:: TDLAT_2 semantic presentation begin theorem :: TDLAT_2:1 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")) ")" ))) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")) ")" ))) ")" ))) ; theorem :: TDLAT_2:2 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")) ")" ) ")" ))) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")) ")" ) ")" ))) ")" ))) ; theorem :: TDLAT_2:3 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))))) ; theorem :: TDLAT_2:4 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))))) ; theorem :: TDLAT_2:5 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")) ")" )))) "holds" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")) ")" ) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:6 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")) ")" ) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")) ")" ) ")" ))))) ; begin notationlet "T" be ($#l1_pre_topc :::"TopSpace":::); let "F" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); synonym :::"Cl"::: "F" for :::"clf"::: "F"; end; theorem :: TDLAT_2:7 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "B")))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) "}" ))) ; theorem :: TDLAT_2:8 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:9 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))) ; theorem :: TDLAT_2:10 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" (Set (Var "F")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "G")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "G")) ")" ))))) ; theorem :: TDLAT_2:11 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "G")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" (Set (Var "F")) ($#k7_subset_1 :::"\"::: ) (Set (Var "G")) ")" ))))) ; theorem :: TDLAT_2:12 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "(" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")))) & (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ))) ")" )))) ; theorem :: TDLAT_2:13 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:14 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:15 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ))))) ; definitionlet "T" be ($#l1_pre_topc :::"TopSpace":::); let "F" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); func :::"Int"::: "F" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "T" means :: TDLAT_2:def 1 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "B")))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) "F") ")" )) ")" )); projectivity (Bool "for" (Set (Var "b1")) "," (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) "iff" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "B")))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) ")" )) ")" ) ")" )) "holds" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) "iff" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "B")))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "b1"))) ")" )) ")" ))) ; end; :: deftheorem defines :::"Int"::: TDLAT_2:def 1 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")))) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "B")))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) ")" )) ")" ))); theorem :: TDLAT_2:16 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "B")))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) "}" ))) ; theorem :: TDLAT_2:17 canceled; theorem :: TDLAT_2:18 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F"))) "is" ($#v1_tops_2 :::"open"::: ) ))) ; theorem :: TDLAT_2:19 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))) ; theorem :: TDLAT_2:20 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "A")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")) ")" ) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: TDLAT_2:21 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G")))) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "G")))))) ; theorem :: TDLAT_2:22 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set "(" (Set (Var "F")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "G")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "G")) ")" ))))) ; theorem :: TDLAT_2:23 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set "(" (Set (Var "F")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "G")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "G")) ")" ))))) ; theorem :: TDLAT_2:24 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "G")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tdlat_2 :::"Int"::: ) (Set "(" (Set (Var "F")) ($#k7_subset_1 :::"\"::: ) (Set (Var "G")) ")" ))))) ; theorem :: TDLAT_2:25 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "(" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ))) & (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")))) ")" )))) ; theorem :: TDLAT_2:26 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")))))) ; theorem :: TDLAT_2:27 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")))))) ; theorem :: TDLAT_2:28 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:29 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:30 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_finset_1 :::"finite"::: ) )) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:31 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "B")) ")" ))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) "}" ))) ; theorem :: TDLAT_2:32 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "B")) ")" ))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) "}" ))) ; theorem :: TDLAT_2:33 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "B")) ")" ) ")" ))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) "}" ))) ; theorem :: TDLAT_2:34 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool "ex" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "B")) ")" ) ")" ))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) "}" ))) ; theorem :: TDLAT_2:35 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:36 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:37 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:38 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:39 