:: WAYBEL10 semantic presentation begin scheme :: WAYBEL10:sch 1 SubrelstrEx{ F1() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) , P1[ ($#m1_hidden :::"set"::: ) ], F2() -> ($#m1_hidden :::"set"::: ) } : (Bool "ex" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set F1 "(" ")" ) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S"))) "iff" (Bool P1[(Set (Var "x"))]) ")" ))) provided (Bool P1[(Set F2 "(" ")" )]) and (Bool (Set F2 "(" ")" ) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F1 "(" ")" ))) proof end; scheme :: WAYBEL10:sch 2 RelstrEq{ F1() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) , F2() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) , P1[ ($#m1_hidden :::"set"::: ) ], P2[ ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ] } : (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F1 "(" ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set F1 "(" ")" )) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F2 "(" ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set F2 "(" ")" )) "#)" )) provided (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" )) "iff" (Bool P1[(Set (Var "x"))]) ")" )) and (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set F2 "(" ")" )) "iff" (Bool P1[(Set (Var "x"))]) ")" )) and (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_orders_2 :::"<="::: ) (Set (Var "b"))) "iff" (Bool P2[(Set (Var "a")) "," (Set (Var "b"))]) ")" )) and (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F2 "(" ")" ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_orders_2 :::"<="::: ) (Set (Var "b"))) "iff" (Bool P2[(Set (Var "a")) "," (Set (Var "b"))]) ")" )) proof end; scheme :: WAYBEL10:sch 3 SubrelstrEq1{ F1() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) , F2() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set F1 "(" ")" ), F3() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set F1 "(" ")" ), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F2 "(" ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set F2 "(" ")" )) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F3 "(" ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set F3 "(" ")" )) "#)" )) provided (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set F2 "(" ")" )) "iff" (Bool P1[(Set (Var "x"))]) ")" )) and (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set F3 "(" ")" )) "iff" (Bool P1[(Set (Var "x"))]) ")" )) proof end; scheme :: WAYBEL10:sch 4 SubrelstrEq2{ F1() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) , F2() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set F1 "(" ")" ), F3() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set F1 "(" ")" ), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F2 "(" ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set F2 "(" ")" )) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F3 "(" ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set F3 "(" ")" )) "#)" )) provided (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set F2 "(" ")" )) "iff" (Bool P1[(Set (Var "x"))]) ")" )) and (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set F3 "(" ")" )) "iff" (Bool P1[(Set (Var "x"))]) ")" )) proof end; theorem :: WAYBEL10:1 (Bool "for" (Set (Var "R")) "," (Set (Var "Q")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "R")) ($#r1_tarski :::"c="::: ) (Set (Var "Q")))) "implies" (Bool (Set (Set (Var "R")) ($#k2_relat_1 :::"~"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "Q")) ($#k2_relat_1 :::"~"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "R")) ($#k2_relat_1 :::"~"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "Q")) ($#k2_relat_1 :::"~"::: ) ))) "implies" (Bool (Set (Var "R")) ($#r1_tarski :::"c="::: ) (Set (Var "Q"))) ")" & "(" (Bool (Bool (Set (Set (Var "R")) ($#k2_relat_1 :::"~"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "Q")))) "implies" (Bool (Set (Var "R")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "Q")) ($#k2_relat_1 :::"~"::: ) )) ")" & "(" (Bool (Bool (Set (Var "R")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "Q")) ($#k2_relat_1 :::"~"::: ) ))) "implies" (Bool (Set (Set (Var "R")) ($#k2_relat_1 :::"~"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "Q"))) ")" ")" )) ; theorem :: WAYBEL10:2 (Bool "for" (Set (Var "L")) "," (Set (Var "S")) "being" ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "S")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")))) "implies" (Bool (Set (Set (Var "S")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "S")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ))) "implies" (Bool (Set (Var "S")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L"))) ")" & "(" (Bool (Bool (Set (Set (Var "S")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")))) "implies" (Bool (Set (Var "S")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) )) ")" & "(" (Bool (Bool (Set (Var "S")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ))) "implies" (Bool (Set (Set (Var "S")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L"))) ")" ")" )) ; theorem :: WAYBEL10:3 (Bool "for" (Set (Var "L")) "," (Set (Var "S")) "being" ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "S")) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")))) "implies" (Bool (Set (Set (Var "S")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) )) ")" & "(" (Bool (Bool (Set (Set (Var "S")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ))) "implies" (Bool (Set (Var "S")) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L"))) ")" & "(" (Bool (Bool (Set (Set (Var "S")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")))) "implies" (Bool (Set (Var "S")) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) )) ")" & "(" (Bool (Bool (Set (Var "S")) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Set (Var "L")) ($#k7_lattice3 :::"opp"::: ) ))) "implies" (Bool (Set (Set (Var "S")) ($#k7_lattice3 :::"opp"::: ) ) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L"))) ")" ")" )) ; definitionlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; let "S" be ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Const "L")); :: original: :::"~"::: redefine func "S" :::"opp"::: -> ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set "L" ($#k7_lattice3 :::"opp"::: ) ); end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set "X" ($#k2_funcop_1 :::"-->"::: ) "L") -> ($#v4_waybel_3 :::"non-Empty"::: ) ; end; registrationlet "S" be ($#l1_orders_2 :::"RelStr"::: ) ; let "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#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"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")) ($#v1_funct_2 :::"quasi_total"::: ) ($#v5_orders_3 :::"monotone"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v6_waybel_1 :::"projection"::: ) -> ($#v11_quantal1 :::"idempotent"::: ) ($#v5_orders_3 :::"monotone"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "S", "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "f" be ($#v5_orders_3 :::"monotone"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "S")) "," (Set (Const "T")); cluster (Set ($#k2_waybel_1 :::"corestr"::: ) "f") -> ($#v5_orders_3 :::"monotone"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k3_struct_0 :::"id"::: ) "L") -> ($#v17_waybel_0 :::"infs-preserving"::: ) ($#v18_waybel_0 :::"sups-preserving"::: ) ; end; theorem :: WAYBEL10:4 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set ($#k4_relat_1 :::"id"::: ) (Set (Var "S"))) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "S")) ")" ) "," (Set (Var "L"))) & (Bool "(" "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "S")) ")" ) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k4_relat_1 :::"id"::: ) (Set (Var "S"))))) "holds" (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) ) ")" ) ")" ))) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV2_FUNCT_1() ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) bbbadV1_PARTFUN1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")) ($#v1_funct_2 :::"quasi_total"::: ) ($#v17_waybel_0 :::"infs-preserving"::: ) ($#v18_waybel_0 :::"sups-preserving"::: ) ($#v7_waybel_1 :::"closure"::: ) ($#v8_waybel_1 :::"kernel"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: WAYBEL10:5 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "c")) "being" ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "c")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_orders_2 :::">="::: ) (Set (Var "x")))))) ; theorem :: WAYBEL10:6 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_tmap_1 :::"|"::: ) (Set (Var "R"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k5_relset_1 :::"|"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_tmap_1 :::"|"::: ) (Set (Var "R")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "x")))) ")" ) ")" )))) ; theorem :: WAYBEL10:7 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) )) "holds" (Bool "for" (Set (Var "R")) "being" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "f")) ($#k2_tmap_1 :::"|"::: ) (Set (Var "R"))) "is" ($#v2_funct_1 :::"one-to-one"::: ) )))) ; registrationlet "S", "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "f" be ($#v5_orders_3 :::"monotone"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "S")) "," (Set (Const "T")); let "R" be ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Const "S")); cluster (Set "f" ($#k2_tmap_1 :::"|"::: ) "R") -> ($#v5_orders_3 :::"monotone"::: ) ; end; theorem :: WAYBEL10:8 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "T")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" bbbadV2_FUNCT_1()) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_funct_1 :::"""::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k2_tmap_1 :::"|"::: ) (Set "(" ($#k1_yellow_2 :::"Image"::: ) (Set "(" (Set (Var "f")) ($#k2_tmap_1 :::"|"::: ) (Set (Var "R")) ")" ) ")" )) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_yellow_2 :::"Image"::: ) (Set "(" (Set (Var "f")) ($#k2_tmap_1 :::"|"::: ) (Set (Var "R")) ")" ) ")" ) "," (Set (Var "R"))) & (Bool (Set (Set (Var "g")) ($#k2_tmap_1 :::"|"::: ) (Set "(" ($#k1_yellow_2 :::"Image"::: ) (Set "(" (Set (Var "f")) ($#k2_tmap_1 :::"|"::: ) (Set (Var "R")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_tmap_1 :::"|"::: ) (Set (Var "R")) ")" ) ($#k2_funct_1 :::"""::: ) )) ")" ))))) ; begin registrationlet "S" be ($#l1_orders_2 :::"RelStr"::: ) ; let "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k7_yellow_1 :::"MonMaps"::: ) "(" "S" "," "T" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; theorem :: WAYBEL10:9 (Bool "for" (Set (Var "S")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_yellow_1 :::"MonMaps"::: ) "(" (Set (Var "S")) "," (Set (Var "T")) ")" ")" )) "iff" (Bool (Set (Var "x")) "is" ($#v5_orders_3 :::"monotone"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T"))) ")" )))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"ClOpers"::: "L" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k7_yellow_1 :::"MonMaps"::: ) "(" "L" "," "L" ")" ) means :: WAYBEL10:def 1 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" "L" "," "L" "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"Element":::) "of" it) "iff" (Bool (Set (Var "f")) "is" ($#v7_waybel_1 :::"closure"::: ) ) ")" )); end; :: deftheorem defines :::"ClOpers"::: WAYBEL10: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 "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k7_yellow_1 :::"MonMaps"::: ) "(" (Set (Var "L")) "," (Set (Var "L")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_waybel10 :::"ClOpers"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b2"))) "iff" (Bool (Set (Var "f")) "is" ($#v7_waybel_1 :::"closure"::: ) ) ")" )) ")" ))); theorem :: WAYBEL10:10 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_waybel10 :::"ClOpers"::: ) (Set (Var "L")) ")" )) "iff" (Bool (Set (Var "x")) "is" ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L"))) ")" ))) ; theorem :: WAYBEL10:11 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "L")) ($#k6_yellow_1 :::"|^"::: ) (Set (Var "X")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool (Set (Var "f")) ($#r1_yellow_2 :::"<="::: ) (Set (Var "g"))) ")" ))))) ; theorem :: WAYBEL10:12 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_waybel10 :::"ClOpers"::: ) (Set (Var "L")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "c1"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "c2")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool (Set (Var "c1")) ($#r1_yellow_2 :::"<="::: ) (Set (Var "c2"))) ")" )))) ; theorem :: WAYBEL10:13 (Bool "for" (Set (Var "L")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "S1")) "," (Set (Var "S2")) "being" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S1"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S2"))))) "holds" (Bool (Set (Var "S1")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "S2"))))) ; theorem :: WAYBEL10:14 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "c1")) "," (Set (Var "c2")) "being" ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "c1")) ($#r1_yellow_2 :::"<="::: ) (Set (Var "c2"))) "iff" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "c2"))) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "c1")))) ")" ))) ; begin definitionlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; func :::"Sub"::: "L" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#l1_orders_2 :::"RelStr"::: ) means :: WAYBEL10:def 2 (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" it) "iff" (Bool (Set (Var "x")) "is" ($#v1_orders_2 :::"strict"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" "L") ")" ) ")" ) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" it "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_orders_2 :::"<="::: ) (Set (Var "b"))) "iff" (Bool "ex" (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool "(" (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "R"))) & (Bool (Set (Var "a")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "R"))) ")" )) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"Sub"::: WAYBEL10:def 2 : (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_waybel10 :::"Sub"::: ) (Set (Var "L")))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b2"))) "iff" (Bool (Set (Var "x")) "is" ($#v1_orders_2 :::"strict"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L"))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b2")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_orders_2 :::"<="::: ) (Set (Var "b"))) "iff" (Bool "ex" (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool "(" (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Var "R"))) & (Bool (Set (Var "a")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "R"))) ")" )) ")" ) ")" ) ")" ) ")" ))); theorem :: WAYBEL10:15 (Bool "for" (Set (Var "L")) "," (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k3_waybel10 :::"Sub"::: ) (Set (Var "L")) ")" ) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "R")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool (Set (Var "x")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "R"))) ")" ))) ; registrationlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k3_waybel10 :::"Sub"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ; end; registrationlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k3_waybel10 :::"Sub"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v3_lattice3 :::"complete"::: ) ; end; registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster ($#v7_yellow_0 :::"infs-inheriting"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) for ($#m1_yellow_0 :::"SubRelStr"::: ) "of" "L"; cluster ($#v8_yellow_0 :::"sups-inheriting"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) for ($#m1_yellow_0 :::"SubRelStr"::: ) "of" "L"; end; definitionlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; mode System of "L" is ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" "L"; end; notationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "S" be ($#m1_yellow_0 :::"System":::) "of" (Set (Const "L")); synonym :::"closure"::: "S" for :::"infs-inheriting"::: ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) "L" ")" )) -> ($#v7_yellow_0 :::"infs-inheriting"::: ) ($#v8_yellow_0 :::"sups-inheriting"::: ) ; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"ClosureSystems"::: "L" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k3_waybel10 :::"Sub"::: ) "L") means :: WAYBEL10:def 3 (Bool "for" (Set (Var "R")) "being" ($#v1_orders_2 :::"strict"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" "L" "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#m1_subset_1 :::"Element":::) "of" it) "iff" (Bool "(" (Bool (Set (Var "R")) "is" ($#v7_yellow_0 :::"infs-inheriting"::: ) ) & (Bool (Set (Var "R")) "is" ($#v4_yellow_0 :::"full"::: ) ) ")" ) ")" )); end; :: deftheorem defines :::"ClosureSystems"::: WAYBEL10:def 3 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k3_waybel10 :::"Sub"::: ) (Set (Var "L"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_waybel10 :::"ClosureSystems"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "R")) "being" ($#v1_orders_2 :::"strict"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b2"))) "iff" (Bool "(" (Bool (Set (Var "R")) "is" ($#v7_yellow_0 :::"infs-inheriting"::: ) ) & (Bool (Set (Var "R")) "is" ($#v4_yellow_0 :::"full"::: ) ) ")" ) ")" )) ")" ))); theorem :: WAYBEL10:16 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_waybel10 :::"ClosureSystems"::: ) (Set (Var "L")) ")" )) "iff" (Bool (Set (Var "x")) "is" ($#v1_orders_2 :::"strict"::: ) ($#v7_yellow_0 :::"closure"::: ) ($#m1_yellow_0 :::"System":::) "of" (Set (Var "L"))) ")" ))) ; theorem :: WAYBEL10:17 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_waybel10 :::"ClosureSystems"::: ) (Set (Var "L")) ")" ) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "R")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool (Set (Var "x")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "R"))) ")" )))) ; begin registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::); let "h" be ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "L")) "," (Set (Const "L")); cluster (Set ($#k1_yellow_2 :::"Image"::: ) "h") -> ($#v7_yellow_0 :::"infs-inheriting"::: ) ; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::); func :::"ClImageMap"::: "L" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_waybel10 :::"ClOpers"::: ) "L" ")" ) "," (Set "(" (Set "(" ($#k4_waybel10 :::"ClosureSystems"::: ) "L" ")" ) ($#k1_waybel10 :::"opp"::: ) ")" ) means :: WAYBEL10:def 4 (Bool "for" (Set (Var "c")) "being" ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" "L" "," "L" "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "c"))))); end; :: deftheorem defines :::"ClImageMap"::: WAYBEL10:def 4 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_waybel10 :::"ClOpers"::: ) (Set (Var "L")) ")" ) "," (Set "(" (Set "(" ($#k4_waybel10 :::"ClosureSystems"::: ) (Set (Var "L")) ")" ) ($#k1_waybel10 :::"opp"::: ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_waybel10 :::"ClImageMap"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "c")) "being" ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "c"))))) ")" ))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "S" be ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Const "L")); func :::"closure_op"::: "S" -> ($#m1_subset_1 :::"Function":::) "of" "L" "," "L" means :: WAYBEL10:def 5 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ")" ) "," "L" ")" ))); end; :: deftheorem defines :::"closure_op"::: WAYBEL10:def 5 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k6_waybel10 :::"closure_op"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")) ")" ) ($#k8_subset_1 :::"/\"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ")" ) "," (Set (Var "L")) ")" ))) ")" )))); registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); let "S" be ($#v7_yellow_0 :::"closure"::: ) ($#m1_yellow_0 :::"System":::) "of" (Set (Const "L")); cluster (Set ($#k6_waybel10 :::"closure_op"::: ) "S") -> ($#v7_waybel_1 :::"closure"::: ) ; end; theorem :: WAYBEL10:18 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "S")) "being" ($#v7_yellow_0 :::"closure"::: ) ($#m1_yellow_0 :::"System":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set "(" ($#k6_waybel10 :::"closure_op"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "S"))) "#)" )))) ; theorem :: WAYBEL10:19 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "c")) "being" ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool (Set ($#k6_waybel10 :::"closure_op"::: ) (Set "(" ($#k1_yellow_2 :::"Image"::: ) (Set (Var "c")) ")" )) ($#r2_funct_2 :::"="::: ) (Set (Var "c"))))) ; registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster (Set ($#k5_waybel10 :::"ClImageMap"::: ) "L") -> bbbadV2_FUNCT_1() ; end; theorem :: WAYBEL10:20 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool (Set (Set "(" ($#k5_waybel10 :::"ClImageMap"::: ) (Set (Var "L")) ")" ) ($#k2_funct_1 :::"""::: ) ) "is" ($#m1_subset_1 :::"Function":::) "of" (Set "(" (Set "(" ($#k4_waybel10 :::"ClosureSystems"::: ) (Set (Var "L")) ")" ) ($#k1_waybel10 :::"opp"::: ) ")" ) "," (Set "(" ($#k2_waybel10 :::"ClOpers"::: ) (Set (Var "L")) ")" ))) ; theorem :: WAYBEL10:21 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "S")) "being" ($#v1_orders_2 :::"strict"::: ) ($#v7_yellow_0 :::"closure"::: ) ($#m1_yellow_0 :::"System":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_waybel10 :::"ClImageMap"::: ) (Set (Var "L")) ")" ) ($#k2_funct_1 :::"""::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k6_waybel10 :::"closure_op"::: ) (Set (Var "S")))))) ; registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster (Set ($#k5_waybel10 :::"ClImageMap"::: ) "L") -> ($#v23_waybel_0 :::"isomorphic"::: ) ; end; theorem :: WAYBEL10:22 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool (Set ($#k2_waybel10 :::"ClOpers"::: ) (Set (Var "L"))) "," (Set (Set "(" ($#k4_waybel10 :::"ClosureSystems"::: ) (Set (Var "L")) ")" ) ($#k1_waybel10 :::"opp"::: ) ) ($#r5_waybel_1 :::"are_isomorphic"::: ) )) ; begin theorem :: WAYBEL10:23 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "S")) "being" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "X")) "is" ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")))) "implies" (Bool (Set (Var "X")) "is" ($#v1_waybel_0 :::"directed"::: ) ) ")" & "(" (Bool (Bool (Set (Var "X")) "is" ($#v2_waybel_0 :::"filtered"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")))) "implies" (Bool (Set (Var "X")) "is" ($#v2_waybel_0 :::"filtered"::: ) ) ")" ")" )))) ; theorem :: WAYBEL10:24 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "S")) "being" ($#v7_yellow_0 :::"closure"::: ) ($#m1_yellow_0 :::"System":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set ($#k6_waybel10 :::"closure_op"::: ) (Set (Var "S"))) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) "iff" (Bool (Set (Var "S")) "is" ($#v4_waybel_0 :::"directed-sups-inheriting"::: ) ) ")" ))) ; theorem :: WAYBEL10:25 