:: WAYBEL15 semantic presentation begin theorem :: WAYBEL15:1 (Bool "for" (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "S")) "being" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "being" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Var "T")) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "R")))))) ; theorem :: WAYBEL15:2 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "holds" (Bool (Set ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "Y")) where Y "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) : (Bool (Set (Var "a")) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) "}" ))) ; theorem :: WAYBEL15:3 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))) ($#r1_orders_2 :::"<="::: ) (Set (Var "a")))) "holds" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")))))) ; theorem :: WAYBEL15:4 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "d")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "g")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set ($#k1_waybel_1 :::"["::: ) (Set (Var "g")) "," (Set (Var "d")) ($#k1_waybel_1 :::"]"::: ) ) "is" ($#v3_waybel_1 :::"Galois"::: ) )) "holds" (Bool (Set (Var "T")) "," (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "d"))) ($#r5_waybel_1 :::"are_isomorphic"::: ) )))) ; theorem :: WAYBEL15:5 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "L3")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) (Bool "for" (Set (Var "g2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "L3")) (Bool "for" (Set (Var "d1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "L1")) (Bool "for" (Set (Var "d2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L3")) "," (Set (Var "L2")) "st" (Bool (Bool (Set ($#k1_waybel_1 :::"["::: ) (Set (Var "g1")) "," (Set (Var "d1")) ($#k1_waybel_1 :::"]"::: ) ) "is" ($#v3_waybel_1 :::"Galois"::: ) ) & (Bool (Set ($#k1_waybel_1 :::"["::: ) (Set (Var "g2")) "," (Set (Var "d2")) ($#k1_waybel_1 :::"]"::: ) ) "is" ($#v3_waybel_1 :::"Galois"::: ) )) "holds" (Bool (Set ($#k1_waybel_1 :::"["::: ) (Set "(" (Set (Var "g2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "g1")) ")" ) "," (Set "(" (Set (Var "d1")) ($#k1_partfun1 :::"*"::: ) (Set (Var "d2")) ")" ) ($#k1_waybel_1 :::"]"::: ) ) "is" ($#v3_waybel_1 :::"Galois"::: ) )))))) ; theorem :: WAYBEL15:6 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "L1")) "st" (Bool (Bool (Set (Var "f1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_funct_1 :::"""::: ) )) & (Bool (Set (Var "f")) "is" ($#v23_waybel_0 :::"isomorphic"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_waybel_1 :::"["::: ) (Set (Var "f")) "," (Set (Var "f1")) ($#k1_waybel_1 :::"]"::: ) ) "is" ($#v3_waybel_1 :::"Galois"::: ) ) & (Bool (Set ($#k1_waybel_1 :::"["::: ) (Set (Var "f1")) "," (Set (Var "f")) ($#k1_waybel_1 :::"]"::: ) ) "is" ($#v3_waybel_1 :::"Galois"::: ) ) ")" )))) ; theorem :: WAYBEL15:7 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X"))) "is" ($#v3_waybel_8 :::"arithmetic"::: ) )) ; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k3_yellow_1 :::"BoolePoset"::: ) "X") -> ($#v3_waybel_8 :::"arithmetic"::: ) ; end; theorem :: WAYBEL15:8 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v23_waybel_0 :::"isomorphic"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L1")) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))))) ; theorem :: WAYBEL15:9 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) "st" (Bool (Bool (Set (Var "L1")) "," (Set (Var "L2")) ($#r5_waybel_1 :::"are_isomorphic"::: ) ) & (Bool (Set (Var "L1")) "is" ($#v3_waybel_3 :::"continuous"::: ) )) "holds" (Bool (Set (Var "L2")) "is" ($#v3_waybel_3 :::"continuous"::: ) )) ; theorem :: WAYBEL15:10 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set (Var "L1")) "," (Set (Var "L2")) ($#r5_waybel_1 :::"are_isomorphic"::: ) ) & (Bool (Set (Var "L1")) "is" ($#v1_yellow_0 :::"lower-bounded"::: ) ) & (Bool (Set (Var "L1")) "is" ($#v3_waybel_8 :::"arithmetic"::: ) )) "holds" (Bool (Set (Var "L2")) "is" ($#v3_waybel_8 :::"arithmetic"::: ) )) ; theorem :: WAYBEL15:11 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "L3")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "L3")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )))) ; begin theorem :: WAYBEL15:12 