:: WAYBEL_6 semantic presentation begin scheme :: WAYBEL_6:sch 1 NonUniqExD1{ F1() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) , F2() -> ($#m1_subset_1 :::"Subset":::) "of" (Set F1 "(" ")" ), F3() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set F1 "(" ")" ), P1[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set F2 "(" ")" ) "," (Set F3 "(" ")" ) "st" (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set F3 "(" ")" )) & (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "e")))) & (Bool P1[(Set (Var "e")) "," (Set (Var "u"))]) ")" )))) provided (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool (Set (Var "e")) ($#r2_hidden :::"in"::: ) (Set F2 "(" ")" ))) "holds" (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set F3 "(" ")" )) & (Bool P1[(Set (Var "e")) "," (Set (Var "u"))]) ")" ))) proof end; definitionlet "L" be ($#l1_orders_2 :::"LATTICE":::); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "A")) "," (Set (Const "A")); let "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); :: original: :::"iter"::: redefine func :::"iter"::: "(" "f" "," "n" ")" -> ($#m1_subset_1 :::"Function":::) "of" "A" "," "A"; end; definitionlet "L" be ($#l1_orders_2 :::"LATTICE":::); let "C", "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "C")) "," (Set (Const "D")); let "c" be ($#m2_subset_1 :::"Element"::: ) "of" (Set (Const "C")); :: original: :::"."::: redefine func "f" :::"."::: "c" -> ($#m1_subset_1 :::"Element":::) "of" "L"; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::); cluster ($#v6_orders_2 :::"strongly_connected"::: ) -> ($#v1_waybel_0 :::"directed"::: ) ($#v2_waybel_0 :::"filtered"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); 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"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_waybel_1 :::"distributive"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v3_waybel_3 :::"continuous"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; theorem :: WAYBEL_6:1 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v19_waybel_0 :::"meet-preserving"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k12_lattice3 :::""/\""::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" )))) ")" ))) ; theorem :: WAYBEL_6:2 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_orders_2 :::"sup-Semilattice":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v20_waybel_0 :::"join-preserving"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k13_lattice3 :::""\/""::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" )))) ")" ))) ; theorem :: WAYBEL_6:3 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "T")) "is" ($#v2_waybel_1 :::"distributive"::: ) ) & (Bool (Set (Var "f")) "is" ($#v19_waybel_0 :::"meet-preserving"::: ) ) & (Bool (Set (Var "f")) "is" ($#v20_waybel_0 :::"join-preserving"::: ) ) & (Bool (Set (Var "f")) "is" bbbadV2_FUNCT_1())) "holds" (Bool (Set (Var "S")) "is" ($#v2_waybel_1 :::"distributive"::: ) ))) ; registrationlet "S", "T" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FUNCT_2((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T")) ($#v18_waybel_0 :::"sups-preserving"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: WAYBEL_6:4 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "f")) "being" ($#v18_waybel_0 :::"sups-preserving"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "T")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) & (Bool (Set (Var "f")) "is" ($#v19_waybel_0 :::"meet-preserving"::: ) ) & (Bool (Set (Var "f")) "is" bbbadV2_FUNCT_1())) "holds" (Bool (Set (Var "S")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ))) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); attr "X" is :::"Open"::: means :: WAYBEL_6:def 1 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "y")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "x"))) ")" ))); end; :: deftheorem defines :::"Open"::: WAYBEL_6: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")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_waybel_6 :::"Open"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "x"))) ")" ))) ")" ))); theorem :: WAYBEL_6:5 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "being" ($#v13_waybel_0 :::"upper"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_waybel_6 :::"Open"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "x"))) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "X")))) ")" ))) ; theorem :: WAYBEL_6:6 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "being" ($#v13_waybel_0 :::"upper"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_waybel_6 :::"Open"::: ) ) "iff" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "x")) ")" ) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "}" )) ")" ))) ; registrationlet "L" be ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_waybel_0 :::"filtered"::: ) ($#v13_waybel_0 :::"upper"::: ) ($#v1_waybel_6 :::"Open"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); end; theorem :: WAYBEL_6:7 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "x"))) "is" ($#v1_waybel_6 :::"Open"::: ) ))) ; theorem :: WAYBEL_6:8 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_3 :::"<<"::: ) (Set (Var "y")))) "holds" (Bool "ex" (Set (Var "F")) "being" ($#v1_waybel_6 :::"Open"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) & (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set ($#k2_waybel_3 :::"wayabove"::: ) (Set (Var "x")))) ")" )))) ; theorem :: WAYBEL_6:9 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "being" ($#v13_waybel_0 :::"upper"::: ) ($#v1_waybel_6 :::"Open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "X")) ($#k3_subset_1 :::"`"::: ) ))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "m")) ($#r3_waybel_4 :::"is_maximal_in"::: ) (Set (Set (Var "X")) ($#k3_subset_1 :::"`"::: ) )) ")" ))))) ; begin definitionlet "G" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "g" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "G")); attr "g" is :::"meet-irreducible"::: means :: WAYBEL_6:def 2 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "G" "holds" (Bool "(" "not" (Bool "g" ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y")))) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) "g") "or" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) "g") ")" )); end; :: deftheorem defines :::"meet-irreducible"::: WAYBEL_6:def 2 : (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v2_waybel_6 :::"meet-irreducible"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool "(" "not" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y")))) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) "or" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" )) ")" ))); notationlet "G" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "g" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "G")); synonym :::"irreducible"::: "g" for :::"meet-irreducible"::: ; end; definitionlet "G" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "g" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "G")); attr "g" is :::"join-irreducible"::: means :: WAYBEL_6:def 3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "G" "holds" (Bool "(" "not" (Bool "g" ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "y")))) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) "g") "or" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) "g") ")" )); end; :: deftheorem defines :::"join-irreducible"::: WAYBEL_6:def 3 : (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v3_waybel_6 :::"join-irreducible"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) "holds" (Bool "(" "not" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "y")))) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) "or" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" )) ")" ))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"IRR"::: "L" -> ($#m1_subset_1 :::"Subset":::) "of" "L" means :: WAYBEL_6:def 4 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#v2_waybel_6 :::"irreducible"::: ) ) ")" )); end; :: deftheorem defines :::"IRR"::: WAYBEL_6:def 4 : (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")) ($#r1_hidden :::"="::: ) (Set ($#k3_waybel_6 :::"IRR"::: ) (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" ($#v2_waybel_6 :::"irreducible"::: ) ) ")" )) ")" ))); theorem :: WAYBEL_6:10 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))) "is" ($#v2_waybel_6 :::"irreducible"::: ) )) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v2_waybel_6 :::"irreducible"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L"); end; theorem :: WAYBEL_6:11 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v2_waybel_6 :::"irreducible"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) ")" ))) ; theorem :: WAYBEL_6:12 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "l")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "l")) ($#k6_domain_1 :::"}"::: ) )) "is" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")))) "holds" (Bool (Set (Var "l")) "is" ($#v2_waybel_6 :::"irreducible"::: ) ))) ; theorem :: WAYBEL_6:13 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) ($#r3_waybel_4 :::"is_maximal_in"::: ) (Set (Set (Var "F")) ($#k3_subset_1 :::"`"::: ) ))) "holds" (Bool (Set (Var "p")) "is" ($#v2_waybel_6 :::"irreducible"::: ) )))) ; theorem :: WAYBEL_6:14 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#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")) "st" (Bool (Bool (Bool "not" (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set (Var "x"))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "p")) "is" ($#v2_waybel_6 :::"irreducible"::: ) ) & (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p"))) & (Bool (Bool "not" (Set (Var "y")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p")))) ")" )))) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); attr "X" is :::"order-generating"::: means :: WAYBEL_6:def 5 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool "(" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) "X") "," "L") & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::"inf"::: ) (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) "X" ")" ))) ")" )); end; :: deftheorem defines :::"order-generating"::: WAYBEL_6:def 5 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_waybel_6 :::"order-generating"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))) "," (Set (Var "L"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::"inf"::: ) (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X")) ")" ))) ")" )) ")" ))); theorem :: WAYBEL_6:15 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_waybel_6 :::"order-generating"::: ) ) "iff" (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "ex" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Set (Var "l")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "Y")) "," (Set (Var "L")) ")" )))) ")" ))) ; theorem :: WAYBEL_6:16 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_waybel_6 :::"order-generating"::: ) ) "iff" (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) & (Bool "(" "for" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y")) "holds" (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "Z")) "," (Set (Var "L")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) ")" )) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Var "Y")))) ")" ))) ; theorem :: WAYBEL_6:17 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_waybel_6 :::"order-generating"::: ) ) "iff" (Bool "for" (Set (Var "l1")) "," (Set (Var "l2")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Bool "not" (Set (Var "l2")) ($#r3_orders_2 :::"<="::: ) (Set (Var "l1"))))) "holds" (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "l1")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p"))) & (Bool (Bool "not" (Set (Var "l2")) ($#r3_orders_2 :::"<="::: ) (Set (Var "p")))) ")" ))) ")" ))) ; theorem :: WAYBEL_6:18 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_waybel_6 :::"IRR"::: ) (Set (Var "L")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) )))) "holds" (Bool (Set (Var "X")) "is" ($#v4_waybel_6 :::"order-generating"::: ) ))) ; theorem :: WAYBEL_6:19 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v4_waybel_6 :::"order-generating"::: ) ) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "Y")) "is" ($#v4_waybel_6 :::"order-generating"::: ) ))) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "l" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); attr "l" is :::"prime"::: means :: WAYBEL_6:def 6 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool "(" "not" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y"))) ($#r1_orders_2 :::"<="::: ) "l") "or" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) "l") "or" (Bool (Set (Var "y")) ($#r1_orders_2 :::"<="::: ) "l") ")" )); end; :: deftheorem defines :::"prime"::: WAYBEL_6:def 6 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#v5_waybel_6 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" "not" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y"))) ($#r1_orders_2 :::"<="::: ) (Set (Var "l"))) "or" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "l"))) "or" (Bool (Set (Var "y")) ($#r1_orders_2 :::"<="::: ) (Set (Var "l"))) ")" )) ")" ))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"PRIME"::: "L" -> ($#m1_subset_1 :::"Subset":::) "of" "L" means :: WAYBEL_6:def 7 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#v5_waybel_6 :::"prime"::: ) ) ")" )); end; :: deftheorem defines :::"PRIME"::: WAYBEL_6:def 7 : (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")) ($#r1_hidden :::"="::: ) (Set ($#k4_waybel_6 :::"PRIME"::: ) (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" ($#v5_waybel_6 :::"prime"::: ) ) ")" )) ")" ))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "l" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); attr "l" is :::"co-prime"::: means :: WAYBEL_6:def 8 (Bool (Set "l" ($#k8_lattice3 :::"~"::: ) ) "is" ($#v5_waybel_6 :::"prime"::: ) ); end; :: deftheorem defines :::"co-prime"::: WAYBEL_6:def 8 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#v6_waybel_6 :::"co-prime"::: ) ) "iff" (Bool (Set (Set (Var "l")) ($#k8_lattice3 :::"~"::: ) ) "is" ($#v5_waybel_6 :::"prime"::: ) ) ")" ))); theorem :: WAYBEL_6:20 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))) "is" ($#v5_waybel_6 :::"prime"::: ) )) ; theorem :: WAYBEL_6:21 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L"))) "is" ($#v6_waybel_6 :::"co-prime"::: ) )) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v5_waybel_6 :::"prime"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L"); end; theorem :: WAYBEL_6:22 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#v5_waybel_6 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "l")) ($#r1_orders_2 :::">="::: ) (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "A"))))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "l")) ($#r1_orders_2 :::">="::: ) (Set (Var "a"))) ")" ))) ")" ))) ; theorem :: WAYBEL_6:23 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"sup-Semilattice":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v6_waybel_6 :::"co-prime"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "A"))))) "holds" (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "a"))) ")" ))) ")" ))) ; theorem :: WAYBEL_6:24 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "l")) "is" ($#v5_waybel_6 :::"prime"::: ) )) "holds" (Bool (Set (Var "l")) "is" ($#v2_waybel_6 :::"irreducible"::: ) ))) ; theorem :: WAYBEL_6:25 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#v5_waybel_6 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) ) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set (Var "p")) ($#r3_orders_2 :::"<="::: ) (Set (Var "l"))) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v19_waybel_0 :::"meet-preserving"::: ) ) & (Bool (Set (Var "f")) "is" ($#v20_waybel_0 :::"join-preserving"::: ) ) ")" ))) ")" ))) ; theorem :: WAYBEL_6:26 (Bool "for" (Set (Var "L")) "being" ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "l")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))))) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#v5_waybel_6 :::"prime"::: ) ) "iff" (Bool (Set (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "l")) ")" ) ($#k3_subset_1 :::"`"::: ) ) "is" ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L"))) ")" ))) ; theorem :: WAYBEL_6:27 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#v5_waybel_6 :::"prime"::: ) ) "iff" (Bool (Set (Var "l")) "is" ($#v2_waybel_6 :::"irreducible"::: ) ) ")" ))) ; theorem :: WAYBEL_6:28 (Bool "for" (Set (Var "L")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool (Set ($#k4_waybel_6 :::"PRIME"::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set ($#k3_waybel_6 :::"IRR"::: ) (Set (Var "L"))))) ; theorem :: WAYBEL_6:29 (Bool "for" (Set (Var "L")) "being" ($#v11_waybel_1 :::"Boolean"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "l")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))))) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#v5_waybel_6 :::"prime"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_orders_2 :::">"::: ) (Set (Var "l")))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))))) ")" ))) ; theorem :: WAYBEL_6:30 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v2_waybel_1 :::"distributive"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "l")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L"))))) "holds" (Bool "(" (Bool (Set (Var "l")) "is" ($#v5_waybel_6 :::"prime"::: ) ) "iff" (Bool "ex" (Set (Var "F")) "being" ($#v1_waybel_6 :::"Open"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set (Var "L")) "st" (Bool (Set (Var "l")) ($#r3_waybel_4 :::"is_maximal_in"::: ) (Set (Set (Var "F")) ($#k3_subset_1 :::"`"::: ) ))) ")" ))) ; theorem :: WAYBEL_6:31 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k5_funct_3 :::"chi"::: ) "(" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ")" ) "is" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" )))) ; theorem :: WAYBEL_6:32 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_funct_3 :::"chi"::: ) "(" (Set "(" (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "p")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) "iff" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "p"))) ")" ))) ; theorem :: WAYBEL_6:33 (Bool "for" (Set (Var "L")) "being" ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) (Bool "for" (Set (Var "p")) "being" ($#v5_waybel_6 :::"prime"::: ) ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k5_funct_3 :::"chi"::: ) "(" (Set "(" (Set "(" ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "p")) ")" ) ($#k3_subset_1 :::"`"::: ) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ")" ) ($#r1_funct_2 :::"="::: ) (Set (Var "f")))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v19_waybel_0 :::"meet-preserving"::: ) ) & (Bool (Set (Var "f")) "is" ($#v20_waybel_0 :::"join-preserving"::: ) ) ")" )))) ; theorem :: WAYBEL_6:34 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool (Set ($#k4_waybel_6 :::"PRIME"::: ) (Set (Var "L"))) "is" ($#v4_waybel_6 :::"order-generating"::: ) )) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_1 :::"distributive"::: ) ) & (Bool (Set (Var "L")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) ")" )) ; theorem :: WAYBEL_6:35 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_1 :::"distributive"::: ) ) "iff" (Bool (Set ($#k4_waybel_6 :::"PRIME"::: ) (Set (Var "L"))) "is" ($#v4_waybel_6 :::"order-generating"::: ) ) ")" )) ; theorem :: WAYBEL_6:36 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_1 :::"distributive"::: ) ) "iff" (Bool (Set (Var "L")) "is" ($#v9_waybel_1 :::"Heyting"::: ) ) ")" )) ; theorem :: WAYBEL_6:37 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (Bool "(" "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "ex" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "l")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "x")) "is" ($#v6_waybel_6 :::"co-prime"::: ) ) ")" ) ")" )) ")" )) "holds" (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "l")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" (Set "(" ($#k1_waybel_3 :::"waybelow"::: ) (Set (Var "l")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k4_waybel_6 :::"PRIME"::: ) (Set "(" (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ")" ) ")" ) ")" ) "," (Set (Var "L")) ")" )))) ; theorem :: WAYBEL_6:38 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v1_waybel_5 :::"completely-distributive"::: ) ) "iff" (Bool "(" (Bool (Set (Var "L")) "is" ($#v3_waybel_3 :::"continuous"::: ) ) & (Bool "(" "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "ex" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "l")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "x")) "is" ($#v6_waybel_6 :::"co-prime"::: ) ) ")" ) ")" )) ")" ) ")" ) ")" )) ; theorem :: WAYBEL_6:39 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v1_waybel_5 :::"completely-distributive"::: ) ) "iff" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_1 :::"distributive"::: ) ) & (Bool (Set (Var "L")) "is" ($#v3_waybel_3 :::"continuous"::: ) ) & (Bool (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#v3_waybel_3 :::"continuous"::: ) ) ")" ) ")" )) ;