECTION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ")" )) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "F")) "," (Set (Var "G")) ($#k2_zfmisc_1 :::":]"::: ) ))))) ; theorem :: TOPGEN_4:26 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k3_topgen_4 :::"UNION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ")" )) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "F")) "," (Set (Var "G")) ($#k2_zfmisc_1 :::":]"::: ) ))))) ; theorem :: TOPGEN_4:27 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) (Set "(" ($#k2_setfam_1 :::"UNION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "F")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "G")) ")" )))) ; theorem :: TOPGEN_4:28 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "G")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "F")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k3_tarski :::"union"::: ) (Set (Var "G")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) (Set "(" ($#k2_setfam_1 :::"UNION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ")" )))) ; theorem :: TOPGEN_4:29 (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_setfam_1 :::"UNION"::: ) "(" (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "F")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: TOPGEN_4:30 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "not" (Bool (Set ($#k2_setfam_1 :::"UNION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool (Set (Var "G")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ; theorem :: TOPGEN_4:31 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "not" (Bool (Set ($#k3_setfam_1 :::"INTERSECTION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool (Set (Var "G")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) ; theorem :: TOPGEN_4:32 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_setfam_1 :::"meet"::: ) (Set "(" ($#k2_setfam_1 :::"UNION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k1_setfam_1 :::"meet"::: ) (Set (Var "G")) ")" )))) ; theorem :: TOPGEN_4:33 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "G")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set ($#k1_setfam_1 :::"meet"::: ) (Set "(" ($#k2_setfam_1 :::"UNION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k1_setfam_1 :::"meet"::: ) (Set (Var "G")) ")" )))) ; theorem :: TOPGEN_4:34 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "G")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Set "(" ($#k1_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k1_setfam_1 :::"meet"::: ) (Set (Var "G")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_setfam_1 :::"meet"::: ) (Set "(" ($#k3_setfam_1 :::"INTERSECTION"::: ) "(" (Set (Var "F")) "," (Set (Var "G")) ")" ")" )))) ; begin definitionlet "T" be ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); attr "A" is :::"F_sigma"::: means :: TOPGEN_4:def 6 (Bool "ex" (Set (Var "F")) "being" ($#v4_card_3 :::"countable"::: ) ($#v2_tops_2 :::"closed"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" "T" "st" (Bool "A" ($#r1_hidden :::"="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))))); end; :: deftheorem defines :::"F_sigma"::: TOPGEN_4:def 6 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v5_topgen_4 :::"F_sigma"::: ) ) "iff" (Bool "ex" (Set (Var "F")) "being" ($#v4_card_3 :::"countable"::: ) ($#v2_tops_2 :::"closed"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))))) ")" ))); definitionlet "T" be ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); attr "A" is :::"G_delta"::: means :: TOPGEN_4:def 7 (Bool "ex" (Set (Var "F")) "being" ($#v4_card_3 :::"countable"::: ) ($#v1_tops_2 :::"open"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" "T" "st" (Bool "A" ($#r1_hidden :::"="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))))); end; :: deftheorem defines :::"G_delta"::: TOPGEN_4:def 7 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v6_topgen_4 :::"G_delta"::: ) ) "iff" (Bool "ex" (Set (Var "F")) "being" ($#v4_card_3 :::"countable"::: ) ($#v1_tops_2 :::"open"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))))) ")" ))); registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v5_topgen_4 :::"F_sigma"::: ) ($#v6_topgen_4 :::"G_delta"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T")); end; theorem :: TOPGEN_4:35 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T"))) "is" ($#v5_topgen_4 :::"F_sigma"::: ) )) ; theorem :: TOPGEN_4:36 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T"))) "is" ($#v6_topgen_4 :::"G_delta"::: ) )) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k2_struct_0 :::"[#]"::: ) "T") -> ($#v5_topgen_4 :::"F_sigma"::: ) ($#v6_topgen_4 :::"G_delta"::: ) ; end; theorem :: TOPGEN_4:37 (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" ($#v5_topgen_4 :::"F_sigma"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k3_subset_1 :::"`"::: ) ) "is" ($#v6_topgen_4 :::"G_delta"::: ) ))) ; theorem :: TOPGEN_4:38 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v6_topgen_4 :::"G_delta"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k3_subset_1 :::"`"::: ) ) "is" ($#v5_topgen_4 :::"F_sigma"::: ) ))) ; theorem :: TOPGEN_4:39 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v5_topgen_4 :::"F_sigma"::: ) ) & (Bool (Set (Var "B")) "is" ($#v5_topgen_4 :::"F_sigma"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B"))) "is" ($#v5_topgen_4 :::"F_sigma"::: ) ))) ; theorem :: TOPGEN_4:40 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v5_topgen_4 :::"F_sigma"::: ) ) & (Bool (Set (Var "B")) "is" ($#v5_topgen_4 :::"F_sigma"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v5_topgen_4 :::"F_sigma"::: ) ))) ; theorem :: TOPGEN_4:41 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v6_topgen_4 :::"G_delta"::: ) ) & (Bool (Set (Var "B")) "is" ($#v6_topgen_4 :::"G_delta"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v6_topgen_4 :::"G_delta"::: ) ))) ; theorem :: TOPGEN_4:42 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v6_topgen_4 :::"G_delta"::: ) ) & (Bool (Set (Var "B")) "is" ($#v6_topgen_4 :::"G_delta"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B"))) "is" ($#v6_topgen_4 :::"G_delta"::: ) ))) ; theorem :: TOPGEN_4:43 (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" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v5_topgen_4 :::"F_sigma"::: ) ))) ; theorem :: TOPGEN_4:44 (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" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v6_topgen_4 :::"G_delta"::: ) ))) ; theorem :: TOPGEN_4:45 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_topmetr :::"R^1"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k3_numbers :::"RAT"::: ) ))) "holds" (Bool (Set (Var "A")) "is" ($#v5_topgen_4 :::"F_sigma"::: ) )) ; begin definitionlet "T" be ($#l1_pre_topc :::"TopSpace":::); attr "T" is :::"T_1/2"::: means :: TOPGEN_4:def 8 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "holds" (Bool (Set ($#k2_topgen_1 :::"Der"::: ) (Set (Var "A"))) "is" ($#v4_pre_topc :::"closed"::: ) )); end; :: deftheorem defines :::"T_1/2"::: TOPGEN_4:def 8 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v7_topgen_4 :::"T_1/2"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_topgen_1 :::"Der"::: ) (Set (Var "A"))) "is" ($#v4_pre_topc :::"closed"::: ) )) ")" )); theorem :: TOPGEN_4:46 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v7_pre_topc :::"T_1"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v7_topgen_4 :::"T_1/2"::: ) )) ; theorem :: TOPGEN_4:47 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v7_topgen_4 :::"T_1/2"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v6_pre_topc :::"T_0"::: ) )) ; registration cluster ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v7_topgen_4 :::"T_1/2"::: ) -> ($#v6_pre_topc :::"T_0"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v7_pre_topc :::"T_1"::: ) -> ($#v7_topgen_4 :::"T_1/2"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); pred "x" :::"is_a_condensation_point_of"::: "A" means :: TOPGEN_4:def 9 (Bool "for" (Set (Var "N")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" "x" "holds" (Bool (Bool "not" (Set (Set (Var "N")) ($#k9_subset_1 :::"/\"::: ) "A") "is" ($#v4_card_3 :::"countable"::: ) ))); end; :: deftheorem defines :::"is_a_condensation_point_of"::: TOPGEN_4:def 9 : (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")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) (Set (Var "A"))) "iff" (Bool "for" (Set (Var "N")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "x")) "holds" (Bool (Bool "not" (Set (Set (Var "N")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "A"))) "is" ($#v4_card_3 :::"countable"::: ) ))) ")" )))); theorem :: TOPGEN_4:48 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "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")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "x")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) (Set (Var "B")))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); func "A" :::"^0"::: -> ($#m1_subset_1 :::"Subset":::) "of" "T" means :: TOPGEN_4:def 10 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) "A") ")" )); end; :: deftheorem defines :::"^0"::: TOPGEN_4:def 10 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k4_topgen_4 :::"^0"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool (Set (Var "x")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) (Set (Var "A"))) ")" )) ")" ))); theorem :: TOPGEN_4:49 (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")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "p")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "p")) ($#r1_topgen_1 :::"is_an_accumulation_point_of"::: ) (Set (Var "A")))))) ; theorem :: TOPGEN_4:50 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "A")) ($#k4_topgen_4 :::"^0"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_topgen_1 :::"Der"::: ) (Set (Var "A")))))) ; theorem :: TOPGEN_4:51 (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")) "holds" (Bool (Set (Set (Var "A")) ($#k4_topgen_4 :::"^0"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "A")) ($#k4_topgen_4 :::"^0"::: ) ")" ))))) ; theorem :: TOPGEN_4:52 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "A")) ($#k4_topgen_4 :::"^0"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k4_topgen_4 :::"^0"::: ) )))) ; theorem :: TOPGEN_4:53 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")))) "or" (Bool (Set (Var "x")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) (Set (Var "A"))) "or" (Bool (Set (Var "x")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) (Set (Var "B"))) ")" )))) ; theorem :: TOPGEN_4:54 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ) ($#k4_topgen_4 :::"^0"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k4_topgen_4 :::"^0"::: ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set (Var "B")) ($#k4_topgen_4 :::"^0"::: ) ")" ))))) ; theorem :: TOPGEN_4:55 (Bool "for" (Set (Var "T")) "being" ($#~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_card_3 :::"countable"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Bool "not" (Set (Var "x")) ($#r1_topgen_4 :::"is_a_condensation_point_of"::: ) (Set (Var "A"))))))) ; theorem :: TOPGEN_4:56 (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" ($#v4_card_3 :::"countable"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k4_topgen_4 :::"^0"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#v4_card_3 :::"countable"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); cluster (Set "A" ($#k4_topgen_4 :::"^0"::: ) ) -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: TOPGEN_4:57 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v5_waybel23 :::"second-countable"::: ) )) "holds" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Basis":::) "of" (Set (Var "T")) "st" (Bool (Set (Var "B")) "is" ($#v4_card_3 :::"countable"::: ) ))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v5_waybel23 :::"second-countable"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; begin registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k1_topgen_4 :::"TotFam"::: ) "T") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_prob_1 :::"compl-closed"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ; end; theorem :: TOPGEN_4:58 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_topgen_3 :::"rng"::: ) (Set (Var "A"))) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v4_card_3 :::"countable"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T"))))) ; theorem :: TOPGEN_4:59 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_topgen_4 :::"all-open-containing"::: ) ) & (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G")))) "holds" (Bool (Set (Var "G")) "is" ($#v1_topgen_4 :::"all-open-containing"::: ) ))) ; theorem :: TOPGEN_4:60 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_topgen_4 :::"all-closed-containing"::: ) ) & (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G")))) "holds" (Bool (Set (Var "G")) "is" ($#v2_topgen_4 :::"all-closed-containing"::: ) ))) ; definitionlet "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; mode SigmaField of "T" is ($#m1_subset_1 :::"SigmaField":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T"); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_prob_1 :::"compl-closed"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v2_topgen_4 :::"all-closed-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ($#v4_topgen_4 :::"closed_for_countable_meets"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); end; theorem :: TOPGEN_4:61 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set ($#k9_prob_1 :::"sigma"::: ) (Set "(" ($#k1_topgen_4 :::"TotFam"::: ) (Set (Var "T")) ")" )) "is" ($#v1_topgen_4 :::"all-open-containing"::: ) ) & (Bool (Set ($#k9_prob_1 :::"sigma"::: ) (Set "(" ($#k1_topgen_4 :::"TotFam"::: ) (Set (Var "T")) ")" )) "is" ($#v1_prob_1 :::"compl-closed"::: ) ) & (Bool (Set ($#k9_prob_1 :::"sigma"::: ) (Set "(" ($#k1_topgen_4 :::"TotFam"::: ) (Set (Var "T")) ")" )) "is" ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ) ")" )) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k9_prob_1 :::"sigma"::: ) (Set "(" ($#k1_topgen_4 :::"TotFam"::: ) "T" ")" )) -> ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_prob_1 :::"compl-closed"::: ) ($#v3_measure1 :::"sigma-additive"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_prob_1 :::"compl-closed"::: ) ($#v4_prob_1 :::"sigma-multiplicative"::: ) -> ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); end; theorem :: TOPGEN_4:62 (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 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_prob_1 :::"compl-closed"::: ) ) & (Bool (Set (Var "F")) "is" ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) )) "holds" (Bool (Set (Var "F")) "is" ($#m1_subset_1 :::"SigmaField":::) "of" (Set (Var "T"))))) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_FINSUB_1() bbbadV2_FINSUB_1() bbbadV3_FINSUB_1() ($#v1_prob_1 :::"compl-closed"::: ) ($#v4_prob_1 :::"sigma-multiplicative"::: ) ($#v3_measure1 :::"sigma-additive"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k1_openlatt :::"Topology_of"::: ) "T") -> ($#v1_tops_2 :::"open"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ; end; theorem :: TOPGEN_4:63 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) (Bool "ex" (Set (Var "Y")) "being" ($#v1_prob_1 :::"compl-closed"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) & (Bool "(" "for" (Set (Var "Z")) "being" ($#v1_prob_1 :::"compl-closed"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) ")" ) ")" )))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); func :::"BorelSets"::: "T" -> ($#v1_prob_1 :::"compl-closed"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" "T" means :: TOPGEN_4:def 11 (Bool "for" (Set (Var "G")) "being" ($#v1_prob_1 :::"compl-closed"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" "T" "holds" (Bool it ($#r1_tarski :::"c="::: ) (Set (Var "G")))); end; :: deftheorem defines :::"BorelSets"::: TOPGEN_4:def 11 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "b2")) "being" ($#v1_prob_1 :::"compl-closed"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_topgen_4 :::"BorelSets"::: ) (Set (Var "T")))) "iff" (Bool "for" (Set (Var "G")) "being" ($#v1_prob_1 :::"compl-closed"::: ) ($#v1_topgen_4 :::"all-open-containing"::: ) ($#v3_topgen_4 :::"closed_for_countable_unions"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set (Var "b2")) ($#r1_tarski :::"c="::: ) (Set (Var "G")))) ")" ))); theorem :: TOPGEN_4:64 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#v2_tops_2 :::"closed"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set ($#k5_topgen_4 :::"BorelSets"::: ) (Set (Var "T")))))) ; theorem :: TOPGEN_4:65 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#v1_tops_2 :::"open"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set ($#k5_topgen_4 :::"BorelSets"::: ) (Set (Var "T")))))) ; theorem :: TOPGEN_4:66 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k5_topgen_4 :::"BorelSets"::: ) (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set ($#k9_prob_1 :::"sigma"::: ) (Set "(" ($#k1_openlatt :::"Topology_of"::: ) (Set (Var "T")) ")" )))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); attr "A" is :::"Borel"::: means :: TOPGEN_4:def 12 (Bool "A" ($#r2_hidden :::"in"::: ) (Set ($#k5_topgen_4 :::"BorelSets"::: ) "T")); end; :: deftheorem defines :::"Borel"::: TOPGEN_4:def 12 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v8_topgen_4 :::"Borel"::: ) ) "iff" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set ($#k5_topgen_4 :::"BorelSets"::: ) (Set (Var "T")))) ")" ))); registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v5_topgen_4 :::"F_sigma"::: ) -> ($#v8_topgen_4 :::"Borel"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T")); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v6_topgen_4 :::"G_delta"::: ) -> ($#v8_topgen_4 :::"Borel"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T")); end;