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_yellow_2 :::"Image"::: ) (Set (Var "f")) ")" ) "holds" (Bool (Set (Set "(" ($#k3_waybel_1 :::"inclusion"::: ) (Set (Var "f")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Var "X")))))) ; theorem :: WAYBEL15:13 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "st" (Bool (Bool (Set (Var "c")) "is" ($#v11_quantal1 :::"idempotent"::: ) ) & (Bool (Set (Var "c")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool (Set ($#k3_waybel_1 :::"inclusion"::: ) (Set (Var "c"))) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ))) ; theorem :: WAYBEL15:14 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#v8_waybel_1 :::"kernel"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "p"))) "is" ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::)))) ; theorem :: WAYBEL15:15 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#v6_waybel_1 :::"projection"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "p"))) "is" ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::)))) ; theorem :: WAYBEL15:16 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_3 :::"continuous"::: ) ) "iff" (Bool "ex" (Set (Var "A")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_8 :::"arithmetic"::: ) ($#l1_orders_2 :::"LATTICE":::)(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "g")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "g")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) ")" ))) ")" )) ; theorem :: WAYBEL15:17 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_3 :::"continuous"::: ) ) "iff" (Bool "ex" (Set (Var "A")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_waybel_8 :::"algebraic"::: ) ($#l1_orders_2 :::"LATTICE":::)(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "A")) "," (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "g")) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Var "g")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) ")" ))) ")" )) ; theorem :: WAYBEL15:18 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "L")) "is" ($#v3_waybel_3 :::"continuous"::: ) )) "implies" (Bool "ex" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "p")) "being" ($#v6_waybel_1 :::"projection"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "st" (Bool "(" (Bool (Set (Var "p")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) & (Bool (Set (Var "L")) "," (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "p"))) ($#r5_waybel_1 :::"are_isomorphic"::: ) ) ")" ))) ")" & "(" (Bool (Bool "ex" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "p")) "being" ($#v6_waybel_1 :::"projection"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "st" (Bool "(" (Bool (Set (Var "p")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) & (Bool (Set (Var "L")) "," (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "p"))) ($#r5_waybel_1 :::"are_isomorphic"::: ) ) ")" )))) "implies" (Bool (Set (Var "L")) "is" ($#v3_waybel_3 :::"continuous"::: ) ) ")" ")" )) ; begin theorem :: WAYBEL15:19 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_waybel_6 :::"PRIME"::: ) (Set "(" (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ")" ))) "iff" (Bool (Set (Var "x")) "is" ($#v6_waybel_6 :::"co-prime"::: ) ) ")" ))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "a" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); attr "a" is :::"atom"::: means :: WAYBEL15:def 1 (Bool "(" (Bool (Set ($#k3_yellow_0 :::"Bottom"::: ) "L") ($#r2_orders_2 :::"<"::: ) "a") & (Bool "(" "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool (Bool (Set ($#k3_yellow_0 :::"Bottom"::: ) "L") ($#r2_orders_2 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_orders_2 :::"<="::: ) "a")) "holds" (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) "a") ")" ) ")" ); end; :: deftheorem defines :::"atom"::: WAYBEL15:def 1 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#v1_waybel15 :::"atom"::: ) ) "iff" (Bool "(" (Bool (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L"))) ($#r2_orders_2 :::"<"::: ) (Set (Var "a"))) & (Bool "(" "for" (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L"))) ($#r2_orders_2 :::"<"::: ) (Set (Var "b"))) & (Bool (Set (Var "b")) ($#r1_orders_2 :::"<="::: ) (Set (Var "a")))) "holds" (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" ) ")" ) ")" ))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"ATOM"::: "L" -> ($#m1_subset_1 :::"Subset":::) "of" "L" means :: WAYBEL15:def 2 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#v1_waybel15 :::"atom"::: ) ) ")" )); end; :: deftheorem defines :::"ATOM"::: WAYBEL15:def 2 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel15 :::"ATOM"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "x")) "is" ($#v1_waybel15 :::"atom"::: ) ) ")" )) ")" ))); theorem :: WAYBEL15:20 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#v1_waybel15 :::"atom"::: ) ) "iff" (Bool "(" (Bool (Set (Var "a")) "is" ($#v6_waybel_6 :::"co-prime"::: ) ) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L")))) ")" ) ")" ))) ; registrationlet "L" be ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster ($#v1_waybel15 :::"atom"::: ) -> ($#v6_waybel_6 :::"co-prime"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L"); end; theorem :: WAYBEL15:21 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool (Set ($#k1_waybel15 :::"ATOM"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_waybel_6 :::"PRIME"::: ) (Set "(" (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: WAYBEL15:22 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "a")) "is" ($#v1_waybel15 :::"atom"::: ) )) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set (Var "x"))) "iff" (Bool (Bool "not" (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set ($#k7_waybel_1 :::"'not'"::: ) (Set (Var "x"))))) ")" ))) ; theorem :: WAYBEL15:23 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" "{" (Set "(" (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "y")) ")" ) where y "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "}" "," (Set (Var "L")) ")" ))))) ; theorem :: WAYBEL15:24 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_waybel15 :::"atom"::: ) ) & (Bool (Set (Var "y")) "is" ($#v1_waybel15 :::"atom"::: ) ) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L")))))) ; theorem :: WAYBEL15:25 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_waybel15 :::"ATOM"::: ) (Set (Var "L"))))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "iff" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_waybel15 :::"atom"::: ) ) & (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "A")))) ")" ) ")" )))) ; theorem :: WAYBEL15:26 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_waybel15 :::"ATOM"::: ) (Set (Var "L")))) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_waybel15 :::"ATOM"::: ) (Set (Var "L"))))) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) "iff" (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X"))) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "Y")))) ")" ))) ; begin theorem :: WAYBEL15:27 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_8 :::"arithmetic"::: ) ) "iff" (Bool "ex" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Var "L")) "," (Set ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X"))) ($#r5_waybel_1 :::"are_isomorphic"::: ) )) ")" )) ; theorem :: WAYBEL15:28 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_8 :::"arithmetic"::: ) ) "iff" (Bool (Set (Var "L")) "is" ($#v2_waybel_8 :::"algebraic"::: ) ) ")" )) ; theorem :: WAYBEL15:29 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_8 :::"arithmetic"::: ) ) "iff" (Bool (Set (Var "L")) "is" ($#v3_waybel_3 :::"continuous"::: ) ) ")" )) ; theorem :: WAYBEL15:30 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_8 :::"arithmetic"::: ) ) "iff" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_3 :::"continuous"::: ) ) & (Bool (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#v3_waybel_3 :::"continuous"::: ) ) ")" ) ")" )) ; theorem :: WAYBEL15:31 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_8 :::"arithmetic"::: ) ) "iff" (Bool (Set (Var "L")) "is" ($#v1_waybel_5 :::"completely-distributive"::: ) ) ")" )) ; theorem :: WAYBEL15:32 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_8 :::"arithmetic"::: ) ) "iff" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_lattice3 :::"complete"::: ) ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "ex" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_waybel15 :::"ATOM"::: ) (Set (Var "L")))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")))) ")" )) ")" ) ")" ) ")" )) ;