:: WAYBEL25 semantic presentation begin theorem :: WAYBEL25:1 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k9_waybel18 :::"Sierpinski_Space"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v4_pre_topc :::"closed"::: ) )) ; theorem :: WAYBEL25:2 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set ($#k9_waybel18 :::"Sierpinski_Space"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool "not" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "p")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; registration cluster (Set ($#k9_waybel18 :::"Sierpinski_Space"::: ) ) -> ($#~v7_pre_topc "non" ($#v7_pre_topc :::"T_1"::: ) ) ; 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"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v4_waybel11 :::"Scott"::: ) -> ($#v6_pre_topc :::"T_0"::: ) for ($#l1_waybel_9 :::"TopRelStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v6_pre_topc :::"T_0"::: ) ($#v2_waybel18 :::"injective"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; registration cluster ($#~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"::: ) for ($#l1_waybel_9 :::"TopRelStr"::: ) ; end; theorem :: WAYBEL25:3 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set ($#k5_yellow_1 :::"product"::: ) (Set "(" (Set (Var "I")) ($#k7_funcop_1 :::"-->"::: ) (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Num 1) ")" ) ")" )) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k3_waybel18 :::"product"::: ) (Set "(" (Set (Var "I")) ($#k7_funcop_1 :::"-->"::: ) (Set ($#k9_waybel18 :::"Sierpinski_Space"::: ) ) ")" ) ")" ))))) ; theorem :: WAYBEL25:4 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "T1")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "L1")) (Bool "for" (Set (Var "T2")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "L2")) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) (Bool "for" (Set (Var "H")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T1")) "," (Set (Var "T2")) "st" (Bool (Bool (Set (Var "h")) ($#r1_funct_2 :::"="::: ) (Set (Var "H"))) & (Bool (Set (Var "h")) "is" ($#v23_waybel_0 :::"isomorphic"::: ) )) "holds" (Bool (Set (Var "H")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) )))))) ; theorem :: WAYBEL25:5 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "T1")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "L1")) (Bool "for" (Set (Var "T2")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "L2")) "st" (Bool (Bool (Set (Var "L1")) "," (Set (Var "L2")) ($#r5_waybel_1 :::"are_isomorphic"::: ) )) "holds" (Bool (Set (Var "T1")) "," (Set (Var "T2")) ($#r2_borsuk_3 :::"are_homeomorphic"::: ) )))) ; theorem :: WAYBEL25:6 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "S")) "is" ($#v2_waybel18 :::"injective"::: ) ) & (Bool (Set (Var "S")) "," (Set (Var "T")) ($#r2_borsuk_3 :::"are_homeomorphic"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v2_waybel18 :::"injective"::: ) )) ; theorem :: WAYBEL25:7 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "T1")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "L1")) (Bool "for" (Set (Var "T2")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "L2")) "st" (Bool (Bool (Set (Var "L1")) "," (Set (Var "L2")) ($#r5_waybel_1 :::"are_isomorphic"::: ) ) & (Bool (Set (Var "T1")) "is" ($#v2_waybel18 :::"injective"::: ) )) "holds" (Bool (Set (Var "T2")) "is" ($#v2_waybel18 :::"injective"::: ) )))) ; definitionlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); redefine pred "X" :::"is_Retract_of"::: "Y" means :: WAYBEL25:def 1 (Bool "ex" (Set (Var "c")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" "X" "," "Y"(Bool "ex" (Set (Var "r")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" "Y" "," "X" "st" (Bool (Set (Set (Var "r")) ($#k1_partfun1 :::"*"::: ) (Set (Var "c"))) ($#r2_funct_2 :::"="::: ) (Set ($#k3_struct_0 :::"id"::: ) "X")))); end; :: deftheorem defines :::"is_Retract_of"::: WAYBEL25:def 1 : (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r1_waybel18 :::"is_Retract_of"::: ) (Set (Var "Y"))) "iff" (Bool "ex" (Set (Var "c")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y"))(Bool "ex" (Set (Var "r")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "Y")) "," (Set (Var "X")) "st" (Bool (Set (Set (Var "r")) ($#k1_partfun1 :::"*"::: ) (Set (Var "c"))) ($#r2_funct_2 :::"="::: ) (Set ($#k3_struct_0 :::"id"::: ) (Set (Var "X")))))) ")" )); theorem :: WAYBEL25:8 