:: WAYBEL_7 semantic presentation begin theorem :: WAYBEL_7:1 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool "(" (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "Y")) "," (Set (Var "L")) ")" )) & (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r1_orders_2 :::">="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "Y")) "," (Set (Var "L")) ")" )) ")" ))) ; theorem :: WAYBEL_7:2 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k9_setfam_1 :::"bool"::: ) (Set (Var "X"))))) ; theorem :: WAYBEL_7:3 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v3_yellow_0 :::"bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v7_struct_0 :::"trivial"::: ) ) "iff" (Bool (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L")))) ")" )) ; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k3_yellow_1 :::"BoolePoset"::: ) "X") -> ($#v11_waybel_1 :::"Boolean"::: ) ; end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k3_yellow_1 :::"BoolePoset"::: ) "X") -> ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ; end; theorem :: WAYBEL_7:4 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v1_subset_1 :::"proper"::: ) ) "iff" (Bool (Bool "not" (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L"))) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) ")" ))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v11_waybel_1 :::"Boolean"::: ) ($#~v1_yellow_3 "non" ($#v1_yellow_3 :::"void"::: ) ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registrationlet "L" be ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"Poset":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_subset_1 :::"proper"::: ) ($#v2_waybel_0 :::"filtered"::: ) ($#v13_waybel_0 :::"upper"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); end; theorem :: WAYBEL_7:5 (Bool "for" (Set (Var "X")) "being" ($#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 ($#k7_waybel_1 :::"'not'"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k6_subset_1 :::"\"::: ) (Set (Var "a")))))) ; theorem :: WAYBEL_7:6 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "Y")) "is" ($#v12_waybel_0 :::"lower"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "y"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) ")" ))) ; theorem :: WAYBEL_7:7 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "Y")) "is" ($#v13_waybel_0 :::"upper"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set (Var "y"))) & (Bool (Set (Var "y")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) ")" ))) ; theorem :: WAYBEL_7:8 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#v12_waybel_0 :::"lower"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "Y")) "is" ($#v1_waybel_0 :::"directed"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "x")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) ")" ))) ; theorem :: WAYBEL_7:9 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#v13_waybel_0 :::"upper"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "Y")) "is" ($#v2_waybel_0 :::"filtered"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "x")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) ")" ))) ; theorem :: WAYBEL_7:10 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v12_waybel_0 :::"lower"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "Y")) "is" ($#v1_waybel_0 :::"directed"::: ) ) "iff" (Bool "for" (Set (Var "Z")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "Z"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) ")" ))) ; theorem :: WAYBEL_7:11 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v13_waybel_0 :::"upper"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "Y")) "is" ($#v2_waybel_0 :::"filtered"::: ) ) "iff" (Bool "for" (Set (Var "Z")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k8_setfam_1 :::"Intersect"::: ) (Set (Var "Z"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y")))) ")" ))) ; begin definitionlet "L" be ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"Poset":::); let "I" be ($#m1_subset_1 :::"Ideal":::) "of" (Set (Const "L")); attr "I" is :::"prime"::: means :: WAYBEL_7:def 1 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool "(" "not" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) "I") "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "I") "or" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "I") ")" )); end; :: deftheorem defines :::"prime"::: WAYBEL_7:def 1 : (Bool "for" (Set (Var "L")) "being" ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v1_waybel_7 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" "not" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) "or" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) ")" )) ")" ))); theorem :: WAYBEL_7:12 (Bool "for" (Set (Var "L")) "being" ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v1_waybel_7 