:: WAYBEL19 semantic presentation begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_9 :::"TopRelStr"::: ) ; attr "T" is :::"lower"::: means :: WAYBEL19:def 1 (Bool "{" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) where x "is" ($#m1_subset_1 :::"Element":::) "of" "T" : (Bool verum) "}" "is" ($#m1_subset_1 :::"prebasis":::) "of" "T"); end; :: deftheorem defines :::"lower"::: WAYBEL19:def 1 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_9 :::"TopRelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_waybel19 :::"lower"::: ) ) "iff" (Bool "{" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) : (Bool verum) "}" "is" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T"))) ")" )); registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) -> (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v1_waybel19 :::"lower"::: ) for ($#l1_waybel_9 :::"TopRelStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel_9 :::"strict"::: ) ($#~v1_yellow_3 "non" ($#v1_yellow_3 :::"void"::: ) ) ($#v1_waybel19 :::"lower"::: ) for ($#l1_waybel_9 :::"TopRelStr"::: ) ; end; theorem :: WAYBEL19:1 (Bool "for" (Set (Var "LL")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "ex" (Set (Var "T")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v1_waybel_9 :::"strict"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "LL")) "st" (Bool (Set (Var "T")) "is" ($#v1_waybel19 :::"lower"::: ) ))) ; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"correct"::: ) ($#v1_waybel_9 :::"strict"::: ) ($#v1_waybel19 :::"lower"::: ) for ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" "R"; end; theorem :: WAYBEL19:2 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) "st" (Bool (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L2"))) "#)" ))) "holds" (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "L1"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "L2"))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"omega"::: "R" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "R" means :: WAYBEL19:def 2 (Bool "for" (Set (Var "T")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v1_waybel19 :::"lower"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" "R" "holds" (Bool it ($#r1_hidden :::"="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))))); end; :: deftheorem defines :::"omega"::: WAYBEL19:def 2 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel19 :::"omega"::: ) (Set (Var "R")))) "iff" (Bool "for" (Set (Var "T")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v1_waybel19 :::"lower"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "R")) "holds" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))))) ")" ))); theorem :: WAYBEL19:3 (Bool "for" (Set (Var "R1")) "," (Set (Var "R2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "R1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "R2"))) "#)" ))) "holds" (Bool (Set ($#k1_waybel19 :::"omega"::: ) (Set (Var "R1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel19 :::"omega"::: ) (Set (Var "R2"))))) ; theorem :: WAYBEL19:4 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k3_subset_1 :::"`"::: ) ) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ))) ; theorem :: WAYBEL19:5 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v12_waybel_0 :::"lower"::: ) ))) ; theorem :: WAYBEL19:6 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v13_waybel_0 :::"upper"::: ) ))) ; theorem :: WAYBEL19:7 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_waybel19 :::"lower"::: ) ) "iff" (Bool "{" (Set "(" (Set "(" ($#k4_waybel_0 :::"uparrow"::: ) (Set (Var "F")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) where F "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool (Set (Var "F")) "is" ($#v1_finset_1 :::"finite"::: ) ) "}" "is" ($#m1_subset_1 :::"Basis":::) "of" (Set (Var "T"))) ")" )) ; theorem :: WAYBEL19:8 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set (Var "f")) ($#r3_waybel_0 :::"preserves_inf_of"::: ) (Set (Var "X"))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))) ; theorem :: WAYBEL19:9 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))) ; theorem :: WAYBEL19:10 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "BB")) "being" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_waybel_0 :::"filtered"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "BB"))) & (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "F"))) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "F")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "F"))) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "F"))))))) ; theorem :: WAYBEL19:11 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) ))) ; theorem :: WAYBEL19:12 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool "(" "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set (Var "f")) ($#r3_waybel_0 :::"preserves_inf_of"::: ) (Set (Var "X"))) ")" )) "holds" (Bool (Set (Var "f")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) ))) ; theorem :: WAYBEL19:13 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")))))) ; definitionmode TopPoset is ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_waybel19 :::"lower"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_pre_topc :::"T_0"::: ) for ($#l1_waybel_9 :::"TopRelStr"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster -> ($#v1_yellow_0 :::"lower-bounded"::: ) for ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" "R"; end; theorem :: WAYBEL19:14 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k7_yellow_3 :::"]"::: ) ) ")" ) ($#k3_subset_1 :::"`"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "s")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ($#k2_zfmisc_1 :::":]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "t")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) )))))) ; theorem :: WAYBEL19:15 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "S9")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v1_waybel19 :::"lower"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "T9")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v1_waybel19 :::"lower"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_waybel19 :::"omega"::: ) (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "S")) "," (Set (Var "T")) ($#k3_yellow_3 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "S9")) "," (Set (Var "T9")) ($#k2_borsuk_1 :::":]"::: ) )))))) ; theorem :: WAYBEL19:16 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopPoset":::) "holds" (Bool (Set ($#k1_waybel19 :::"omega"::: ) (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "S")) "," (Set (Var "T")) ($#k3_yellow_3 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "S")) "," (Set (Var "T")) ($#k2_borsuk_1 :::":]"::: ) )))) ; theorem :: WAYBEL19:17 (Bool "for" (Set (Var "T")) "," (Set (Var "T2")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopLattice":::) "st" (Bool (Bool (Set (Var "T2")) "is" ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "T")) "," (Set (Var "T")) ($#k3_yellow_3 :::":]"::: ) ))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T2")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_waybel_2 :::"inf_op"::: ) (Set (Var "T"))))) "holds" (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ))) ; begin scheme :: WAYBEL19:sch 1 TopInd{ F1() -> ($#l1_waybel_9 :::"TopLattice":::), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool P1[(Set (Var "A"))])) provided (Bool "ex" (Set (Var "K")) "being" ($#m1_subset_1 :::"prebasis":::) "of" (Set F1 "(" ")" ) "st" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "K")))) "holds" (Bool P1[(Set (Var "A"))]))) and (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool P1[(Set (Var "A"))]) ")" )) "holds" (Bool P1[(Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F")))])) and (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "A1"))]) & (Bool P1[(Set (Var "A2"))])) "holds" (Bool P1[(Set (Set (Var "A1")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "A2")))])) and (Bool P1[(Set ($#k2_struct_0 :::"[#]"::: ) (Set F1 "(" ")" ))]) proof end; theorem :: WAYBEL19:18 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L2"))) "#)" )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L1")) "holds" (Bool "(" (Bool (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) "is" ($#v1_waybel_0 :::"directed"::: ) ) & (Bool (Bool "not" (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) ")" ) ")" ) & (Bool (Set (Var "L1")) "is" ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) )) "holds" (Bool (Set (Var "L2")) "is" ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) )) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"Poset":::); cluster -> ($#v3_waybel_3 :::"continuous"::: ) for ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" "T"; end; theorem :: WAYBEL19:19 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "R")) "being" ($#m3_yellow_9 :::"Refinement"::: ) "of" (Set (Var "T")) "," (Set (Var "S")) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool (Bool "(" (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T")))) "or" (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S")))) ")" )) "holds" (Bool (Set (Var "W")) "is" ($#v3_pre_topc :::"open"::: ) )))) ; theorem :: WAYBEL19:20 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "R")) "being" ($#m3_yellow_9 :::"Refinement"::: ) "of" (Set (Var "T")) "," (Set (Var "S")) (Bool "for" (Set (Var "V")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "V"))) & (Bool (Set (Var "V")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "W")) "is" ($#v3_pre_topc :::"open"::: ) ))))) ; theorem :: WAYBEL19:21 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))))) "holds" (Bool "for" (Set (Var "R")) "being" ($#m3_yellow_9 :::"Refinement"::: ) "of" (Set (Var "T")) "," (Set (Var "S")) (Bool "for" (Set (Var "V")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "V"))) & (Bool (Set (Var "V")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool (Set (Var "W")) "is" ($#v4_pre_topc :::"closed"::: ) ))))) ; theorem :: WAYBEL19:22 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "K")) "," (Set (Var "O")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "K")) ($#r1_tarski :::"c="::: ) (Set (Var "O"))) & (Bool (Set (Var "O")) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "K")) "is" ($#m1_subset_1 :::"Basis":::) "of" (Set (Var "T")))) "implies" (Bool (Set (Var "O")) "is" ($#m1_subset_1 :::"Basis":::) "of" (Set (Var "T"))) ")" & "(" (Bool (Bool (Set (Var "K")) "is" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T")))) "implies" (Bool (Set (Var "O")) "is" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T"))) ")" ")" ))) ; theorem :: WAYBEL19:23 (Bool "for" (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T1"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T2"))))) "holds" (Bool "for" (Set (Var "T")) "being" ($#m3_yellow_9 :::"Refinement"::: ) "of" (Set (Var "T1")) "," (Set (Var "T2")) (Bool "for" (Set (Var "B1")) "being" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T1")) (Bool "for" (Set (Var "B2")) "being" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T2")) "holds" (Bool (Set (Set (Var "B1")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "B2"))) "is" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T"))))))) ; theorem :: WAYBEL19:24 (Bool "for" (Set (Var "T1")) "," (Set (Var "S1")) "," (Set (Var "T2")) "," (Set (Var "S2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "R1")) "being" ($#m3_yellow_9 :::"Refinement"::: ) "of" (Set (Var "T1")) "," (Set (Var "S1")) (Bool "for" (Set (Var "R2")) "being" ($#m3_yellow_9 :::"Refinement"::: ) "of" (Set (Var "T2")) "," (Set (Var "S2")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T1")) "," (Set (Var "T2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S1")) "," (Set (Var "S2")) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R1")) "," (Set (Var "R2")) "st" (Bool (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "f")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_pre_topc :::"continuous"::: ) )) "holds" (Bool (Set (Var "h")) "is" ($#v5_pre_topc :::"continuous"::: ) ))))))) ; theorem :: WAYBEL19:25 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "K")) "being" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "K")))) "holds" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))))))) ; theorem :: WAYBEL19:26 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "S")))) "holds" (Bool (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "S"))))))) ; theorem :: WAYBEL19:27 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" (Set (Var "X")) ($#k2_yellow_9 :::"+id"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set (Var "X")))) & (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" (Set (Var "X")) ($#k3_yellow_9 :::"opp+id"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set (Var "X")))) ")" ))) ; theorem :: WAYBEL19:28 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )))) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) ; attr "T" is :::"Lawson"::: means :: WAYBEL19:def 3 (Bool (Set (Set "(" ($#k1_waybel19 :::"omega"::: ) "T" ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k5_waybel11 :::"sigma"::: ) "T" ")" )) "is" ($#m1_subset_1 :::"prebasis":::) "of" "T"); end; :: deftheorem defines :::"Lawson"::: WAYBEL19:def 3 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v2_waybel19 :::"Lawson"::: ) ) "iff" (Bool (Set (Set "(" ($#k1_waybel19 :::"omega"::: ) (Set (Var "T")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k5_waybel11 :::"sigma"::: ) (Set (Var "T")) ")" )) "is" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T"))) ")" )); theorem :: WAYBEL19:29 (Bool "for" (Set (Var "R")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "LL")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v1_waybel19 :::"lower"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "S")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "being" ($#v2_pre_topc :::"correct"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v2_waybel19 :::"Lawson"::: ) ) "iff" (Bool (Set (Var "T")) "is" ($#m3_yellow_9 :::"Refinement"::: ) "of" (Set (Var "S")) "," (Set (Var "LL"))) ")" ))))) ; registrationlet "R" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"correct"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) bbbadV3_YELLOW_0() ($#v24_waybel_0 :::"up-complete"::: ) ($#v25_waybel_0 :::"/\-complete"::: ) ($#v1_waybel_9 :::"strict"::: ) ($#~v1_yellow_3 "non" ($#v1_yellow_3 :::"void"::: ) ) ($#v2_waybel19 :::"Lawson"::: ) for ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" "R"; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel_9 :::"strict"::: ) ($#v4_waybel11 :::"Scott"::: ) ($#~v1_yellow_3 "non" ($#v1_yellow_3 :::"void"::: ) ) for ($#l1_waybel_9 :::"TopRelStr"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) bbbadV3_YELLOW_0() ($#v3_waybel_3 :::"continuous"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#v25_waybel_0 :::"/\-complete"::: ) ($#v1_waybel_9 :::"strict"::: ) ($#~v1_yellow_3 "non" ($#v1_yellow_3 :::"void"::: ) ) ($#v2_waybel19 :::"Lawson"::: ) for ($#l1_waybel_9 :::"TopRelStr"::: ) ; end; theorem :: WAYBEL19:30 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#l1_waybel_9 :::"TopLattice":::) "holds" (Bool (Set (Set "(" ($#k5_waybel11 :::"sigma"::: ) (Set (Var "T")) ")" ) ($#k2_xboole_0 :::"\/"::: ) "{" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) : (Bool verum) "}" ) "is" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T")))) ; theorem :: WAYBEL19:31 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#l1_waybel_9 :::"TopLattice":::) "holds" (Bool (Set (Set "(" ($#k5_waybel11 :::"sigma"::: ) (Set (Var "T")) ")" ) ($#k2_xboole_0 :::"\/"::: ) "{" (Set "(" (Set (Var "W")) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ")" ) where W "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")), x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) : (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set ($#k5_waybel11 :::"sigma"::: ) (Set (Var "T")))) "}" ) "is" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T")))) ; theorem :: WAYBEL19:32 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#l1_waybel_9 :::"TopLattice":::) "holds" (Bool "{" (Set "(" (Set (Var "W")) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k4_waybel_0 :::"uparrow"::: ) (Set (Var "F")) ")" ) ")" ) where W, F "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool "(" (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set ($#k5_waybel11 :::"sigma"::: ) (Set (Var "T")))) & (Bool (Set (Var "F")) "is" ($#v1_finset_1 :::"finite"::: ) ) ")" ) "}" "is" ($#m1_subset_1 :::"Basis":::) "of" (Set (Var "T")))) ; definitionlet "T" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); func :::"lambda"::: "T" -> ($#m1_subset_1 :::"Subset-Family":::) "of" "T" means :: WAYBEL19:def 4 (Bool "for" (Set (Var "S")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" "T" "holds" (Bool it ($#r1_hidden :::"="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))))); end; :: deftheorem defines :::"lambda"::: WAYBEL19:def 4 : (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_waybel19 :::"lambda"::: ) (Set (Var "T")))) "iff" (Bool "for" (Set (Var "S")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "T")) "holds" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))))) ")" ))); theorem :: WAYBEL19:33 (Bool "for" (Set (Var "R")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool (Set ($#k2_waybel19 :::"lambda"::: ) (Set (Var "R"))) ($#r1_hidden :::"="::: ) (Set ($#k1_cantor_1 :::"UniCl"::: ) (Set "(" ($#k2_cantor_1 :::"FinMeetCl"::: ) (Set "(" (Set "(" ($#k5_waybel11 :::"sigma"::: ) (Set (Var "R")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_waybel19 :::"omega"::: ) (Set (Var "R")) ")" ) ")" ) ")" )))) ; theorem :: WAYBEL19:34 (Bool "for" (Set (Var "R")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "T")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v1_waybel19 :::"lower"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "S")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "M")) "being" ($#m3_yellow_9 :::"Refinement"::: ) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool (Set ($#k2_waybel19 :::"lambda"::: ) (Set (Var "R"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "M")))))))) ; theorem :: WAYBEL19:35 (Bool "for" (Set (Var "T")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#v1_waybel19 :::"lower"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v3_waybel11 :::"property(S)"::: ) ))) ; theorem :: WAYBEL19:36 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v3_waybel11 :::"property(S)"::: ) ))) ; theorem :: WAYBEL19:37 (Bool "for" (Set (Var "S")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v4_waybel11 :::"Scott"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "T")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "C")) "is" ($#v3_pre_topc :::"open"::: ) ))))) ; theorem :: WAYBEL19:38 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ))) ; theorem :: WAYBEL19:39 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k3_subset_1 :::"`"::: ) ) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x")) ")" ) ($#k3_subset_1 :::"`"::: ) ) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_subset_1 :::"`"::: ) ) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ))) ; theorem :: WAYBEL19:40 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "x"))) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Set "(" ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "x")) ")" ) ($#k3_subset_1 :::"`"::: ) ) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ))) ; theorem :: WAYBEL19:41 (Bool "for" (Set (Var "S")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v4_waybel11 :::"Scott"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "T")) "being" ($#v2_pre_topc :::"correct"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "being" ($#v13_waybel_0 :::"upper"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "C")) "is" ($#v3_pre_topc :::"open"::: ) ))))) ; theorem :: WAYBEL19:42 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "A")) "being" ($#v12_waybel_0 :::"lower"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool (Set (Var "A")) "is" ($#v2_waybel11 :::"closed_under_directed_sups"::: ) ) ")" ))) ; theorem :: WAYBEL19:43 (Bool "for" (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel19 :::"Lawson"::: ) ($#l1_waybel_9 :::"TopLattice":::) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_waybel_0 :::"filtered"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k10_yellow_6 :::"Lim"::: ) (Set "(" (Set (Var "F")) ($#k3_yellow_9 :::"opp+id"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k2_yellow_0 :::"inf"::: ) (Set (Var "F")) ")" ) ($#k6_domain_1 :::"}"::: ) )))) ; registration cluster ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel19 :::"Lawson"::: ) -> ($#v7_pre_topc :::"T_1"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v1_compts_1 :::"compact"::: ) for ($#l1_waybel_9 :::"TopRelStr"::: ) ; end; registration cluster ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#v2_waybel19 :::"Lawson"::: ) -> ($#v8_pre_topc :::"Hausdorff"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) for ($#l1_waybel_9 :::"TopRelStr"::: ) ; end;