:: TEX_1 semantic presentation begin theorem :: TEX_1:1 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "X")) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set (Var "D")) ")" ) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set (Var "C"))) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "C")) "is" ($#v3_pre_topc :::"open"::: ) )))) ; theorem :: TEX_1:2 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "X")) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set (Var "D")) ")" ) "st" (Bool (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set (Var "C"))) & (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool (Set (Var "C")) "is" ($#v4_pre_topc :::"closed"::: ) )))) ; theorem :: TEX_1:3 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "X")) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set "(" (Set (Var "D")) ($#k3_subset_1 :::"`"::: ) ")" ) ")" ) "st" (Bool (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "D")))) "holds" (Bool (Set (Var "C")) "is" ($#v4_pre_topc :::"closed"::: ) )))) ; theorem :: TEX_1:4 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "X")) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set (Var "D")) ")" ) "st" (Bool (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool (Set (Var "D")) "is" ($#v1_tops_1 :::"dense"::: ) )) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v1_tops_1 :::"dense"::: ) ) & (Bool (Set (Var "C")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" )))) ; theorem :: TEX_1:5 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "X")) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set (Var "D")) ")" ) "st" (Bool (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set (Var "C"))) & (Bool (Set (Var "D")) "is" ($#v1_tops_1 :::"dense"::: ) )) "holds" (Bool (Set (Var "C")) "is" ($#v1_tops_3 :::"everywhere_dense"::: ) )))) ; theorem :: TEX_1:6 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "X")) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set "(" (Set (Var "D")) ($#k3_subset_1 :::"`"::: ) ")" ) ")" ) "st" (Bool (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool (Set (Var "D")) "is" ($#v2_tops_1 :::"boundary"::: ) )) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v2_tops_1 :::"boundary"::: ) ) & (Bool (Set (Var "C")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" )))) ; theorem :: TEX_1:7 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "X")) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set "(" (Set (Var "D")) ($#k3_subset_1 :::"`"::: ) ")" ) ")" ) "st" (Bool (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "D"))) & (Bool (Set (Var "D")) "is" ($#v2_tops_1 :::"boundary"::: ) )) "holds" (Bool (Set (Var "C")) "is" ($#v3_tops_1 :::"nowhere_dense"::: ) )))) ; begin definitionlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; redefine attr "Y" is :::"trivial"::: means :: TEX_1:def 1 (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" "Y" "st" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d")) ($#k6_domain_1 :::"}"::: ) ))); end; :: deftheorem defines :::"trivial"::: TEX_1:def 1 : (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) "holds" (Bool "(" (Bool (Set (Var "Y")) "is" ($#v7_struct_0 :::"trivial"::: ) ) "iff" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "Y")) "st" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d")) ($#k6_domain_1 :::"}"::: ) ))) ")" )); registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v1_pre_topc :::"strict"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v1_pre_topc :::"strict"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: TEX_1:8 (Bool "for" (Set (Var "Y")) "being" (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Bool "not" (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Var "Y")) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) )) "holds" (Bool (Set (Var "Y")) "is" ($#v2_pre_topc :::"TopSpace-like"::: ) )) ; registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) -> (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v1_tdlat_3 :::"discrete"::: ) ($#v2_tdlat_3 :::"anti-discrete"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_tdlat_3 :::"discrete"::: ) ($#v2_tdlat_3 :::"anti-discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; registration cluster ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) for ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; begin definitionlet "D" be ($#m1_hidden :::"set"::: ) ; func :::"cobool"::: "D" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "D" equals :: TEX_1:def 2 (Set ($#k2_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," "D" ($#k2_tarski :::"}"::: ) ); end; :: deftheorem defines :::"cobool"::: TEX_1:def 2 : (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_tex_1 :::"cobool"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "D")) ($#k2_tarski :::"}"::: ) ))); registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_tex_1 :::"cobool"::: ) "D") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "D" be ($#m1_hidden :::"set"::: ) ; func :::"ADTS"::: "D" -> ($#l1_pre_topc :::"TopStruct"::: ) equals :: TEX_1:def 3 (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" "D" "," (Set "(" ($#k1_tex_1 :::"cobool"::: ) "D" ")" ) "#)" ); end; :: deftheorem defines :::"ADTS"::: TEX_1:def 3 : (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k2_tex_1 :::"ADTS"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set (Var "D")) "," (Set "(" ($#k1_tex_1 :::"cobool"::: ) (Set (Var "D")) ")" ) "#)" ))); registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k2_tex_1 :::"ADTS"::: ) "D") -> ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ; end; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k2_tex_1 :::"ADTS"::: ) "D") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; theorem :: TEX_1:9 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "A")) "is" ($#v1_xboole_0 :::"empty"::: ) )) "implies" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "A")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "implies" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))) ")" ")" ))) ; theorem :: TEX_1:10 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "implies" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" & "(" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "implies" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))) ")" ")" ))) ; theorem :: TEX_1:11 (Bool "for" (Set (Var "X")) "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 "X")) "st" (Bool (Bool (Bool "not" (Set (Var "A")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))) ")" )) "holds" (Bool (Set (Var "X")) "is" ($#v2_tdlat_3 :::"anti-discrete"::: ) )) ; theorem :: TEX_1:12 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" )) "holds" (Bool (Set (Var "X")) "is" ($#v2_tdlat_3 :::"anti-discrete"::: ) )) ; theorem :: TEX_1:13 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) ) ")" & "(" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "implies" (Bool (Set (Var "A")) "is" ($#v2_tops_1 :::"boundary"::: ) ) ")" ")" ))) ; theorem :: TEX_1:14 (Bool "for" (Set (Var "X")) "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 "X")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "holds" (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) ) ")" )) "holds" (Bool (Set (Var "X")) "is" ($#v2_tdlat_3 :::"anti-discrete"::: ) )) ; theorem :: TEX_1:15 (Bool "for" (Set (Var "X")) "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 "X")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "holds" (Bool (Set (Var "A")) "is" ($#v2_tops_1 :::"boundary"::: ) ) ")" )) "holds" (Bool (Set (Var "X")) "is" ($#v2_tdlat_3 :::"anti-discrete"::: ) )) ; registrationlet "D" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_compts_1 :::"1TopSp"::: ) "D") -> ($#v1_tdlat_3 :::"discrete"::: ) ; end; theorem :: TEX_1:16 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" ))) ; theorem :: TEX_1:17 (Bool "for" (Set (Var "X")) "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 "X")) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set (Var "X")) "is" ($#v1_tdlat_3 :::"discrete"::: ) )) ; theorem :: TEX_1:18 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set (Var "X")) "is" ($#v1_tdlat_3 :::"discrete"::: ) )) ; theorem :: TEX_1:19 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k2_tex_1 :::"ADTS"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set ($#k1_compts_1 :::"1TopSp"::: ) (Set (Var "D")))) "iff" (Bool "ex" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Set (Var "D")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ))) ")" )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_tdlat_3 :::"discrete"::: ) ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) for ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; begin definitionlet "D" be ($#m1_hidden :::"set"::: ) ; let "d0" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); func :::"STS"::: "(" "D" "," "d0" ")" -> ($#l1_pre_topc :::"TopStruct"::: ) equals :: TEX_1:def 4 (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" "D" "," (Set "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) "D" ")" ) ($#k7_subset_1 :::"\"::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" "D" : (Bool "(" (Bool "d0" ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) "D") ")" ) "}" ")" ) "#)" ); end; :: deftheorem defines :::"STS"::: TEX_1:def 4 : (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool (Set ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set (Var "D")) "," (Set "(" (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set (Var "D")) ")" ) ($#k7_subset_1 :::"\"::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "D")) : (Bool "(" (Bool (Set (Var "d0")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set (Var "D"))) ")" ) "}" ")" ) "#)" )))); registrationlet "D" be ($#m1_hidden :::"set"::: ) ; let "d0" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); cluster (Set ($#k3_tex_1 :::"STS"::: ) "(" "D" "," "d0" ")" ) -> ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ; end; registrationlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "d0" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); cluster (Set ($#k3_tex_1 :::"STS"::: ) "(" "D" "," "d0" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; theorem :: TEX_1:20 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "implies" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "A")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) )) "implies" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" ")" )))) ; theorem :: TEX_1:21 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Bool "not" (Set (Set (Var "D")) ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_tops_1 :::"boundary"::: ) ) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "A")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_tops_1 :::"boundary"::: ) )) "implies" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) )) ")" ")" )))) ; theorem :: TEX_1:22 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "D")) ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) )))) "implies" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" & "(" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set (Var "D"))) & (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "implies" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "D")) ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ))) ")" ")" )))) ; theorem :: TEX_1:23 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "st" (Bool (Bool (Bool "not" (Set (Set (Var "D")) ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set (Var "D")) ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) )))) "implies" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) ) ")" ) ")" & "(" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set (Var "D"))) & (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) )) "implies" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set (Var "D")) ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ))) ")" ")" )))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: TEX_1:24 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" )) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" )) "#)" )) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "implies" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "A")) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) )) "implies" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" ")" ) ")" ) ")" ) ")" )))) ; theorem :: TEX_1:25 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" )) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" )) "#)" )) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "D")) ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) )))) "implies" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" & "(" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set (Var "D"))) & (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "implies" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "D")) ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ))) ")" ")" ) ")" ) ")" ) ")" )))) ; theorem :: TEX_1:26 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" )) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" )) "#)" )) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ))) ")" ) ")" ) ")" )))) ; theorem :: TEX_1:27 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" )) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" )) "#)" )) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set (Var "D")))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ))) ")" ) ")" ) ")" )))) ; theorem :: TEX_1:28 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_tex_1 :::"ADTS"::: ) (Set (Var "D")))) "iff" (Bool (Set (Var "D")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) )) ")" ))) ; theorem :: TEX_1:29 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) "holds" (Bool "(" (Bool (Set ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_compts_1 :::"1TopSp"::: ) (Set (Var "D")))) "iff" (Bool (Set (Var "D")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) )) ")" ))) ; theorem :: TEX_1:30 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "d0")) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool (Set ($#k1_compts_1 :::"1TopSp"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "D")) "," (Set (Var "d0")) ")" ")" ) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set (Var "A"))))))) ; begin definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); redefine attr "X" is :::"discrete"::: means :: TEX_1:def 5 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" "X" "holds" (Bool (Bool "not" (Set (Var "A")) "is" ($#v2_tops_1 :::"boundary"::: ) ))); end; :: deftheorem defines :::"discrete"::: TEX_1:def 5 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_tdlat_3 :::"discrete"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Bool "not" (Set (Var "A")) "is" ($#v2_tops_1 :::"boundary"::: ) ))) ")" )); theorem :: TEX_1:31 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_tdlat_3 :::"discrete"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "holds" (Bool "not" (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) ))) ")" )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#~v3_tdlat_3 "non" ($#v3_tdlat_3 :::"almost_discrete"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); redefine attr "X" is :::"almost_discrete"::: means :: TEX_1:def 6 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" "X" "holds" (Bool (Bool "not" (Set (Var "A")) "is" ($#v3_tops_1 :::"nowhere_dense"::: ) ))); end; :: deftheorem defines :::"almost_discrete"::: TEX_1:def 6 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Bool "not" (Set (Var "A")) "is" ($#v3_tops_1 :::"nowhere_dense"::: ) ))) ")" )); theorem :: TEX_1:32 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) "holds" (Bool "not" (Bool (Set (Var "A")) "is" ($#v1_tops_3 :::"everywhere_dense"::: ) ))) ")" )) ; theorem :: TEX_1:33 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "X")) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) )) "iff" (Bool "ex" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_tops_1 :::"boundary"::: ) ) & (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" )) ")" )) ; theorem :: TEX_1:34 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "X")) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) )) "iff" (Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))) & (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) ) & (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" )) ")" )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: TEX_1:35 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "c0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) "holds" (Bool "(" (Bool (Bool "not" (Set (Set (Var "C")) ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "c0")) ($#k6_domain_1 :::"}"::: ) )) "is" ($#v1_xboole_0 :::"empty"::: ) )) "iff" (Bool "not" (Bool (Set ($#k3_tex_1 :::"STS"::: ) "(" (Set (Var "C")) "," (Set (Var "c0")) ")" ) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) )) ")" ))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#~v3_tdlat_3 "non" ($#v3_tdlat_3 :::"almost_discrete"::: ) ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: TEX_1:36 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_tops_1 :::"boundary"::: ) )) "holds" (Bool "not" (Bool (Set (Set (Var "X")) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set "(" (Set (Var "A")) ($#k3_subset_1 :::"`"::: ) ")" )) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) )))) ; theorem :: TEX_1:37 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))) & (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) )) "holds" (Bool "not" (Bool (Set (Set (Var "X")) ($#k6_tmap_1 :::"modified_with_respect_to"::: ) (Set (Var "A"))) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) )))) ;