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) ")" ))) ")" ))) ; registrationlet "L" be ($#l1_orders_2 :::"LATTICE":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#v12_waybel_0 :::"lower"::: ) ($#v1_waybel_7 :::"prime"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); end; theorem :: WAYBEL_7:13 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L2"))) "#)" ))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) "is" ($#v1_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L1")))) "holds" (Bool (Set (Var "x")) "is" ($#v1_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L2"))))) ; definitionlet "L" be ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::); let "F" be ($#m1_subset_1 :::"Filter":::) "of" (Set (Const "L")); attr "F" is :::"prime"::: means :: WAYBEL_7:def 2 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool "(" "not" (Bool (Set (Set (Var "x")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) "F") "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "F") "or" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "F") ")" )); end; :: deftheorem defines :::"prime"::: WAYBEL_7:def 2 : (Bool "for" (Set (Var "L")) "being" ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_waybel_7 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" "not" (Bool (Set (Set (Var "x")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) "or" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) ")" ))); theorem :: WAYBEL_7:14 (Bool "for" (Set (Var "L")) "being" ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_waybel_7 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" ))) ")" ))) ; registrationlet "L" be ($#l1_orders_2 :::"LATTICE":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_waybel_0 :::"filtered"::: ) ($#v13_waybel_0 :::"upper"::: ) ($#v2_waybel_7 :::"prime"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); end; theorem :: WAYBEL_7:15 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L1"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L1"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L2"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L2"))) "#)" ))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) "is" ($#v2_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L1")))) "holds" (Bool (Set (Var "x")) "is" ($#v2_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L2"))))) ; theorem :: WAYBEL_7:16 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L"))) "iff" (Bool (Set (Var "x")) "is" ($#v2_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ")" )) ")" ))) ; theorem :: WAYBEL_7:17 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v2_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L"))) "iff" (Bool (Set (Var "x")) "is" ($#v1_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Ideal":::) "of" (Set "(" (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ")" )) ")" ))) ; theorem :: WAYBEL_7:18 (Bool "for" (Set (Var "L")) "being" ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v1_waybel_7 :::"prime"::: ) ) "iff" (Bool "(" (Bool (Set (Set (Var "I")) ($#k3_subset_1 :::"`"::: ) ) "is" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L"))) "or" (Bool (Set (Set (Var "I")) ($#k3_subset_1 :::"`"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ) ")" ))) ; theorem :: WAYBEL_7:19 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v1_waybel_7 :::"prime"::: ) ) "iff" (Bool (Set (Var "I")) ($#r2_hidden :::"in"::: ) (Set ($#k4_waybel_6 :::"PRIME"::: ) (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) ")" ))) ")" ))) ; theorem :: WAYBEL_7:20 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_waybel_7 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) "or" (Bool (Set ($#k7_waybel_1 :::"'not'"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) ")" ))) ; theorem :: WAYBEL_7:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_waybel_7 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) "or" (Bool (Set (Set (Var "X")) ($#k6_subset_1 :::"\"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )) ")" ))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::); let "F" be ($#m1_subset_1 :::"Filter":::) "of" (Set (Const "L")); attr "F" is :::"ultra"::: means :: WAYBEL_7:def 3 (Bool "(" (Bool "F" "is" ($#v1_subset_1 :::"proper"::: ) ) & (Bool "(" "for" (Set (Var "G")) "being" ($#m1_subset_1 :::"Filter":::) "of" "L" "holds" (Bool "(" "not" (Bool "F" ($#r1_tarski :::"c="::: ) (Set (Var "G"))) "or" (Bool "F" ($#r1_hidden :::"="::: ) (Set (Var "G"))) "or" (Bool (Set (Var "G")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"ultra"::: WAYBEL_7:def 3 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v3_waybel_7 :::"ultra"::: ) ) "iff" (Bool "(" (Bool (Set (Var "F")) "is" ($#v1_subset_1 :::"proper"::: ) ) & (Bool "(" "for" (Set (Var "G")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "holds" (Bool "(" "not" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G"))) "or" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set (Var "G"))) "or" (Bool (Set (Var "G")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L")))) ")" ) ")" ) ")" ) ")" ))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_waybel_0 :::"filtered"::: ) ($#v13_waybel_0 :::"upper"::: ) ($#v3_waybel_7 :::"ultra"::: ) -> ($#v1_subset_1 :::"proper"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); end; theorem :: WAYBEL_7:22 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "F")) "is" ($#v1_subset_1 :::"proper"::: ) ) & (Bool (Set (Var "F")) "is" ($#v2_waybel_7 :::"prime"::: ) ) ")" ) "iff" (Bool (Set (Var "F")) "is" ($#v3_waybel_7 :::"ultra"::: ) ) ")" ))) ; theorem :: WAYBEL_7:23 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "I")) ($#r1_subset_1 :::"misses"::: ) (Set (Var "F")))) "holds" (Bool "ex" (Set (Var "P")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "P")) "is" ($#v1_waybel_7 :::"prime"::: ) ) & (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set (Var "P"))) & (Bool (Set (Var "P")) ($#r1_subset_1 :::"misses"::: ) (Set (Var "F"))) ")" ))))) ; theorem :: WAYBEL_7:24 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))))) "holds" (Bool "ex" (Set (Var "P")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "P")) "is" ($#v1_waybel_7 :::"prime"::: ) ) & (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set (Var "P"))) & (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) ")" ))))) ; theorem :: WAYBEL_7:25 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "I")) ($#r1_subset_1 :::"misses"::: ) (Set (Var "F")))) "holds" (Bool "ex" (Set (Var "P")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "P")) "is" ($#v2_waybel_7 :::"prime"::: ) ) & (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "P"))) & (Bool (Set (Var "I")) ($#r1_subset_1 :::"misses"::: ) (Set (Var "P"))) ")" ))))) ; theorem :: WAYBEL_7:26 (Bool "for" (Set (Var "L")) "being" ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) (Bool "ex" (Set (Var "G")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G"))) & (Bool (Set (Var "G")) "is" ($#v3_waybel_7 :::"ultra"::: ) ) ")" )))) ; begin definitionlet "T" be ($#l1_pre_topc :::"TopSpace":::); let "F", "x" be ($#m1_hidden :::"set"::: ) ; pred "x" :::"is_a_cluster_point_of"::: "F" "," "T" means :: WAYBEL_7:def 4 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool "x" ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) "F")) "holds" (Bool (Set (Var "A")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "B"))))); pred "x" :::"is_a_convergence_point_of"::: "F" "," "T" means :: WAYBEL_7:def 5 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "T" "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool "x" ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) "F")); end; :: deftheorem defines :::"is_a_cluster_point_of"::: WAYBEL_7:def 4 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_waybel_7 :::"is_a_cluster_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "A")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "B"))))) ")" ))); :: deftheorem defines :::"is_a_convergence_point_of"::: WAYBEL_7:def 5 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) ")" ))); registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_waybel_0 :::"filtered"::: ) ($#v13_waybel_0 :::"upper"::: ) ($#v3_waybel_7 :::"ultra"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) "X" ")" ))); end; theorem :: WAYBEL_7:27 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#v3_waybel_7 :::"ultra"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_waybel_7 :::"is_a_cluster_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) "iff" (Bool (Set (Var "p")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) ")" )))) ; theorem :: WAYBEL_7:28 (Bool "for" (Set (Var "T")) "being" ($#~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 :::"<<"::: ) (Set (Var "y")))) "holds" (Bool "for" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "y"))) & (Bool (Set (Var "p")) ($#r1_waybel_7 :::"is_a_cluster_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) ")" ))))) ; theorem :: WAYBEL_7:29 (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")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y")))) "holds" (Bool "for" (Set (Var "F")) "being" ($#v3_waybel_7 :::"ultra"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "y"))) & (Bool (Set (Var "p")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) ")" ))))) ; theorem :: WAYBEL_7:30 (Bool "for" (Set (Var "T")) "being" ($#~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_tarski :::"c="::: ) (Set (Var "y"))) & (Bool "(" "for" (Set (Var "F")) "being" ($#v3_waybel_7 :::"ultra"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "y"))) & (Bool (Set (Var "p")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) ")" )) ")" )) "holds" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))))) ; theorem :: WAYBEL_7:31 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"prebasis":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#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")) ($#r1_tarski :::"c="::: ) (Set (Var "y")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))) "iff" (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "B")) "st" (Bool (Bool (Set (Var "y")) ($#r1_tarski :::"c="::: ) (Set ($#k3_tarski :::"union"::: ) (Set (Var "F"))))) "holds" (Bool "ex" (Set (Var "G")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "F")) "st" (Bool (Set (Var "x")) ($#r1_tarski :::"c="::: ) (Set ($#k3_tarski :::"union"::: ) (Set (Var "G")))))) ")" )))) ; theorem :: WAYBEL_7:32 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y"))) "iff" (Bool "for" (Set (Var "P")) "being" ($#v1_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "P"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "P")))) ")" ))) ; theorem :: WAYBEL_7:33 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v5_waybel_6 :::"prime"::: ) )) "holds" (Bool (Set ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "p"))) "is" ($#v1_waybel_7 :::"prime"::: ) ))) ; begin definitionlet "L" be ($#l1_orders_2 :::"LATTICE":::); let "p" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); attr "p" is :::"pseudoprime"::: means :: WAYBEL_7:def 6 (Bool "ex" (Set (Var "P")) "being" ($#v1_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Ideal":::) "of" "L" "st" (Bool "p" ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "P"))))); end; :: deftheorem defines :::"pseudoprime"::: WAYBEL_7:def 6 : (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v4_waybel_7 :::"pseudoprime"::: ) ) "iff" (Bool "ex" (Set (Var "P")) "being" ($#v1_waybel_7 :::"prime"::: ) ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "P"))))) ")" ))); theorem :: WAYBEL_7:34 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v5_waybel_6 :::"prime"::: ) )) "holds" (Bool (Set (Var "p")) "is" ($#v4_waybel_7 :::"pseudoprime"::: ) ))) ; theorem :: WAYBEL_7:35 (Bool "for" (Set (Var "L")) "being" ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v4_waybel_7 :::"pseudoprime"::: ) )) "holds" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "A"))) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "p")))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p"))) ")" ))))) ; theorem :: WAYBEL_7:36 (Bool "for" (Set (Var "L")) "being" ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")))) "or" "not" (Bool (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))) "is" ($#v1_waybel_3 :::"compact"::: ) ) ")" ) & (Bool "(" "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "A"))) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "p")))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p"))) ")" )) ")" )) "holds" (Bool (Set ($#k4_waybel_0 :::"uparrow"::: ) (Set "(" ($#k12_waybel_0 :::"fininfs"::: ) (Set "(" (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "p")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) ")" )) ($#r1_subset_1 :::"misses"::: ) (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "p")))))) ; theorem :: WAYBEL_7:37 (Bool "for" (Set (Var "L")) "being" ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))) "is" ($#v1_waybel_3 :::"compact"::: ) )) "holds" (Bool "(" (Bool "(" "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "A"))) ($#r1_waybel_3 :::"<<"::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")))) ")" )) ")" ) & (Bool (Set ($#k4_waybel_0 :::"uparrow"::: ) (Set "(" ($#k12_waybel_0 :::"fininfs"::: ) (Set "(" (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set "(" ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")) ")" ) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) ")" )) ($#r2_subset_1 :::"meets"::: ) (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set "(" ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")) ")" ))) ")" )) ; theorem :: WAYBEL_7:38 (Bool "for" (Set (Var "L")) "being" ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k4_waybel_0 :::"uparrow"::: ) (Set "(" ($#k12_waybel_0 :::"fininfs"::: ) (Set "(" (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "p")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) ")" )) ($#r1_subset_1 :::"misses"::: ) (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "p"))))) "holds" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "A"))) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "p")))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p"))) ")" ))))) ; theorem :: WAYBEL_7:39 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k4_waybel_0 :::"uparrow"::: ) (Set "(" ($#k12_waybel_0 :::"fininfs"::: ) (Set "(" (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "p")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) ")" )) ($#r1_subset_1 :::"misses"::: ) (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Var "p")) "is" ($#v4_waybel_7 :::"pseudoprime"::: ) ))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); attr "R" is :::"multiplicative"::: means :: WAYBEL_7:def 7 (Bool "for" (Set (Var "a")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "a")) "," (Set (Var "x")) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R") & (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "a")) "," (Set (Var "y")) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R")) "holds" (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "a")) "," (Set "(" (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y")) ")" ) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R")); end; :: deftheorem defines :::"multiplicative"::: WAYBEL_7:def 7 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v5_waybel_7 :::"multiplicative"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "a")) "," (Set (Var "x")) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "a")) "," (Set (Var "y")) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) "holds" (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "a")) "," (Set "(" (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y")) ")" ) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) ")" ))); registrationlet "L" be ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"sup-Semilattice":::); let "R" be ($#v5_waybel_4 :::"auxiliary"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "L")); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); cluster (Set "R" ($#k6_waybel_4 :::"-above"::: ) "x") -> ($#v13_waybel_0 :::"upper"::: ) ; end; theorem :: WAYBEL_7:40 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#v5_waybel_4 :::"auxiliary"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v5_waybel_7 :::"multiplicative"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "R")) ($#k6_waybel_4 :::"-above"::: ) (Set (Var "x"))) "is" ($#v2_waybel_0 :::"filtered"::: ) )) ")" ))) ; theorem :: WAYBEL_7:41 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#v5_waybel_4 :::"auxiliary"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v5_waybel_7 :::"multiplicative"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "a")) "," (Set (Var "x")) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "b")) "," (Set (Var "y")) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) "holds" (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set "(" (Set (Var "a")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "b")) ")" ) "," (Set "(" (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "y")) ")" ) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) ")" ))) ; theorem :: WAYBEL_7:42 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#v5_waybel_4 :::"auxiliary"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v5_waybel_7 :::"multiplicative"::: ) ) "iff" (Bool "for" (Set (Var "S")) "being" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Var "R")))) "holds" (Bool (Set (Var "S")) "is" ($#v5_yellow_0 :::"meet-inheriting"::: ) )) ")" ))) ; theorem :: WAYBEL_7:43 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#v5_waybel_4 :::"auxiliary"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v5_waybel_7 :::"multiplicative"::: ) ) "iff" (Bool (Set (Set (Var "R")) ($#k7_waybel_4 :::"-below"::: ) ) "is" ($#v19_waybel_0 :::"meet-preserving"::: ) ) ")" ))) ; theorem :: WAYBEL_7:44 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set (Set (Var "L")) ($#k1_waybel_4 :::"-waybelow"::: ) ) "is" ($#v5_waybel_7 :::"multiplicative"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "p")) "is" ($#v4_waybel_7 :::"pseudoprime"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" "not" (Bool (Set (Set (Var "a")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "b"))) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "p"))) "or" (Bool (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p"))) "or" (Bool (Set (Var "b")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p"))) ")" )) ")" ))) ; theorem :: WAYBEL_7:45 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set (Set (Var "L")) ($#k1_waybel_4 :::"-waybelow"::: ) ) "is" ($#v5_waybel_7 :::"multiplicative"::: ) )) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v4_waybel_7 :::"pseudoprime"::: ) )) "holds" (Bool (Set (Var "p")) "is" ($#v5_waybel_6 :::"prime"::: ) ))) ; theorem :: WAYBEL_7:46 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_waybel_1 :::"distributive"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v4_waybel_7 :::"pseudoprime"::: ) )) "holds" (Bool (Set (Var "p")) "is" ($#v5_waybel_6 :::"prime"::: ) ) ")" )) "holds" (Bool (Set (Set (Var "L")) ($#k1_waybel_4 :::"-waybelow"::: ) ) "is" ($#v5_waybel_7 :::"multiplicative"::: ) )) ;