:: WAYBEL_3 semantic presentation begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x", "y" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); pred "x" :::"is_way_below"::: "y" means :: WAYBEL_3:def 1 (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" "L" "st" (Bool (Bool "y" ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D"))))) "holds" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool "(" (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool "x" ($#r3_orders_2 :::"<="::: ) (Set (Var "d"))) ")" ))); end; :: deftheorem defines :::"is_way_below"::: WAYBEL_3:def 1 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"is_way_below"::: ) (Set (Var "y"))) "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 (Var "L")) "st" (Bool (Bool (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D"))))) "holds" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set (Var "D"))) & (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "d"))) ")" ))) ")" ))); notationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x", "y" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); synonym "x" :::"<<"::: "y" for "x" :::"is_way_below"::: "y"; synonym "y" :::">>"::: "x" for "x" :::"is_way_below"::: "y"; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); attr "x" is :::"compact"::: means :: WAYBEL_3:def 2 (Bool "x" ($#r1_waybel_3 :::"is_way_below"::: ) "x"); end; :: deftheorem defines :::"compact"::: WAYBEL_3:def 2 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_waybel_3 :::"compact"::: ) ) "iff" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"is_way_below"::: ) (Set (Var "x"))) ")" ))); notationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); synonym :::"isolated_from_below"::: "x" for :::"compact"::: ; end; theorem :: WAYBEL_3:1 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y")))) "holds" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y"))))) ; theorem :: WAYBEL_3:2 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "u")) "," (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "u")) ($#r3_orders_2 :::"<="::: ) (Set (Var "x"))) & (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))) & (Bool (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set (Var "z")))) "holds" (Bool (Set (Var "u")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "z"))))) ; theorem :: WAYBEL_3:3 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) "st" (Bool (Bool "(" (Bool (Set (Var "L")) "is" ($#v1_lattice3 :::"with_suprema"::: ) ) "or" (Bool (Set (Var "L")) "is" ($#v25_waybel_0 :::"/\-complete"::: ) ) ")" )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "z"))) & (Bool (Set (Var "y")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "z")))) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k7_domain_1 :::"}"::: ) ) "," (Set (Var "L"))) & (Bool (Set (Set (Var "x")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "y"))) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "z"))) ")" ))) ; theorem :: WAYBEL_3:4 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L"))) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "x"))))) ; theorem :: WAYBEL_3:5 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))) & (Bool (Set (Var "y")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "z")))) "holds" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "z"))))) ; theorem :: WAYBEL_3:6 (Bool "for" (Set (Var "L")) "being" ($#~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")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))) & (Bool (Set (Var "x")) ($#r1_waybel_3 :::">>"::: ) (Set (Var "y")))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); func :::"waybelow"::: "x" -> ($#m1_subset_1 :::"Subset":::) "of" "L" equals :: WAYBEL_3:def 3 "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Element":::) "of" "L" : (Bool (Set (Var "y")) ($#r1_waybel_3 :::"<<"::: ) "x") "}" ; func :::"wayabove"::: "x" -> ($#m1_subset_1 :::"Subset":::) "of" "L" equals :: WAYBEL_3:def 4 "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Element":::) "of" "L" : (Bool (Set (Var "y")) ($#r1_waybel_3 :::">>"::: ) "x") "}" ; end; :: deftheorem defines :::"waybelow"::: WAYBEL_3:def 3 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool (Set (Var "y")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "x"))) "}" ))); :: deftheorem defines :::"wayabove"::: WAYBEL_3:def 4 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool (Set (Var "y")) ($#r1_waybel_3 :::">>"::: ) (Set (Var "x"))) "}" ))); theorem :: WAYBEL_3:7 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "y")))) "iff" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))) ")" ))) ; theorem :: WAYBEL_3:8 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "y")))) "iff" (Bool (Set (Var "x")) ($#r1_waybel_3 :::">>"::: ) (Set (Var "y"))) ")" ))) ; theorem :: WAYBEL_3:9 (Bool "for" (Set (Var "L")) "being" ($#~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" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "x")) ($#r2_lattice3 :::"is_>=_than"::: ) (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x")))))) ; theorem :: WAYBEL_3:10 (Bool "for" (Set (Var "L")) "being" ($#~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" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "x")) ($#r1_lattice3 :::"is_<=_than"::: ) (Set ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "x")))))) ; theorem :: WAYBEL_3:11 (Bool "for" (Set (Var "L")) "being" ($#~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" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x")))) & (Bool (Set ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")))) ")" ))) ; theorem :: WAYBEL_3:12 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y")))) "holds" (Bool "(" (Bool (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "y")))) & (Bool (Set ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "y"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "x")))) ")" ))) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); cluster (Set ($#k1_waybel_3 :::"waybelow"::: ) "x") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); cluster (Set ($#k1_waybel_3 :::"waybelow"::: ) "x") -> ($#v12_waybel_0 :::"lower"::: ) ; cluster (Set ($#k2_waybel_3 :::"wayabove"::: ) "x") -> ($#v13_waybel_0 :::"upper"::: ) ; end; registrationlet "L" be ($#l1_orders_2 :::"sup-Semilattice":::); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); cluster (Set ($#k1_waybel_3 :::"waybelow"::: ) "x") -> ($#v1_waybel_0 :::"directed"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v25_waybel_0 :::"/\-complete"::: ) ($#l1_orders_2 :::"Poset":::); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); cluster (Set ($#k1_waybel_3 :::"waybelow"::: ) "x") -> ($#v1_waybel_0 :::"directed"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v16_waybel_0 :::"connected"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster -> ($#v1_waybel_0 :::"directed"::: ) ($#v2_waybel_0 :::"filtered"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L"))); end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v16_waybel_0 :::"connected"::: ) ($#v24_waybel_0 :::"up-complete"::: ) -> ($#v3_lattice3 :::"complete"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) bbbadV2_ORDERS_2() ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v16_waybel_0 :::"connected"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; theorem :: WAYBEL_3:13 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Chain":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_orders_2 :::"<"::: ) (Set (Var "y")))) "holds" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))))) ; theorem :: WAYBEL_3:14 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Bool "not" (Set (Var "x")) "is" ($#v1_waybel_3 :::"compact"::: ) )) & (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y")))) "holds" (Bool (Set (Var "x")) ($#r2_orders_2 :::"<"::: ) (Set (Var "y"))))) ; theorem :: WAYBEL_3:15 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L"))) "is" ($#v1_waybel_3 :::"compact"::: ) )) ; theorem :: WAYBEL_3:16 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D"))) ($#r2_hidden :::"in"::: ) (Set (Var "D"))))) ; theorem :: WAYBEL_3:17 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Poset":::) "st" (Bool (Bool (Set (Var "L")) "is" ($#v8_struct_0 :::"finite"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "x")) "is" ($#v1_waybel_3 :::"isolated_from_below"::: ) ))) ; begin theorem :: WAYBEL_3:18 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y")))) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X"))))) "holds" (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "A")))) ")" ))))) ; theorem :: WAYBEL_3:19 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X"))))) "holds" (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "A")))) ")" )) ")" )) "holds" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))))) ; theorem :: WAYBEL_3:20 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y")))) "holds" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))))) ; theorem :: WAYBEL_3:21 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool "(" "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) ")" )) "holds" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))))) ; theorem :: WAYBEL_3:22 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set (Var "L")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))) "iff" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) ")" ))) ; theorem :: WAYBEL_3:23 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "x")) "is" ($#v1_waybel_3 :::"compact"::: ) ) ")" ) "iff" (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "not" (Bool (Set (Var "x")) ($#r2_orders_2 :::"<"::: ) (Set (Var "y")))) ")" ) ")" ))) ")" )) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; attr "L" is :::"satisfying_axiom_of_approximation"::: means :: WAYBEL_3:def 5 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x")) ")" )))); end; :: deftheorem defines :::"satisfying_axiom_of_approximation"::: WAYBEL_3:def 5 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x")) ")" )))) ")" )); registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) -> (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; attr "L" is :::"continuous"::: means :: WAYBEL_3:def 6 (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool "(" (Bool (Bool "not" (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) "is" ($#v1_waybel_0 :::"directed"::: ) ) ")" ) ")" ) & (Bool "L" "is" ($#v24_waybel_0 :::"up-complete"::: ) ) & (Bool "L" "is" ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) ) ")" ); end; :: deftheorem defines :::"continuous"::: WAYBEL_3:def 6 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_3 :::"continuous"::: ) ) "iff" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Bool "not" (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) "is" ($#v1_waybel_0 :::"directed"::: ) ) ")" ) ")" ) & (Bool (Set (Var "L")) "is" ($#v24_waybel_0 :::"up-complete"::: ) ) & (Bool (Set (Var "L")) "is" ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) ) ")" ) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v3_waybel_3 :::"continuous"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) -> ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_3 :::"continuous"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) bbbadV2_ORDERS_2() ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); cluster (Set ($#k1_waybel_3 :::"waybelow"::: ) "x") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ; end; theorem :: WAYBEL_3:24 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Bool "not" (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) "is" ($#v1_xboole_0 :::"empty"::: ) )) & (Bool (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) "is" ($#v1_waybel_0 :::"directed"::: ) ) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y"))))) "holds" (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "x"))) & (Bool (Bool "not" (Set (Var "u")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y")))) ")" ))) ")" )) ; theorem :: WAYBEL_3:25 (Bool "for" (Set (Var "L")) "being" ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "y")))) ")" ))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v16_waybel_0 :::"connected"::: ) -> ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; theorem :: WAYBEL_3:26 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "x")) "is" ($#v1_waybel_3 :::"compact"::: ) ) ")" )) "holds" (Bool (Set (Var "L")) "is" ($#v2_waybel_3 :::"satisfying_axiom_of_approximation"::: ) )) ; begin definitionlet "f" be ($#m1_hidden :::"Relation":::); attr "f" is :::"non-Empty"::: means :: WAYBEL_3:def 7 (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "f"))) "holds" (Bool "not" (Bool (Set (Var "S")) "is" ($#v2_struct_0 :::"empty"::: ) ))); attr "f" is :::"reflexive-yielding"::: means :: WAYBEL_3:def 8 (Bool "for" (Set (Var "S")) "being" ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "f"))) "holds" (Bool (Set (Var "S")) "is" ($#v3_orders_2 :::"reflexive"::: ) )); end; :: deftheorem defines :::"non-Empty"::: WAYBEL_3:def 7 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v4_waybel_3 :::"non-Empty"::: ) ) "iff" (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool "not" (Bool (Set (Var "S")) "is" ($#v2_struct_0 :::"empty"::: ) ))) ")" )); :: deftheorem defines :::"reflexive-yielding"::: WAYBEL_3:def 8 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v5_waybel_3 :::"reflexive-yielding"::: ) ) "iff" (Bool "for" (Set (Var "S")) "being" ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "S")) "is" ($#v3_orders_2 :::"reflexive"::: ) )) ")" )); registrationlet "I" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("I") ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#v5_waybel_3 :::"reflexive-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set ($#k5_yellow_1 :::"product"::: ) "J") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "I")); :: original: :::"."::: redefine func "J" :::"."::: "i" -> ($#l1_orders_2 :::"RelStr"::: ) ; end; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "I")); cluster (Set "J" ($#k1_funct_1 :::"."::: ) "i") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set ($#k5_yellow_1 :::"product"::: ) "J") -> ($#v1_monoid_0 :::"constituted-Functions"::: ) ; end; definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Const "J")) ")" ); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "I")); :: original: :::"."::: redefine func "x" :::"."::: "i" -> ($#m1_subset_1 :::"Element":::) "of" (Set "(" "J" ($#k3_waybel_3 :::"."::: ) "i" ")" ); end; definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "I")); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Const "J")) ")" ); :: original: :::"pi"::: redefine func :::"pi"::: "(" "X" "," "i" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "J" ($#k3_waybel_3 :::"."::: ) "i" ")" ); end; theorem :: WAYBEL_3:27 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Var "J")) ")" )) "iff" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "I"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "x")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i")) ")" )) ")" ) ")" ) ")" )))) ; theorem :: WAYBEL_3:28 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Var "J")) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "x")) ($#k4_waybel_3 :::"."::: ) (Set (Var "i"))) ($#r1_orders_2 :::"<="::: ) (Set (Set (Var "y")) ($#k4_waybel_3 :::"."::: ) (Set (Var "i"))))) ")" )))) ; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v5_waybel_3 :::"reflexive-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "I")); cluster (Set "J" ($#k1_funct_1 :::"."::: ) "i") -> ($#v3_orders_2 :::"reflexive"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#v5_waybel_3 :::"reflexive-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set ($#k5_yellow_1 :::"product"::: ) "J") -> ($#v3_orders_2 :::"reflexive"::: ) ; end; theorem :: WAYBEL_3:29 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v4_orders_2 :::"transitive"::: ) ) ")" )) "holds" (Bool (Set ($#k5_yellow_1 :::"product"::: ) (Set (Var "J"))) "is" ($#v4_orders_2 :::"transitive"::: ) ))) ; theorem :: WAYBEL_3:30 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v5_orders_2 :::"antisymmetric"::: ) ) ")" )) "holds" (Bool (Set ($#k5_yellow_1 :::"product"::: ) (Set (Var "J"))) "is" ($#v5_orders_2 :::"antisymmetric"::: ) ))) ; theorem :: WAYBEL_3:31 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#v5_waybel_3 :::"reflexive-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::)) ")" )) "holds" (Bool (Set ($#k5_yellow_1 :::"product"::: ) (Set (Var "J"))) "is" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::)))) ; theorem :: WAYBEL_3:32 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#v5_waybel_3 :::"reflexive-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::)) ")" )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Var "J")) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" ) ($#k4_waybel_3 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k5_waybel_3 :::"pi"::: ) "(" (Set (Var "X")) "," (Set (Var "i")) ")" ")" ))))))) ; theorem :: WAYBEL_3:33 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#v5_waybel_3 :::"reflexive-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::)) ")" )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Var "J")) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "x")) ($#k4_waybel_3 :::"."::: ) (Set (Var "i"))) ($#r1_waybel_3 :::"<<"::: ) (Set (Set (Var "y")) ($#k4_waybel_3 :::"."::: ) (Set (Var "i")))) ")" ) & (Bool "ex" (Set (Var "K")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "I")) "st" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "st" (Bool (Bool (Bool "not" (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "x")) ($#k4_waybel_3 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set "(" (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i")) ")" ))))) ")" ) ")" )))) ; begin theorem :: WAYBEL_3:34 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"is_way_below"::: ) (Set (Var "y")))) "holds" (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tops_2 :::"open"::: ) ) & (Bool (Set (Var "y")) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))))) "holds" (Bool "ex" (Set (Var "G")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "F")) "st" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set ($#k3_tarski :::"union"::: ) (Set (Var "G")))))))) ; theorem :: WAYBEL_3:35 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v1_tops_2 :::"open"::: ) ) & (Bool (Set (Var "y")) ($#r1_tarski :::"c="::: ) (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))))) "holds" (Bool "ex" (Set (Var "G")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "F")) "st" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set ($#k3_tarski :::"union"::: ) (Set (Var "G"))))) ")" )) "holds" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"is_way_below"::: ) (Set (Var "y"))))) ; theorem :: WAYBEL_3:36 (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 "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_waybel_3 :::"compact"::: ) ) "iff" (Bool (Set (Var "X")) "is" ($#v2_compts_1 :::"compact"::: ) ) ")" )))) ; theorem :: WAYBEL_3:37 (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 "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))))) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_waybel_3 :::"compact"::: ) ) "iff" (Bool (Set (Var "T")) "is" ($#v1_compts_1 :::"compact"::: ) ) ")" ))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); attr "T" is :::"locally-compact"::: means :: WAYBEL_3:def 9 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "X")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool "ex" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "Y")))) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "Y")) "is" ($#v2_compts_1 :::"compact"::: ) ) ")" )))); end; :: deftheorem defines :::"locally-compact"::: WAYBEL_3:def 9 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v6_waybel_3 :::"locally-compact"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "X")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool "ex" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_tops_1 :::"Int"::: ) (Set (Var "Y")))) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "Y")) "is" ($#v2_compts_1 :::"compact"::: ) ) ")" )))) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v8_pre_topc :::"T_2"::: ) ($#v1_compts_1 :::"compact"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v9_pre_topc :::"regular"::: ) ($#v10_pre_topc :::"normal"::: ) ($#v6_waybel_3 :::"locally-compact"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v8_pre_topc :::"T_2"::: ) ($#v1_compts_1 :::"compact"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; theorem :: WAYBEL_3:38 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) ")" ) "st" (Bool (Bool "ex" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set (Var "y"))) & (Bool (Set (Var "Z")) "is" ($#v2_compts_1 :::"compact"::: ) ) ")" ))) "holds" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))))) ; theorem :: WAYBEL_3:39 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v6_waybel_3 :::"locally-compact"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y")))) "holds" (Bool "ex" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "Z"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set (Var "y"))) & (Bool (Set (Var "Z")) "is" ($#v2_compts_1 :::"compact"::: ) ) ")" )))) ; theorem :: WAYBEL_3:40 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v6_waybel_3 :::"locally-compact"::: ) ) & (Bool (Set (Var "T")) "is" ($#v8_pre_topc :::"T_2"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y")))) "holds" (Bool "ex" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "Z"))) ($#r1_tarski :::"c="::: ) (Set (Var "y"))) & (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "Z"))) "is" ($#v2_compts_1 :::"compact"::: ) ) ")" )))) ; theorem :: WAYBEL_3:41 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v9_pre_topc :::"regular"::: ) ) & (Bool (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "X")))) "is" ($#v3_waybel_3 :::"continuous"::: ) )) "holds" (Bool (Set (Var "X")) "is" ($#v6_waybel_3 :::"locally-compact"::: ) )) ; theorem :: WAYBEL_3:42 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool (Set (Var "T")) "is" ($#v6_waybel_3 :::"locally-compact"::: ) )) "holds" (Bool (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T")))) "is" ($#v3_waybel_3 :::"continuous"::: ) )) ;