:: TEX_2 semantic presentation begin theorem :: TEX_2:1 (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "B")) "being" (Num 1) ($#v3_card_1 :::"-element"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B"))))) ; theorem :: TEX_2:2 (Bool "for" (Set (Var "A")) "being" (Num 1) ($#v3_card_1 :::"-element"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Bool "not" (Set (Set (Var "A")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "B"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))))) ; registrationlet "S" be (Num 1) ($#v3_card_1 :::"-element"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_subset_1 :::"proper"::: ) -> ($#v1_xboole_0 :::"empty"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "S"); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) -> ($#~v1_subset_1 "non" ($#v1_subset_1 :::"proper"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "S"); end; theorem :: TEX_2:3 (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "y")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v1_subset_1 :::"proper"::: ) )) "holds" (Bool "not" (Bool (Set (Var "S")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )))) ; theorem :: TEX_2:4 (Bool "for" (Set (Var "S")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "S")) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "y")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v1_subset_1 :::"proper"::: ) ))) ; registrationlet "S" be (Num 1) ($#v3_card_1 :::"-element"::: ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_subset_1 "non" ($#v1_subset_1 :::"proper"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_zfmisc_1 :::"trivial"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "S"); end; registrationlet "S" be ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_zfmisc_1 :::"trivial"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_subset_1 :::"proper"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "S"); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_subset_1 "non" ($#v1_subset_1 :::"proper"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "S"); end; registrationlet "S" be ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_zfmisc_1 :::"trivial"::: ) ($#v1_subset_1 :::"proper"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "S"); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#~v1_subset_1 "non" ($#v1_subset_1 :::"proper"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) "S"); end; theorem :: TEX_2:5 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "y")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v1_subset_1 :::"proper"::: ) )) "holds" (Bool "not" (Bool (Set (Var "Y")) "is" ($#v7_struct_0 :::"trivial"::: ) )))) ; theorem :: TEX_2:6 (Bool "for" (Set (Var "Y")) "being" ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "Y")) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "y")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v1_subset_1 :::"proper"::: ) ))) ; registrationlet "Y" be (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_subset_1 "non" ($#v1_subset_1 :::"proper"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_subset_1 "non" ($#v1_subset_1 :::"proper"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_zfmisc_1 :::"trivial"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")); end; registrationlet "Y" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_zfmisc_1 :::"trivial"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_subset_1 :::"proper"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_subset_1 "non" ($#v1_subset_1 :::"proper"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")); end; registrationlet "Y" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_zfmisc_1 :::"trivial"::: ) ($#v1_subset_1 :::"proper"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#~v1_subset_1 "non" ($#v1_subset_1 :::"proper"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")); end; registrationlet "Y" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_zfmisc_1 :::"trivial"::: ) ($#v1_subset_1 :::"proper"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")); end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "X")); cluster (Set "A" ($#k3_subset_1 :::"`"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; begin theorem :: TEX_2:7 (Bool "for" (Set (Var "Y0")) "," (Set (Var "Y1")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y1"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y1"))) "#)" )) & (Bool (Set (Var "Y0")) "is" ($#v2_pre_topc :::"TopSpace-like"::: ) )) "holds" (Bool (Set (Var "Y1")) "is" ($#v2_pre_topc :::"TopSpace-like"::: ) )) ; definitionlet "Y" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "IT" be ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "Y")); attr "IT" is :::"proper"::: means :: TEX_2:def 1 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Y" "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "IT"))) "holds" (Bool (Set (Var "A")) "is" ($#v1_subset_1 :::"proper"::: ) )); end; :: deftheorem defines :::"proper"::: TEX_2:def 1 : (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "IT")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_tex_2 :::"proper"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "IT"))))) "holds" (Bool (Set (Var "A")) "is" ($#v1_subset_1 :::"proper"::: ) )) ")" ))); theorem :: TEX_2:8 (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "Y0")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v1_subset_1 :::"proper"::: ) ) "iff" (Bool (Set (Var "Y0")) "is" ($#v1_tex_2 :::"proper"::: ) ) ")" )))) ; theorem :: TEX_2:9 (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "Y0")) "," (Set (Var "Y1")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y1"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y1"))) "#)" )) & (Bool (Set (Var "Y0")) "is" ($#v1_tex_2 :::"proper"::: ) )) "holds" (Bool (Set (Var "Y1")) "is" ($#v1_tex_2 :::"proper"::: ) ))) ; theorem :: TEX_2:10 (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "Y0")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))))) "holds" (Bool "not" (Bool (Set (Var "Y0")) "is" ($#v1_tex_2 :::"proper"::: ) )))) ; registrationlet "Y" be (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_tex_2 "non" ($#v1_tex_2 :::"proper"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_tex_2 "non" ($#v1_tex_2 :::"proper"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; registrationlet "Y" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_tex_2 "non" ($#v1_tex_2 :::"proper"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; registrationlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#~v1_tex_2 "non" ($#v1_tex_2 :::"proper"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; theorem :: TEX_2:11 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "Y0")) "being" ($#~v1_tex_2 "non" ($#v1_tex_2 :::"proper"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) "holds" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y"))) "#)" )))) ; registrationlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v1_tdlat_3 :::"discrete"::: ) -> ($#v2_pre_topc :::"TopSpace-like"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#v2_tdlat_3 :::"anti-discrete"::: ) -> ($#v2_pre_topc :::"TopSpace-like"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#~v2_pre_topc "non" ($#v2_pre_topc :::"TopSpace-like"::: ) ) -> ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#~v2_pre_topc "non" ($#v2_pre_topc :::"TopSpace-like"::: ) ) -> ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; theorem :: TEX_2:12 (Bool "for" (Set (Var "Y0")) "," (Set (Var "Y1")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y1"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y1"))) "#)" )) & (Bool (Set (Var "Y0")) "is" ($#v1_tdlat_3 :::"discrete"::: ) )) "holds" (Bool (Set (Var "Y1")) "is" ($#v1_tdlat_3 :::"discrete"::: ) )) ; theorem :: TEX_2:13 (Bool "for" (Set (Var "Y0")) "," (Set (Var "Y1")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y1"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y1"))) "#)" )) & (Bool (Set (Var "Y0")) "is" ($#v2_tdlat_3 :::"anti-discrete"::: ) )) "holds" (Bool (Set (Var "Y1")) "is" ($#v2_tdlat_3 :::"anti-discrete"::: ) )) ; registrationlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v1_tdlat_3 :::"discrete"::: ) -> ($#v3_tdlat_3 :::"almost_discrete"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#~v3_tdlat_3 "non" ($#v3_tdlat_3 :::"almost_discrete"::: ) ) -> ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#v2_tdlat_3 :::"anti-discrete"::: ) -> ($#v3_tdlat_3 :::"almost_discrete"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#~v3_tdlat_3 "non" ($#v3_tdlat_3 :::"almost_discrete"::: ) ) -> ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; theorem :: TEX_2:14 (Bool "for" (Set (Var "Y0")) "," (Set (Var "Y1")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y1"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y1"))) "#)" )) & (Bool (Set (Var "Y0")) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) )) "holds" (Bool (Set (Var "Y1")) "is" ($#v3_tdlat_3 :::"almost_discrete"::: ) )) ; registrationlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#v2_tdlat_3 :::"anti-discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; definitionlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "y" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "Y")); func :::"Sspace"::: "y" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y" means :: TEX_2:def 2 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) "y" ($#k6_domain_1 :::"}"::: ) )); end; :: deftheorem defines :::"Sspace"::: TEX_2:def 2 : (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) (Bool "for" (Set (Var "b3")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_tex_2 :::"Sspace"::: ) (Set (Var "y")))) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "y")) ($#k6_domain_1 :::"}"::: ) )) ")" )))); registrationlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v1_pre_topc :::"strict"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; registrationlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "y" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "Y")); cluster (Set ($#k1_tex_2 :::"Sspace"::: ) "y") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v1_pre_topc :::"strict"::: ) ; end; theorem :: TEX_2:15 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) "holds" (Bool "(" (Bool (Set ($#k1_tex_2 :::"Sspace"::: ) (Set (Var "y"))) "is" ($#v1_tex_2 :::"proper"::: ) ) "iff" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "y")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v1_subset_1 :::"proper"::: ) ) ")" ))) ; theorem :: TEX_2:16 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set ($#k1_tex_2 :::"Sspace"::: ) (Set (Var "y"))) "is" ($#v1_tex_2 :::"proper"::: ) )) "holds" (Bool "not" (Bool (Set (Var "Y")) "is" ($#v7_struct_0 :::"trivial"::: ) )))) ; registrationlet "Y" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v1_pre_topc :::"strict"::: ) ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; registrationlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; theorem :: TEX_2:17 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "Y0")) "being" (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) "st" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_tex_2 :::"Sspace"::: ) (Set (Var "y")) ")" )) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set "(" ($#k1_tex_2 :::"Sspace"::: ) (Set (Var "y")) ")" )) "#)" ))))) ; theorem :: TEX_2:18 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set ($#k1_tex_2 :::"Sspace"::: ) (Set (Var "y"))) "is" ($#v2_pre_topc :::"TopSpace-like"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_tex_2 :::"Sspace"::: ) (Set (Var "y"))) "is" ($#v1_tdlat_3 :::"discrete"::: ) ) & (Bool (Set ($#k1_tex_2 :::"Sspace"::: ) (Set (Var "y"))) "is" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) ")" ))) ; registrationlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#v2_tdlat_3 :::"anti-discrete"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "X")); cluster (Set ($#k1_tex_2 :::"Sspace"::: ) "x") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); 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 :::"anti-discrete"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "X")); cluster (Set ($#k1_tex_2 :::"Sspace"::: ) "x") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v1_tdlat_3 :::"discrete"::: ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v1_tex_2 "non" ($#v1_tex_2 :::"proper"::: ) ) -> ($#v1_borsuk_1 :::"closed"::: ) ($#v1_tsep_1 :::"open"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v1_tsep_1 "non" ($#v1_tsep_1 :::"open"::: ) ) -> ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v1_borsuk_1 "non" ($#v1_borsuk_1 :::"closed"::: ) ) -> ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_borsuk_1 :::"closed"::: ) ($#v1_tsep_1 :::"open"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_borsuk_1 :::"closed"::: ) ($#v1_tsep_1 :::"open"::: ) ($#v1_tdlat_3 :::"discrete"::: ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tex_2 :::"proper"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_borsuk_1 "non" ($#v1_borsuk_1 :::"closed"::: ) ) ($#~v1_tsep_1 "non" ($#v1_tsep_1 :::"open"::: ) ) ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v1_tdlat_3 :::"discrete"::: ) ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#~v1_borsuk_1 "non" ($#v1_borsuk_1 :::"closed"::: ) ) ($#~v1_tsep_1 "non" ($#v1_tsep_1 :::"open"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; begin definitionlet "Y" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "IT" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "Y")); attr "IT" is :::"discrete"::: means :: TEX_2:def 3 (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Y" "st" (Bool (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) "IT")) "holds" (Bool "ex" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Y" "st" (Bool "(" (Bool (Set (Var "G")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set "IT" ($#k9_subset_1 :::"/\"::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) ")" ))); end; :: deftheorem defines :::"discrete"::: TEX_2:def 3 : (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "IT")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_tex_2 :::"discrete"::: ) ) "iff" (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set (Var "IT")))) "holds" (Bool "ex" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool "(" (Bool (Set (Var "G")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Set (Var "IT")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) ")" ))) ")" ))); definitionlet "Y" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "Y")); redefine attr "A" is :::"discrete"::: means :: TEX_2:def 4 (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Y" "st" (Bool (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) "A")) "holds" (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Y" "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set "A" ($#k9_subset_1 :::"/\"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) ")" ))); end; :: deftheorem defines :::"discrete"::: TEX_2:def 4 : (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) "iff" (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) ")" ))) ")" ))); theorem :: TEX_2:19 (Bool "for" (Set (Var "Y0")) "," (Set (Var "Y1")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "D0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y0")) (Bool "for" (Set (Var "D1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y1")) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y1"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y1"))) "#)" )) & (Bool (Set (Var "D0")) ($#r1_hidden :::"="::: ) (Set (Var "D1"))) & (Bool (Set (Var "D0")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool (Set (Var "D1")) "is" ($#v2_tex_2 :::"discrete"::: ) )))) ; theorem :: TEX_2:20 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "Y0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) "iff" (Bool (Set (Var "Y0")) "is" ($#v1_tdlat_3 :::"discrete"::: ) ) ")" )))) ; theorem :: TEX_2:21 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) "iff" (Bool (Set (Var "Y")) "is" ($#v1_tdlat_3 :::"discrete"::: ) ) ")" ))) ; theorem :: TEX_2:22 (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool (Set (Var "B")) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:23 (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) "or" (Bool (Set (Var "B")) "is" ($#v2_tex_2 :::"discrete"::: ) ) ")" )) "holds" (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B"))) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:24 (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool "(" "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "P")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "Q")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Set (Var "P")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Q"))) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ) ")" )) "holds" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) & (Bool (Set (Var "B")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:25 (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool "(" "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "P")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "Q")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "P")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Set (Var "P")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Q"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ) ")" )) "holds" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) & (Bool (Set (Var "B")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:26 (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool "(" (Bool (Set (Var "G")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ")" ))))) ; theorem :: TEX_2:27 (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ")" ))))) ; theorem :: TEX_2:28 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A0")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A0")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool "ex" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v1_tdlat_3 :::"discrete"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "st" (Bool (Set (Var "A0")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X0"))))))) ; theorem :: TEX_2:29 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:30 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:31 (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 "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "G")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ")" )) ")" )) "holds" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:32 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) & (Bool (Set (Var "B")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:33 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) & (Bool (Set (Var "B")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:34 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v1_tops_3 :::"everywhere_dense"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ))) ; theorem :: TEX_2:35 (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")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) "iff" (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "D")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "D")))) ")" ))) ; theorem :: TEX_2:36 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: TEX_2:37 (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" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:38 (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" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) "iff" (Bool (Set (Var "A")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) ) ")" ))) ; definitionlet "Y" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "IT" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "Y")); attr "IT" is :::"maximal_discrete"::: means :: TEX_2:def 5 (Bool "(" (Bool "IT" "is" ($#v2_tex_2 :::"discrete"::: ) ) & (Bool "(" "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Y" "st" (Bool (Bool (Set (Var "D")) "is" ($#v2_tex_2 :::"discrete"::: ) ) & (Bool "IT" ($#r1_tarski :::"c="::: ) (Set (Var "D")))) "holds" (Bool "IT" ($#r1_hidden :::"="::: ) (Set (Var "D"))) ")" ) ")" ); end; :: deftheorem defines :::"maximal_discrete"::: TEX_2:def 5 : (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "IT")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) ) "iff" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_tex_2 :::"discrete"::: ) ) & (Bool "(" "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "D")) "is" ($#v2_tex_2 :::"discrete"::: ) ) & (Bool (Set (Var "IT")) ($#r1_tarski :::"c="::: ) (Set (Var "D")))) "holds" (Bool (Set (Var "IT")) ($#r1_hidden :::"="::: ) (Set (Var "D"))) ")" ) ")" ) ")" ))); theorem :: TEX_2:39 (Bool "for" (Set (Var "Y0")) "," (Set (Var "Y1")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "D0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y0")) (Bool "for" (Set (Var "D1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y1")) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y1"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y1"))) "#)" )) & (Bool (Set (Var "D0")) ($#r1_hidden :::"="::: ) (Set (Var "D1"))) & (Bool (Set (Var "D0")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )) "holds" (Bool (Set (Var "D1")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )))) ; theorem :: TEX_2:40 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#v1_xboole_0 :::"empty"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Bool "not" (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )))) ; theorem :: TEX_2:41 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) ))) ; theorem :: TEX_2:42 (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")) "is" ($#v1_tops_1 :::"dense"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) ))) ; theorem :: TEX_2:43 (Bool "for" (Set (Var "X")) "being" ($#~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 (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) ) "iff" (Bool (Bool "not" (Set (Var "A")) "is" ($#v1_subset_1 :::"proper"::: ) )) ")" ))) ; theorem :: TEX_2:44 (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" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) ) "iff" (Bool (Set (Var "A")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) ) ")" ))) ; definitionlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "IT" be ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "Y")); attr "IT" is :::"maximal_discrete"::: means :: TEX_2:def 6 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "Y" "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "IT"))) "holds" (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )); end; :: deftheorem defines :::"maximal_discrete"::: TEX_2:def 6 : (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "IT")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v4_tex_2 :::"maximal_discrete"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "IT"))))) "holds" (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )) ")" ))); theorem :: TEX_2:45 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "Y0")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) ) "iff" (Bool (Set (Var "Y0")) "is" ($#v4_tex_2 :::"maximal_discrete"::: ) ) ")" )))) ; registrationlet "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_tex_2 :::"maximal_discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_tex_2 "non" ($#v4_tex_2 :::"maximal_discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "Y"; end; theorem :: TEX_2:46 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "X0")) "is" ($#v4_tex_2 :::"maximal_discrete"::: ) ) "iff" (Bool "(" (Bool (Set (Var "X0")) "is" ($#v1_tdlat_3 :::"discrete"::: ) ) & (Bool "(" "for" (Set (Var "Y0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "X0")) "is" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y0")))) "holds" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "X0"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y0"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y0"))) "#)" )) ")" ) ")" ) ")" ))) ; theorem :: TEX_2:47 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A0")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A0")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )) "holds" (Bool "ex" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "X0")) "is" ($#v4_tex_2 :::"maximal_discrete"::: ) ) & (Bool (Set (Var "A0")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X0")))) ")" )))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v4_tex_2 :::"maximal_discrete"::: ) -> ($#~v1_tex_2 "non" ($#v1_tex_2 :::"proper"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#v1_tex_2 :::"proper"::: ) -> ($#~v4_tex_2 "non" ($#v4_tex_2 :::"maximal_discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v1_tex_2 "non" ($#v1_tex_2 :::"proper"::: ) ) -> ($#v4_tex_2 :::"maximal_discrete"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v4_tex_2 "non" ($#v4_tex_2 :::"maximal_discrete"::: ) ) -> ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_tex_2 :::"maximal_discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_tex_2 "non" ($#v4_tex_2 :::"maximal_discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_tex_2 :::"maximal_discrete"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_tex_2 "non" ($#v4_tex_2 :::"maximal_discrete"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; begin scheme :: TEX_2:sch 1 ExChoiceFCol{ F1() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) , F2() -> ($#m1_subset_1 :::"Subset-Family":::) "of" (Set F1 "(" ")" ), P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set F2 "(" ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F1 "(" ")" )) "st" (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool P1[(Set (Var "S")) "," (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "S")))]))) provided (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set F1 "(" ")" ) "st" (Bool P1[(Set (Var "S")) "," (Set (Var "x"))]))) proof end; theorem :: TEX_2:48 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ")" ) where a "is" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) : (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "}" )))) ; theorem :: TEX_2:49 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "b")) ($#k6_domain_1 :::"}"::: ) )))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "b")) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: TEX_2:50 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "b")) ($#k6_domain_1 :::"}"::: ) ))) "or" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "b")) ($#k6_domain_1 :::"}"::: ) ))) ")" ))) ; theorem :: TEX_2:51 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "F")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ")" )) ")" )) "holds" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:52 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ")" )) "holds" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ))) ; theorem :: TEX_2:53 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set (Var "b")))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "b")) ($#k6_domain_1 :::"}"::: ) )))) ")" ))) ; theorem :: TEX_2:54 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) )) ")" ))) ")" ))) ; theorem :: TEX_2:55 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) "or" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ) & (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )) "holds" (Bool "not" (Bool (Set (Var "A")) "is" ($#v1_subset_1 :::"proper"::: ) )))) ; theorem :: TEX_2:56 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v1_tops_1 :::"dense"::: ) ))) ; theorem :: TEX_2:57 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) )) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ")" ) where a "is" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) : (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "}" ) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))))) ; theorem :: TEX_2:58 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) )) ")" ))) ")" ))) ; theorem :: TEX_2:59 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_tex_2 :::"discrete"::: ) )) "holds" (Bool "ex" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "M"))) & (Bool (Set (Var "M")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) ) ")" )))) ; theorem :: TEX_2:60 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "ex" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Set (Var "M")) "is" ($#v3_tex_2 :::"maximal_discrete"::: ) ))) ; theorem :: TEX_2:61 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) (Bool "ex" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "Y0")) "is" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X0"))) & (Bool (Set (Var "X0")) "is" ($#v4_tex_2 :::"maximal_discrete"::: ) ) ")" )))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_tdlat_3 "non" ($#v1_tdlat_3 :::"discrete"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_tex_2 :::"maximal_discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tex_2 :::"proper"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v1_tex_2 "non" ($#v1_tex_2 :::"proper"::: ) ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_tex_2 "non" ($#v4_tex_2 :::"maximal_discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v2_tdlat_3 "non" ($#v2_tdlat_3 :::"anti-discrete"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_tex_2 :::"maximal_discrete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v4_tex_2 "non" ($#v4_tex_2 :::"maximal_discrete"::: ) ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v4_tex_2 :::"maximal_discrete"::: ) for ($#m1_pre_topc :::"SubSpace"::: ) "of" "X"; end; begin theorem :: TEX_2:62 (Bool "for" (Set (Var "X")) "being" ($#v1_tdlat_3 :::"discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: TEX_2:63 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool "(" "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )) ")" )) "holds" (Bool (Set (Var "X")) "is" ($#v1_tdlat_3 :::"discrete"::: ) )) ; theorem :: TEX_2:64 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_tdlat_3 :::"anti-discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )))) ; theorem :: TEX_2:65 (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )) ")" )) "holds" (Bool (Set (Var "Y")) "is" ($#v2_tdlat_3 :::"anti-discrete"::: ) )) ; theorem :: TEX_2:66 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) (Bool "ex" (Set (Var "r")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "X0")) "st" (Bool (Set (Var "r")) "is" ($#v3_borsuk_1 :::"being_a_retraction"::: ) )))) ; theorem :: TEX_2:67 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Var "X0")) ($#r1_borsuk_1 :::"is_a_retract_of"::: ) (Set (Var "X"))))) ; theorem :: TEX_2:68 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_tex_2 :::"maximal_discrete"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) (Bool "ex" (Set (Var "r")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "X0")) "st" (Bool (Set (Var "r")) "is" ($#v3_borsuk_1 :::"being_a_retraction"::: ) )))) ; theorem :: TEX_2:69 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_tex_2 :::"maximal_discrete"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Var "X0")) ($#r1_borsuk_1 :::"is_a_retract_of"::: ) (Set (Var "X"))))) ; theorem :: TEX_2:70 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_tex_2 :::"maximal_discrete"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "r")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "X0")) "st" (Bool (Bool (Set (Var "r")) "is" ($#v3_borsuk_1 :::"being_a_retraction"::: ) )) "holds" (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X0")) (Bool "for" (Set (Var "E")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set (Var "E")))) "holds" (Bool (Set (Set (Var "r")) ($#k8_relset_1 :::"""::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "E"))))))))) ; theorem :: TEX_2:71 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_tex_2 :::"maximal_discrete"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "r")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "X0")) "st" (Bool (Bool (Set (Var "r")) "is" ($#v3_borsuk_1 :::"being_a_retraction"::: ) )) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X0")) (Bool "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "b")))) "holds" (Bool (Set (Set (Var "r")) ($#k8_relset_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "b")) ($#k6_domain_1 :::"}"::: ) )))))))) ; theorem :: TEX_2:72 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) (Bool "ex" (Set (Var "r")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "X0")) "st" (Bool (Set (Var "r")) "is" ($#v3_borsuk_1 :::"being_a_retraction"::: ) )))) ; theorem :: TEX_2:73 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_tdlat_3 :::"almost_discrete"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X0")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_tdlat_3 :::"discrete"::: ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Var "X0")) ($#r1_borsuk_1 :::"is_a_retract_of"::: ) (Set (Var "X"))))) ;