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "," (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) "#)" )) & (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "X"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "Y"))) "#)" )) & (Bool (Set (Var "S")) ($#r1_waybel18 :::"is_Retract_of"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "T")) ($#r1_waybel18 :::"is_Retract_of"::: ) (Set (Var "Y")))) ; theorem :: WAYBEL25:9 (Bool "for" (Set (Var "T")) "," (Set (Var "S1")) "," (Set (Var "S2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set (Var "S1")) "," (Set (Var "S2")) ($#r2_borsuk_3 :::"are_homeomorphic"::: ) ) & (Bool (Set (Var "S1")) ($#r1_waybel18 :::"is_Retract_of"::: ) (Set (Var "T")))) "holds" (Bool (Set (Var "S2")) ($#r1_waybel18 :::"is_Retract_of"::: ) (Set (Var "T")))) ; theorem :: WAYBEL25:10 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v2_waybel18 :::"injective"::: ) ) & (Bool (Set (Var "S")) ($#r1_waybel18 :::"is_Retract_of"::: ) (Set (Var "T")))) "holds" (Bool (Set (Var "S")) "is" ($#v2_waybel18 :::"injective"::: ) )) ; theorem :: WAYBEL25:11 (Bool "for" (Set (Var "J")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_waybel18 :::"injective"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "J")) "is" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "Y")))) "holds" (Bool (Set (Var "J")) ($#r1_waybel18 :::"is_Retract_of"::: ) (Set (Var "Y"))))) ; theorem :: WAYBEL25:12 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "T")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "L")) "holds" (Bool (Set (Var "T")) "is" ($#v2_waybel18 :::"injective"::: ) ))) ; registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster ($#v4_waybel11 :::"Scott"::: ) -> ($#v2_waybel18 :::"injective"::: ) for ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" "L"; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_waybel18 :::"injective"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "T") "#)" ) -> ($#v2_waybel18 :::"injective"::: ) ; end; begin definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; func :::"Omega"::: "T" -> ($#v1_waybel_9 :::"strict"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) means :: WAYBEL25:def 2 (Bool "(" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" it) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "T") "#)" )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" it "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool "ex" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool "(" (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "Y")))) ")" )) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"Omega"::: WAYBEL25:def 2 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "b2")) "being" ($#v1_waybel_9 :::"strict"::: ) ($#l1_waybel_9 :::"TopRelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T")))) "iff" (Bool "(" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "b2"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) "#)" )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b2")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool "ex" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "Y")))) ")" )) ")" ) ")" ) ")" ) ")" ))); registrationlet "T" be ($#v2_struct_0 :::"empty"::: ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster (Set ($#k1_waybel25 :::"Omega"::: ) "T") -> ($#v2_struct_0 :::"empty"::: ) ($#v1_waybel_9 :::"strict"::: ) ; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster (Set ($#k1_waybel25 :::"Omega"::: ) "T") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_waybel_9 :::"strict"::: ) ; end; registrationlet "T" be ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k1_waybel25 :::"Omega"::: ) "T") -> ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_waybel_9 :::"strict"::: ) ; end; registrationlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; cluster (Set ($#k1_waybel25 :::"Omega"::: ) "T") -> ($#v3_orders_2 :::"reflexive"::: ) ($#v1_waybel_9 :::"strict"::: ) ; end; registrationlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; cluster (Set ($#k1_waybel25 :::"Omega"::: ) "T") -> ($#v4_orders_2 :::"transitive"::: ) ($#v1_waybel_9 :::"strict"::: ) ; end; registrationlet "T" be ($#l1_pre_topc :::"T_0-TopSpace":::); cluster (Set ($#k1_waybel25 :::"Omega"::: ) "T") -> ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_waybel_9 :::"strict"::: ) ; end; theorem :: WAYBEL25:13 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) "#)" ))) "holds" (Bool (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T"))))) ; theorem :: WAYBEL25:14 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set "(" ($#k3_waybel18 :::"product"::: ) (Set "(" (Set (Var "M")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "T")) ")" ) ")" ) ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set "(" ($#k3_waybel18 :::"product"::: ) (Set "(" (Set (Var "M")) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "T")) ")" ) ")" ) ")" )) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set "(" (Set (Var "M")) ($#k7_funcop_1 :::"-->"::: ) (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T")) ")" ) ")" ) ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set "(" (Set (Var "M")) ($#k7_funcop_1 :::"-->"::: ) (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T")) ")" ) ")" ) ")" )) "#)" )))) ; theorem :: WAYBEL25:15 (Bool "for" (Set (Var "S")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v4_waybel11 :::"Scott"::: ) ($#l1_waybel_9 :::"TopLattice":::) "holds" (Bool (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#g1_waybel_9 :::"TopRelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))) "#)" ))) ; theorem :: WAYBEL25:16 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "S")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Var "L")) "holds" (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "S")) ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "S")) ")" )) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L"))) "#)" )))) ; registrationlet "S" be ($#v3_lattice3 :::"complete"::: ) ($#v4_waybel11 :::"Scott"::: ) ($#l1_waybel_9 :::"TopLattice":::); cluster (Set ($#k1_waybel25 :::"Omega"::: ) "S") -> ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel_9 :::"strict"::: ) ; end; theorem :: WAYBEL25:17 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "S"))) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T")))))) ; theorem :: WAYBEL25:18 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "S")) ")" ) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T")) ")" ) "st" (Bool (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "h")) "is" ($#v3_tops_2 :::"being_homeomorphism"::: ) )) "holds" (Bool (Set (Var "g")) "is" ($#v23_waybel_0 :::"isomorphic"::: ) )))) ; theorem :: WAYBEL25:19 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "S")) "," (Set (Var "T")) ($#r1_borsuk_3 :::"are_homeomorphic"::: ) )) "holds" (Bool (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "S"))) "," (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T"))) ($#r5_waybel_1 :::"are_isomorphic"::: ) )) ; theorem :: WAYBEL25:20 (Bool "for" (Set (Var "T")) "being" ($#v2_waybel18 :::"injective"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) "holds" (Bool (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T"))) "is" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::))) ; registrationlet "T" be ($#v2_waybel18 :::"injective"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::); cluster (Set ($#k1_waybel25 :::"Omega"::: ) "T") -> ($#v3_lattice3 :::"complete"::: ) ($#v1_waybel_9 :::"strict"::: ) ($#v3_waybel_3 :::"continuous"::: ) ; end; theorem :: WAYBEL25:21 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) "holds" (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) ))) ; registrationlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v5_pre_topc :::"continuous"::: ) -> ($#v5_orders_3 :::"monotone"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) "X" ")" )) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) "Y" ")" )) ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: WAYBEL25:22 (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 :::"Element":::) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T")) ")" ) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x")))))) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Const "T")) ")" ); cluster (Set ($#k2_pre_topc :::"Cl"::: ) (Set ($#k6_domain_1 :::"{"::: ) "x" ($#k6_domain_1 :::"}"::: ) )) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#v12_waybel_0 :::"lower"::: ) ; cluster (Set ($#k5_waybel_0 :::"downarrow"::: ) "x") -> ($#v4_pre_topc :::"closed"::: ) ; end; theorem :: WAYBEL25:23 (Bool "for" (Set (Var "X")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "X")) ")" ) "holds" (Bool (Set (Var "A")) "is" ($#v13_waybel_0 :::"upper"::: ) ))) ; registrationlet "T" be ($#l1_pre_topc :::"TopSpace":::); cluster ($#v3_pre_topc :::"open"::: ) -> ($#v13_waybel_0 :::"upper"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) "T" ")" ))); end; definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "S", "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "N" be ($#l1_waybel_0 :::"net":::) "of" (Set (Const "T")); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "I")); assume (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "T"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set (Const "S")) ($#k6_yellow_1 :::"|^"::: ) (Set (Const "I")) ")" ))) ; func :::"commute"::: "(" "N" "," "i" "," "S" ")" -> ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"net":::) "of" "S" means :: WAYBEL25:def 3 (Bool "(" (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" it) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "N") "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" "N") "#)" )) & (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_funct_6 :::"commute"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" "N") ")" ) ($#k1_funct_1 :::"."::: ) "i")) ")" ); end; :: deftheorem defines :::"commute"::: WAYBEL25:def 3 : (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set (Var "S")) ($#k6_yellow_1 :::"|^"::: ) (Set (Var "I")) ")" )))) "holds" (Bool "for" (Set (Var "b6")) "being" ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set ($#k2_waybel25 :::"commute"::: ) "(" (Set (Var "N")) "," (Set (Var "i")) "," (Set (Var "S")) ")" )) "iff" (Bool "(" (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b6"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "b6"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "N"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "N"))) "#)" )) & (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "b6"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_funct_6 :::"commute"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ) ")" )))))); theorem :: WAYBEL25:24 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "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 ($#k3_waybel24 :::"ContMaps"::: ) "(" (Set (Var "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" ($#k2_waybel25 :::"commute"::: ) "(" (Set (Var "N")) "," (Set (Var "x")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) ($#k2_yellow_6 :::"."::: ) (Set (Var "i")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))))))) ; theorem :: WAYBEL25:25 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#v10_waybel_0 :::"eventually-directed"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set ($#k3_waybel24 :::"ContMaps"::: ) "(" (Set (Var "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k2_waybel25 :::"commute"::: ) "(" (Set (Var "N")) "," (Set (Var "x")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) )))) ; registrationlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "N" be ($#v10_waybel_0 :::"eventually-directed"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set ($#k3_waybel24 :::"ContMaps"::: ) "(" (Set (Const "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Const "Y")) ")" ) ")" ); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "X")); cluster (Set ($#k2_waybel25 :::"commute"::: ) "(" "N" "," "x" "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) "Y" ")" ) ")" ) -> ($#v6_waybel_0 :::"strict"::: ) ($#v10_waybel_0 :::"eventually-directed"::: ) ; end; registrationlet "X", "Y" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#v7_waybel_0 :::"directed"::: ) -> ($#v1_yellow14 :::"Function-yielding"::: ) for ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set ($#k3_waybel24 :::"ContMaps"::: ) "(" "X" "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) "Y" ")" ) ")" ); end; theorem :: WAYBEL25:26 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"T_0-TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set ($#k3_waybel24 :::"ContMaps"::: ) "(" (Set (Var "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool ($#r1_waybel_9 :::"ex_sup_of"::: ) (Set ($#k2_waybel25 :::"commute"::: ) "(" (Set (Var "N")) "," (Set (Var "x")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" )) ")" )) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N")))) "," (Set (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ($#k6_yellow_1 :::"|^"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))))))) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); attr "T" is :::"monotone-convergence"::: means :: WAYBEL25:def 4 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) "T" ")" ) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "D")) "," (Set ($#k1_waybel25 :::"Omega"::: ) "T")) & (Bool "(" "for" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D"))) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool (Set (Var "D")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "V"))) ")" ) ")" )); end; :: deftheorem defines :::"monotone-convergence"::: WAYBEL25:def 4 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_waybel25 :::"monotone-convergence"::: ) ) "iff" (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T")) ")" ) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "D")) "," (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T")))) & (Bool "(" "for" (Set (Var "V")) "being" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D"))) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool (Set (Var "D")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "V"))) ")" ) ")" )) ")" )); theorem :: WAYBEL25:27 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) "#)" )) & (Bool (Set (Var "S")) "is" ($#v1_waybel25 :::"monotone-convergence"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v1_waybel25 :::"monotone-convergence"::: ) )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_struct_0 :::"trivial"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v6_pre_topc :::"T_0"::: ) -> ($#v1_waybel25 :::"monotone-convergence"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: WAYBEL25:28 (Bool "for" (Set (Var "S")) "being" ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"T_0-TopSpace":::) "st" (Bool (Bool (Set (Var "S")) "," (Set (Var "T")) ($#r2_borsuk_3 :::"are_homeomorphic"::: ) )) "holds" (Bool (Set (Var "T")) "is" ($#v1_waybel25 :::"monotone-convergence"::: ) ))) ; theorem :: WAYBEL25:29 (Bool "for" (Set (Var "S")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v4_waybel11 :::"Scott"::: ) ($#l1_waybel_9 :::"TopLattice":::) "holds" (Bool (Set (Var "S")) "is" ($#v1_waybel25 :::"monotone-convergence"::: ) )) ; registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster ($#v4_waybel11 :::"Scott"::: ) -> ($#v4_waybel11 :::"Scott"::: ) ($#v1_waybel25 :::"monotone-convergence"::: ) for ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" "L"; end; registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); let "S" be ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set (Const "L")); cluster (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "S") "#)" ) -> ($#v1_waybel25 :::"monotone-convergence"::: ) ; end; theorem :: WAYBEL25:30 (Bool "for" (Set (Var "X")) "being" ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) "holds" (Bool (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "X"))) "is" ($#v24_waybel_0 :::"up-complete"::: ) )) ; registrationlet "X" be ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::); cluster (Set ($#k1_waybel25 :::"Omega"::: ) "X") -> ($#v24_waybel_0 :::"up-complete"::: ) ($#v1_waybel_9 :::"strict"::: ) ; end; theorem :: WAYBEL25:31 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#v10_waybel_0 :::"eventually-directed"::: ) ($#l1_waybel_0 :::"prenet":::) "of" (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "X"))) "holds" (Bool ($#r1_waybel_9 :::"ex_sup_of"::: ) (Set (Var "N"))))) ; theorem :: WAYBEL25:32 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#v10_waybel_0 :::"eventually-directed"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "X"))) "holds" (Bool (Set ($#k1_waybel_2 :::"sup"::: ) (Set (Var "N"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))))) ; theorem :: WAYBEL25:33 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#v10_waybel_0 :::"eventually-directed"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "X"))) "holds" (Bool (Set (Var "N")) "is" ($#v3_yellow_6 :::"convergent"::: ) ))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#v7_waybel_0 :::"directed"::: ) ($#v10_waybel_0 :::"eventually-directed"::: ) -> ($#v10_waybel_0 :::"eventually-directed"::: ) ($#v3_yellow_6 :::"convergent"::: ) for ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set ($#k1_waybel25 :::"Omega"::: ) "X"); end; theorem :: WAYBEL25:34 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool "(" "for" (Set (Var "N")) "being" ($#v10_waybel_0 :::"eventually-directed"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "X"))) "holds" (Bool "(" (Bool ($#r1_waybel_9 :::"ex_sup_of"::: ) (Set (Var "N"))) & (Bool (Set ($#k1_waybel_2 :::"sup"::: ) (Set (Var "N"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))) ")" ) ")" )) "holds" (Bool (Set (Var "X")) "is" ($#v1_waybel25 :::"monotone-convergence"::: ) )) ; theorem :: WAYBEL25:35 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#l1_pre_topc :::"T_0-TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) "holds" (Bool (Set (Var "f")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"TopSpace":::); let "Y" be ($#l1_pre_topc :::"T_0-TopSpace":::); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v5_pre_topc :::"continuous"::: ) -> ($#v22_waybel_0 :::"directed-sups-preserving"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) "X" ")" )) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k1_waybel25 :::"Omega"::: ) "Y" ")" )) ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: WAYBEL25:36 (Bool "for" (Set (Var "T")) "being" ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) (Bool "for" (Set (Var "R")) "being" ($#l1_pre_topc :::"T_0-TopSpace":::) "st" (Bool (Bool (Set (Var "R")) ($#r1_waybel18 :::"is_Retract_of"::: ) (Set (Var "T")))) "holds" (Bool (Set (Var "R")) "is" ($#v1_waybel25 :::"monotone-convergence"::: ) ))) ; theorem :: WAYBEL25:37 (Bool "for" (Set (Var "T")) "being" ($#v2_waybel18 :::"injective"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) (Bool "for" (Set (Var "S")) "being" ($#v4_waybel11 :::"Scott"::: ) ($#m1_yellow_9 :::"TopAugmentation"::: ) "of" (Set ($#k1_waybel25 :::"Omega"::: ) (Set (Var "T"))) "holds" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) "#)" )))) ; theorem :: WAYBEL25:38 (Bool "for" (Set (Var "T")) "being" ($#v2_waybel18 :::"injective"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_compts_1 :::"compact"::: ) ) & (Bool (Set (Var "T")) "is" ($#v6_waybel_3 :::"locally-compact"::: ) ) & (Bool (Set (Var "T")) "is" ($#v4_yellow_8 :::"sober"::: ) ) ")" )) ; theorem :: WAYBEL25:39 (Bool "for" (Set (Var "T")) "being" ($#v2_waybel18 :::"injective"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) "holds" (Bool (Set (Var "T")) "is" ($#v1_waybel25 :::"monotone-convergence"::: ) )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v6_pre_topc :::"T_0"::: ) ($#v2_waybel18 :::"injective"::: ) -> ($#v1_waybel25 :::"monotone-convergence"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: WAYBEL25:40 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set ($#k3_waybel24 :::"ContMaps"::: ) "(" (Set (Var "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) ")" ) "," (Set "(" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ($#k6_yellow_1 :::"|^"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" )) & (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N")))) "," (Set (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ($#k6_yellow_1 :::"|^"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N")))))) "holds" (Bool (Set (Var "g")) ($#r1_yellow_2 :::"<="::: ) (Set (Var "f"))))))) ; theorem :: WAYBEL25:41 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set ($#k3_waybel24 :::"ContMaps"::: ) "(" (Set (Var "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k2_waybel25 :::"commute"::: ) "(" (Set (Var "N")) "," (Set (Var "a")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) ) ")" ) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) ")" ) "," (Set "(" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ($#k6_yellow_1 :::"|^"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ))) "holds" (Bool (Set (Set (Var "f")) ($#k2_yellow_6 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_2 :::"sup"::: ) (Set "(" ($#k2_waybel25 :::"commute"::: ) "(" (Set (Var "N")) "," (Set (Var "x")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ")" )))))))) ; theorem :: WAYBEL25:42 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set ($#k3_waybel24 :::"ContMaps"::: ) "(" (Set (Var "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k2_waybel25 :::"commute"::: ) "(" (Set (Var "N")) "," (Set (Var "x")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) ) ")" )) "holds" (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) ")" ) "," (Set "(" (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ($#k6_yellow_1 :::"|^"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ) "is" ($#v5_pre_topc :::"continuous"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")))))) ; theorem :: WAYBEL25:43 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#v1_waybel25 :::"monotone-convergence"::: ) ($#l1_pre_topc :::"T_0-TopSpace":::) "holds" (Bool (Set ($#k3_waybel24 :::"ContMaps"::: ) "(" (Set (Var "X")) "," (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ")" ) "is" ($#v4_waybel_0 :::"directed-sups-inheriting"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Set "(" ($#k1_waybel25 :::"Omega"::: ) (Set (Var "Y")) ")" ) ($#k6_yellow_1 :::"|^"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))))) ;