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "h")) "being" ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) "iff" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "h"))) "is" ($#v4_waybel_0 :::"directed-sups-inheriting"::: ) ) ")" ))) ; registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); let "S" be ($#v7_yellow_0 :::"closure"::: ) ($#v4_waybel_0 :::"directed-sups-inheriting"::: ) ($#m1_yellow_0 :::"System":::) "of" (Set (Const "L")); cluster (Set ($#k6_waybel10 :::"closure_op"::: ) "S") -> ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ; end; registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); let "h" be ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "L")) "," (Set (Const "L")); cluster (Set ($#k1_yellow_2 :::"Image"::: ) "h") -> ($#v4_waybel_0 :::"directed-sups-inheriting"::: ) ; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"DsupClOpers"::: "L" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k2_waybel10 :::"ClOpers"::: ) "L") means :: WAYBEL10:def 6 (Bool "for" (Set (Var "f")) "being" ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" "L" "," "L" "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"Element":::) "of" it) "iff" (Bool (Set (Var "f")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) ")" )); end; :: deftheorem defines :::"DsupClOpers"::: WAYBEL10:def 6 : (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 "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k2_waybel10 :::"ClOpers"::: ) (Set (Var "L"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k7_waybel10 :::"DsupClOpers"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "f")) "being" ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b2"))) "iff" (Bool (Set (Var "f")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) ")" )) ")" ))); theorem :: WAYBEL10:26 (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" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_waybel10 :::"DsupClOpers"::: ) (Set (Var "L")) ")" )) "iff" (Bool (Set (Var "x")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ($#v7_waybel_1 :::"closure"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L"))) ")" ))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"Subalgebras"::: "L" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k4_waybel10 :::"ClosureSystems"::: ) "L") means :: WAYBEL10:def 7 (Bool "for" (Set (Var "R")) "being" ($#v1_orders_2 :::"strict"::: ) ($#v7_yellow_0 :::"closure"::: ) ($#m1_yellow_0 :::"System":::) "of" "L" "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#m1_subset_1 :::"Element":::) "of" it) "iff" (Bool (Set (Var "R")) "is" ($#v4_waybel_0 :::"directed-sups-inheriting"::: ) ) ")" )); end; :: deftheorem defines :::"Subalgebras"::: WAYBEL10:def 7 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set ($#k4_waybel10 :::"ClosureSystems"::: ) (Set (Var "L"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k8_waybel10 :::"Subalgebras"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "R")) "being" ($#v1_orders_2 :::"strict"::: ) ($#v7_yellow_0 :::"closure"::: ) ($#m1_yellow_0 :::"System":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b2"))) "iff" (Bool (Set (Var "R")) "is" ($#v4_waybel_0 :::"directed-sups-inheriting"::: ) ) ")" )) ")" ))); theorem :: WAYBEL10:27 (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_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k8_waybel10 :::"Subalgebras"::: ) (Set (Var "L")) ")" )) "iff" (Bool (Set (Var "x")) "is" ($#v1_orders_2 :::"strict"::: ) ($#v7_yellow_0 :::"closure"::: ) ($#v4_waybel_0 :::"directed-sups-inheriting"::: ) ($#m1_yellow_0 :::"System":::) "of" (Set (Var "L"))) ")" ))) ; theorem :: WAYBEL10:28 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set "(" (Set "(" ($#k5_waybel10 :::"ClImageMap"::: ) (Set (Var "L")) ")" ) ($#k2_tmap_1 :::"|"::: ) (Set "(" ($#k7_waybel10 :::"DsupClOpers"::: ) (Set (Var "L")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k8_waybel10 :::"Subalgebras"::: ) (Set (Var "L")) ")" ) ($#k1_waybel10 :::"opp"::: ) ))) ; registrationlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster (Set ($#k2_waybel_1 :::"corestr"::: ) (Set "(" (Set "(" ($#k5_waybel10 :::"ClImageMap"::: ) "L" ")" ) ($#k2_tmap_1 :::"|"::: ) (Set "(" ($#k7_waybel10 :::"DsupClOpers"::: ) "L" ")" ) ")" )) -> ($#v23_waybel_0 :::"isomorphic"::: ) ; end; theorem :: WAYBEL10:29 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool (Set ($#k7_waybel10 :::"DsupClOpers"::: ) (Set (Var "L"))) "," (Set (Set "(" ($#k8_waybel10 :::"Subalgebras"::: ) (Set (Var "L")) ")" ) ($#k1_waybel10 :::"opp"::: ) ) ($#r5_waybel_1 :::"are_isomorphic"::: ) )) ;