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:40 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:41 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ))))) ; theorem :: TDLAT_2:42 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ))))) ; theorem :: TDLAT_2:43 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ))))) ; theorem :: TDLAT_2:44 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ))))) ; theorem :: TDLAT_2:45 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:46 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ))))) ; theorem :: TDLAT_2:47 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:48 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:49 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ))))) ; theorem :: TDLAT_2:50 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ))))) ; theorem :: TDLAT_2:51 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ))))) ; theorem :: TDLAT_2:52 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ))))) ; theorem :: TDLAT_2:53 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:54 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:55 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:56 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set "(" ($#k1_pcomps_1 :::"Cl"::: ) (Set "(" ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))))) ; theorem :: TDLAT_2:57 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set (Var "A")) ")" ))) ")" )) "holds" (Bool "(" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ))) & (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))) ")" ))) ; theorem :: TDLAT_2:58 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" )) "holds" (Bool "(" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")))) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ))) ")" ))) ; begin theorem :: TDLAT_2:59 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) "is" ($#v4_tops_1 :::"condensed"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ) ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B"))) "iff" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" ))) ; theorem :: TDLAT_2:60 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_tops_1 :::"condensed"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" ) ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "A"))) "iff" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" ))) ; theorem :: TDLAT_2:61 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v5_tops_1 :::"closed_condensed"::: ) ) & (Bool (Set (Var "B")) "is" ($#v5_tops_1 :::"closed_condensed"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "B")))) "iff" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" ))) ; theorem :: TDLAT_2:62 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v6_tops_1 :::"open_condensed"::: ) ) & (Bool (Set (Var "B")) "is" ($#v6_tops_1 :::"open_condensed"::: ) )) "holds" (Bool "(" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "B")))) "iff" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" ))) ; theorem :: TDLAT_2:63 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v5_tops_1 :::"closed_condensed"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "A"))))) ; theorem :: TDLAT_2:64 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) "is" ($#v6_tops_1 :::"open_condensed"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "B"))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "IT" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); attr "IT" is :::"domains-family"::: means :: TDLAT_2:def 2 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) "IT")) "holds" (Bool (Set (Var "A")) "is" ($#v4_tops_1 :::"condensed"::: ) )); end; :: deftheorem defines :::"domains-family"::: TDLAT_2:def 2 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "IT")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))) "holds" (Bool (Set (Var "A")) "is" ($#v4_tops_1 :::"condensed"::: ) )) ")" ))); theorem :: TDLAT_2:65 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set ($#k1_tdlat_1 :::"Domains_of"::: ) (Set (Var "T")))) "iff" (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) ) ")" ))) ; theorem :: TDLAT_2:66 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) )) "holds" (Bool "(" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ))) & (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))) ")" ))) ; theorem :: TDLAT_2:67 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")))) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ))) ")" ))) ; theorem :: TDLAT_2:68 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) )) "holds" (Bool (Set (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) "is" ($#v4_tops_1 :::"condensed"::: ) ))) ; theorem :: TDLAT_2:69 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_tops_1 :::"condensed"::: ) ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" ) ")" ))) ; theorem :: TDLAT_2:70 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) )) "holds" (Bool (Set (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) "is" ($#v4_tops_1 :::"condensed"::: ) ))) ; theorem :: TDLAT_2:71 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" ) & (Bool "(" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_tops_1 :::"condensed"::: ) ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ) ")" )))) ")" ) ")" ))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "IT" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); attr "IT" is :::"closed-domains-family"::: means :: TDLAT_2:def 3 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) "IT")) "holds" (Bool (Set (Var "A")) "is" ($#v5_tops_1 :::"closed_condensed"::: ) )); end; :: deftheorem defines :::"closed-domains-family"::: TDLAT_2:def 3 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "IT")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v2_tdlat_2 :::"closed-domains-family"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))) "holds" (Bool (Set (Var "A")) "is" ($#v5_tops_1 :::"closed_condensed"::: ) )) ")" ))); theorem :: TDLAT_2:72 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set ($#k5_tdlat_1 :::"Closed_Domains_of"::: ) (Set (Var "T")))) "iff" (Bool (Set (Var "F")) "is" ($#v2_tdlat_2 :::"closed-domains-family"::: ) ) ")" ))) ; theorem :: TDLAT_2:73 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tdlat_2 :::"closed-domains-family"::: ) )) "holds" (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) ))) ; theorem :: TDLAT_2:74 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tdlat_2 :::"closed-domains-family"::: ) )) "holds" (Bool (Set (Var "F")) "is" ($#v2_tops_2 :::"closed"::: ) ))) ; theorem :: TDLAT_2:75 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) )) "holds" (Bool (Set ($#k1_pcomps_1 :::"Cl"::: ) (Set (Var "F"))) "is" ($#v2_tdlat_2 :::"closed-domains-family"::: ) ))) ; theorem :: TDLAT_2:76 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tdlat_2 :::"closed-domains-family"::: ) )) "holds" (Bool "(" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" )) "is" ($#v5_tops_1 :::"closed_condensed"::: ) ) & (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" )) "is" ($#v5_tops_1 :::"closed_condensed"::: ) ) ")" ))) ; theorem :: TDLAT_2:77 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v5_tops_1 :::"closed_condensed"::: ) ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" ) ")" ))) ; theorem :: TDLAT_2:78 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "F")) "is" ($#v2_tops_2 :::"closed"::: ) )) "implies" (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) ")" & (Bool "(" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v5_tops_1 :::"closed_condensed"::: ) ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" )))) ")" ) ")" ))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "IT" be ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Const "T")); attr "IT" is :::"open-domains-family"::: means :: TDLAT_2:def 4 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) "IT")) "holds" (Bool (Set (Var "A")) "is" ($#v6_tops_1 :::"open_condensed"::: ) )); end; :: deftheorem defines :::"open-domains-family"::: TDLAT_2:def 4 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "IT")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "IT")) "is" ($#v3_tdlat_2 :::"open-domains-family"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "IT")))) "holds" (Bool (Set (Var "A")) "is" ($#v6_tops_1 :::"open_condensed"::: ) )) ")" ))); theorem :: TDLAT_2:79 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set ($#k9_tdlat_1 :::"Open_Domains_of"::: ) (Set (Var "T")))) "iff" (Bool (Set (Var "F")) "is" ($#v3_tdlat_2 :::"open-domains-family"::: ) ) ")" ))) ; theorem :: TDLAT_2:80 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v3_tdlat_2 :::"open-domains-family"::: ) )) "holds" (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) ))) ; theorem :: TDLAT_2:81 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v3_tdlat_2 :::"open-domains-family"::: ) )) "holds" (Bool (Set (Var "F")) "is" ($#v1_tops_2 :::"open"::: ) ))) ; theorem :: TDLAT_2:82 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) )) "holds" (Bool (Set ($#k1_tdlat_2 :::"Int"::: ) (Set (Var "F"))) "is" ($#v3_tdlat_2 :::"open-domains-family"::: ) ))) ; theorem :: TDLAT_2:83 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v3_tdlat_2 :::"open-domains-family"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" )) "is" ($#v6_tops_1 :::"open_condensed"::: ) ) & (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" )) "is" ($#v6_tops_1 :::"open_condensed"::: ) ) ")" ))) ; theorem :: TDLAT_2:84 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "F")) "is" ($#v1_tops_2 :::"open"::: ) )) "implies" (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" )))) ")" & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v6_tops_1 :::"open_condensed"::: ) ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) ")" ) ")" ))) ; theorem :: TDLAT_2:85 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" ) & (Bool "(" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "or" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v6_tops_1 :::"open_condensed"::: ) ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" )))) ")" ) ")" ))) ; begin theorem :: TDLAT_2:86 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_tdlat_1 :::"Domains_of"::: ) (Set (Var "T"))))) ; theorem :: TDLAT_2:87 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_tdlat_1 :::"Domains_of"::: ) (Set (Var "T"))) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "(" (Bool (Set (Set (Var "a")) ($#k1_lattices :::""\/""::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ) ")" ) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ))) & (Bool (Set (Set (Var "a")) ($#k2_lattices :::""/\""::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" ) ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" ))) ")" )))) ; theorem :: TDLAT_2:88 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set ($#k5_lattices :::"Bottom"::: ) (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "T")))) & (Bool (Set ($#k6_lattices :::"Top"::: ) (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")))) ")" )) ; theorem :: TDLAT_2:89 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_tdlat_1 :::"Domains_of"::: ) (Set (Var "T"))) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_lattices :::"[="::: ) (Set (Var "b"))) "iff" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )))) ; theorem :: TDLAT_2:90 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool "(" (Bool (Set (Var "X")) ($#r4_lattice3 :::"is_less_than"::: ) (Set (Var "a"))) & (Bool "(" "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r4_lattice3 :::"is_less_than"::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "a")) ($#r1_lattices :::"[="::: ) (Set (Var "b"))) ")" ) ")" )))) ; theorem :: TDLAT_2:91 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T"))) "is" ($#v4_lattice3 :::"complete"::: ) )) ; theorem :: TDLAT_2:92 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "F")))) "holds" (Bool (Set ($#k15_lattice3 :::""\/""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" ) ")" )))))) ; theorem :: TDLAT_2:93 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tdlat_2 :::"domains-family"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "F")))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k16_lattice3 :::""/\""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k16_lattice3 :::""/\""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")))) ")" ")" )))) ; begin theorem :: TDLAT_2:94 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_tdlat_1 :::"Closed_Domains_of"::: ) (Set (Var "T"))))) ; theorem :: TDLAT_2:95 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_tdlat_1 :::"Closed_Domains_of"::: ) (Set (Var "T"))) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "(" (Bool (Set (Set (Var "a")) ($#k1_lattices :::""\/""::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")))) & (Bool (Set (Set (Var "a")) ($#k2_lattices :::""/\""::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")) ")" ) ")" ))) ")" )))) ; theorem :: TDLAT_2:96 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set ($#k5_lattices :::"Bottom"::: ) (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "T")))) & (Bool (Set ($#k6_lattices :::"Top"::: ) (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")))) ")" )) ; theorem :: TDLAT_2:97 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_tdlat_1 :::"Closed_Domains_of"::: ) (Set (Var "T"))) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_lattices :::"[="::: ) (Set (Var "b"))) "iff" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )))) ; theorem :: TDLAT_2:98 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool "(" (Bool (Set (Var "X")) ($#r4_lattice3 :::"is_less_than"::: ) (Set (Var "a"))) & (Bool "(" "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r4_lattice3 :::"is_less_than"::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "a")) ($#r1_lattices :::"[="::: ) (Set (Var "b"))) ")" ) ")" )))) ; theorem :: TDLAT_2:99 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T"))) "is" ($#v4_lattice3 :::"complete"::: ) )) ; theorem :: TDLAT_2:100 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tdlat_2 :::"closed-domains-family"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "F")))) "holds" (Bool (Set ($#k15_lattice3 :::""\/""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" )))))) ; theorem :: TDLAT_2:101 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tdlat_2 :::"closed-domains-family"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "F")))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k16_lattice3 :::""/\""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k16_lattice3 :::""/\""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k8_tdlat_1 :::"Closed_Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")))) ")" ")" )))) ; theorem :: TDLAT_2:102 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tdlat_2 :::"closed-domains-family"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "F")))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k16_lattice3 :::""/\""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ) ")" ))) ")" & "(" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k16_lattice3 :::""/\""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")))) ")" ")" )))) ; theorem :: TDLAT_2:103 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_tdlat_1 :::"Open_Domains_of"::: ) (Set (Var "T"))))) ; theorem :: TDLAT_2:104 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_tdlat_1 :::"Open_Domains_of"::: ) (Set (Var "T"))) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "(" (Bool (Set (Set (Var "a")) ($#k1_lattices :::""\/""::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" ) ")" ))) & (Bool (Set (Set (Var "a")) ($#k2_lattices :::""/\""::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B")))) ")" )))) ; theorem :: TDLAT_2:105 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set ($#k5_lattices :::"Bottom"::: ) (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "T")))) & (Bool (Set ($#k6_lattices :::"Top"::: ) (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")))) ")" )) ; theorem :: TDLAT_2:106 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_tdlat_1 :::"Open_Domains_of"::: ) (Set (Var "T"))) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Var "A"))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_lattices :::"[="::: ) (Set (Var "b"))) "iff" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) ")" )))) ; theorem :: TDLAT_2:107 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool "(" (Bool (Set (Var "X")) ($#r4_lattice3 :::"is_less_than"::: ) (Set (Var "a"))) & (Bool "(" "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r4_lattice3 :::"is_less_than"::: ) (Set (Var "b")))) "holds" (Bool (Set (Var "a")) ($#r1_lattices :::"[="::: ) (Set (Var "b"))) ")" ) ")" )))) ; theorem :: TDLAT_2:108 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T"))) "is" ($#v4_lattice3 :::"complete"::: ) )) ; theorem :: TDLAT_2:109 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v3_tdlat_2 :::"open-domains-family"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "F")))) "holds" (Bool (Set ($#k15_lattice3 :::""\/""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" )))))) ; theorem :: TDLAT_2:110 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v3_tdlat_2 :::"open-domains-family"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "F")))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k16_lattice3 :::""/\""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F")) ")" ))) ")" & "(" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) "implies" (Bool (Set ($#k16_lattice3 :::""/\""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k12_tdlat_1 :::"Open_Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")))) ")" ")" )))) ; theorem :: TDLAT_2:111 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v3_tdlat_2 :::"open-domains-family"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "F")))) "holds" (Bool (Set ($#k15_lattice3 :::""\/""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k4_tdlat_1 :::"Domains_Lattice"::: ) (Set (Var "T")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set "(" ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")) ")" ) ")" )))))) ;