:: WAYBEL20 semantic presentation begin theorem :: WAYBEL20:1 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "X")) ")" ) "holds" (Bool (Set ($#k9_xtuple_0 :::"proj1"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k10_xtuple_0 :::"proj2"::: ) (Set (Var "S")))))) ; theorem :: WAYBEL20:2 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "Y")) "holds" (Bool (Set (Set ($#k16_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "f")) ($#k16_funct_3 :::":]"::: ) ) ($#k8_relset_1 :::"""::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set (Var "Y")) ")" )) "is" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set (Var "X"))))) ; definitionlet "L1", "L2", "T1", "T2" be ($#l1_orders_2 :::"RelStr"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "L1")) "," (Set (Const "T1")); let "g" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "L2")) "," (Set (Const "T2")); :: original: :::"[:"::: redefine func :::"[:":::"f" "," "g":::":]"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) "L1" "," "L2" ($#k3_yellow_3 :::":]"::: ) ) "," (Set ($#k3_yellow_3 :::"[:"::: ) "T1" "," "T2" ($#k3_yellow_3 :::":]"::: ) ); end; theorem :: WAYBEL20:3 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"proj1"::: ) (Set "(" (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k9_xtuple_0 :::"proj1"::: ) (Set (Var "X")) ")" ))) & (Bool (Set ($#k10_xtuple_0 :::"proj2"::: ) (Set "(" (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "g")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k10_xtuple_0 :::"proj2"::: ) (Set (Var "X")) ")" ))) ")" ))) ; theorem :: WAYBEL20:4 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f")) ")" ) "," (Set "(" ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "g")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"proj1"::: ) (Set "(" (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k9_xtuple_0 :::"proj1"::: ) (Set (Var "X")) ")" ))) & (Bool (Set ($#k10_xtuple_0 :::"proj2"::: ) (Set "(" (Set ($#k15_funct_3 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k15_funct_3 :::":]"::: ) ) ($#k7_relat_1 :::".:"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k7_relat_1 :::".:"::: ) (Set "(" ($#k10_xtuple_0 :::"proj2"::: ) (Set (Var "X")) ")" ))) ")" ))) ; theorem :: WAYBEL20:5 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "S")))) "holds" (Bool (Set (Var "S")) "is" ($#v2_yellow_0 :::"upper-bounded"::: ) )) ; theorem :: WAYBEL20:6 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set (Var "S")))) "holds" (Bool (Set (Var "S")) "is" ($#v1_yellow_0 :::"lower-bounded"::: ) )) ; theorem :: WAYBEL20:7 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L1")) "," (Set (Var "L2")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "D")) "," (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L1")) "," (Set (Var "L2")) ($#k3_yellow_3 :::":]"::: ) ))) "holds" (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set ($#k7_yellow_3 :::"["::: ) (Set "(" ($#k2_yellow_0 :::"inf"::: ) (Set "(" ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")) ")" ) ")" ) "," (Set "(" ($#k2_yellow_0 :::"inf"::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ) ")" ) ($#k7_yellow_3 :::"]"::: ) )))) ; theorem :: WAYBEL20:8 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L1")) "," (Set (Var "L2")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "D")) "," (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L1")) "," (Set (Var "L2")) ($#k3_yellow_3 :::":]"::: ) ))) "holds" (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D"))) ($#r1_hidden :::"="::: ) (Set ($#k7_yellow_3 :::"["::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")) ")" ) ")" ) "," (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ) ")" ) ($#k7_yellow_3 :::"]"::: ) )))) ; theorem :: WAYBEL20:9 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "T1")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "T2")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) )) "holds" (Bool (Set ($#k1_waybel20 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k1_waybel20 :::":]"::: ) ) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) )))) ; theorem :: WAYBEL20:10 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "T1")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "T2")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) )) "holds" (Bool (Set ($#k1_waybel20 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k1_waybel20 :::":]"::: ) ) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) )))) ; theorem :: WAYBEL20:11 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "T1")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "T2")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) )) "holds" (Bool (Set ($#k1_waybel20 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k1_waybel20 :::":]"::: ) ) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) )))) ; theorem :: WAYBEL20:12 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "T1")) "," (Set (Var "T2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "T1")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "T2")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool (Set ($#k1_waybel20 :::"[:"::: ) (Set (Var "f")) "," (Set (Var "g")) ($#k1_waybel20 :::":]"::: ) ) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )))) ; theorem :: WAYBEL20:13 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) & (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "X")) "," (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ))) "holds" (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))))) ; theorem :: WAYBEL20:14 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) & (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ))) "holds" (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))))) ; theorem :: WAYBEL20:15 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set (Var "L")) "," (Set (Var "M")) ($#r5_waybel_1 :::"are_isomorphic"::: ) ) & (Bool (Set (Var "L")) "is" ($#v3_orders_2 :::"reflexive"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v3_orders_2 :::"reflexive"::: ) )) ; theorem :: WAYBEL20:16 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set (Var "L")) "," (Set (Var "M")) ($#r5_waybel_1 :::"are_isomorphic"::: ) ) & (Bool (Set (Var "L")) "is" ($#v4_orders_2 :::"transitive"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v4_orders_2 :::"transitive"::: ) )) ; theorem :: WAYBEL20:17 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set (Var "L")) "," (Set (Var "M")) ($#r5_waybel_1 :::"are_isomorphic"::: ) ) & (Bool (Set (Var "L")) "is" ($#v5_orders_2 :::"antisymmetric"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v5_orders_2 :::"antisymmetric"::: ) )) ; theorem :: WAYBEL20:18 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set (Var "L")) "," (Set (Var "M")) ($#r5_waybel_1 :::"are_isomorphic"::: ) ) & (Bool (Set (Var "L")) "is" ($#v3_lattice3 :::"complete"::: ) )) "holds" (Bool (Set (Var "M")) "is" ($#v3_lattice3 :::"complete"::: ) )) ; theorem :: WAYBEL20:19 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "k")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) )) "holds" (Bool (Set ($#k2_waybel_1 :::"corestr"::: ) (Set (Var "k"))) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) ))) ; theorem :: WAYBEL20:20 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "k")) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) )) "holds" (Bool (Set ($#k2_waybel_1 :::"corestr"::: ) (Set (Var "k"))) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) ))) ; theorem :: WAYBEL20:21 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "k")) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) )) "holds" (Bool (Set ($#k2_waybel_1 :::"corestr"::: ) (Set (Var "k"))) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) ))) ; theorem :: WAYBEL20:22 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "k")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool (Set ($#k2_waybel_1 :::"corestr"::: ) (Set (Var "k"))) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ))) ; theorem :: WAYBEL20:23 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) ))) ; theorem :: WAYBEL20:24 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v2_waybel_0 :::"filtered"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "X"))) "is" ($#v2_waybel_0 :::"filtered"::: ) )))) ; theorem :: WAYBEL20:25 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "L3")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "L3")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) )))) ; theorem :: WAYBEL20:26 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "L3")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "L3")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) )))) ; theorem :: WAYBEL20:27 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "L3")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "L3")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) )))) ; theorem :: WAYBEL20:28 (Bool "for" (Set (Var "L1")) "," (Set (Var "L2")) "," (Set (Var "L3")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L1")) "," (Set (Var "L2")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L2")) "," (Set (Var "L3")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) & (Bool (Set (Var "g")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )))) ; begin theorem :: WAYBEL20:29 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) ) ")" )) "holds" (Bool (Set ($#k5_yellow_1 :::"product"::: ) (Set (Var "J"))) "is" ($#v1_yellow_0 :::"lower-bounded"::: ) ))) ; theorem :: WAYBEL20:30 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) ) ")" )) "holds" (Bool (Set ($#k5_yellow_1 :::"product"::: ) (Set (Var "J"))) "is" ($#v2_yellow_0 :::"upper-bounded"::: ) ))) ; theorem :: WAYBEL20:31 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) ) ")" )) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set "(" ($#k3_yellow_0 :::"Bottom"::: ) (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Var "J")) ")" ) ")" ) ($#k4_waybel_3 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set "(" (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i")) ")" )))))) ; theorem :: WAYBEL20:32 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) ) ")" )) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set "(" ($#k4_yellow_0 :::"Top"::: ) (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Var "J")) ")" ) ")" ) ($#k4_waybel_3 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set "(" (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i")) ")" )))))) ; theorem :: WAYBEL20:33 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#v5_waybel_3 :::"reflexive-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "J")) ($#k3_waybel_3 :::"."::: ) (Set (Var "i"))) "is" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::)) ")" )) "holds" (Bool (Set ($#k5_yellow_1 :::"product"::: ) (Set (Var "J"))) "is" ($#v3_waybel_3 :::"continuous"::: ) ))) ; begin theorem :: WAYBEL20:34 (Bool "for" (Set (Var "L")) "," (Set (Var "T")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "g")) "being" ($#m1_waybel16 :::"CLHomomorphism"::: ) "of" (Set (Var "L")) "," (Set (Var "T")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_waybel20 :::"[:"::: ) (Set (Var "g")) "," (Set (Var "g")) ($#k1_waybel20 :::":]"::: ) ) ($#k8_relset_1 :::"""::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ")" )))) "holds" (Bool (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "S"))) "is" ($#m1_yellow_0 :::"CLSubFrame":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ))))) ; definitionlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; let "R" be ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Const "L")) "," (Set (Const "L")) ($#k3_yellow_3 :::":]"::: ) ); assume (Bool (Set (Const "R")) "is" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L")))) ; func :::"EqRel"::: "R" -> ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") equals :: WAYBEL20:def 1 "R"; end; :: deftheorem defines :::"EqRel"::: WAYBEL20:def 1 : (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "R")) "is" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) "holds" (Bool (Set ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R"))) ($#r1_hidden :::"="::: ) (Set (Var "R"))))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "R" be ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Const "L")) "," (Set (Const "L")) ($#k3_yellow_3 :::":]"::: ) ); attr "R" is :::"CLCongruence"::: means :: WAYBEL20:def 2 (Bool "(" (Bool "R" "is" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")) & (Bool (Set ($#k5_yellow_0 :::"subrelstr"::: ) "R") "is" ($#m1_yellow_0 :::"CLSubFrame":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) "L" "," "L" ($#k3_yellow_3 :::":]"::: ) )) ")" ); end; :: deftheorem defines :::"CLCongruence"::: WAYBEL20:def 2 : (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 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v1_waybel20 :::"CLCongruence"::: ) ) "iff" (Bool "(" (Bool (Set (Var "R")) "is" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L")))) & (Bool (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "R"))) "is" ($#m1_yellow_0 :::"CLSubFrame":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) )) ")" ) ")" ))); theorem :: WAYBEL20:35 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "R")) "is" ($#v1_waybel20 :::"CLCongruence"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k7_yellow_3 :::"["::: ) (Set "(" ($#k2_yellow_0 :::"inf"::: ) (Set "(" ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ) "," (Set (Var "x")) ")" ")" ) ")" ) "," (Set (Var "x")) ($#k7_yellow_3 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))))) ; definitionlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::); let "R" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Const "L")) "," (Set (Const "L")) ($#k3_yellow_3 :::":]"::: ) ); assume (Bool (Set (Const "R")) "is" ($#v1_waybel20 :::"CLCongruence"::: ) ) ; func :::"kernel_op"::: "R" -> ($#v8_waybel_1 :::"kernel"::: ) ($#m1_subset_1 :::"Function":::) "of" "L" "," "L" means :: WAYBEL20:def 3 (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 :::"inf"::: ) (Set "(" ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k2_waybel20 :::"EqRel"::: ) "R" ")" ) "," (Set (Var "x")) ")" ")" )))); end; :: deftheorem defines :::"kernel_op"::: WAYBEL20:def 3 : (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "R")) "is" ($#v1_waybel20 :::"CLCongruence"::: ) )) "holds" (Bool "for" (Set (Var "b3")) "being" ($#v8_waybel_1 :::"kernel"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_waybel20 :::"kernel_op"::: ) (Set (Var "R")))) "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 :::"inf"::: ) (Set "(" ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ) "," (Set (Var "x")) ")" ")" )))) ")" )))); theorem :: WAYBEL20:36 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "R")) "is" ($#v1_waybel20 :::"CLCongruence"::: ) )) "holds" (Bool "(" (Bool (Set ($#k3_waybel20 :::"kernel_op"::: ) (Set (Var "R"))) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) & (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_waybel20 :::"[:"::: ) (Set "(" ($#k3_waybel20 :::"kernel_op"::: ) (Set (Var "R")) ")" ) "," (Set "(" ($#k3_waybel20 :::"kernel_op"::: ) (Set (Var "R")) ")" ) ($#k1_waybel20 :::":]"::: ) ) ($#k8_relset_1 :::"""::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ")" ))) ")" ))) ; theorem :: WAYBEL20:37 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) (Bool "for" (Set (Var "k")) "being" ($#v8_waybel_1 :::"kernel"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "k")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) & (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_waybel20 :::"[:"::: ) (Set (Var "k")) "," (Set (Var "k")) ($#k1_waybel20 :::":]"::: ) ) ($#k8_relset_1 :::"""::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ")" )))) "holds" (Bool "ex" (Set (Var "LR")) "being" ($#v1_orders_2 :::"strict"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "LR"))) ($#r1_hidden :::"="::: ) (Set ($#k8_eqrel_1 :::"Class"::: ) (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ))) & (Bool (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "LR"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k1_domain_1 :::"["::: ) (Set "(" ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ) "," (Set (Var "x")) ")" ")" ) "," (Set "(" ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ) "," (Set (Var "y")) ")" ")" ) ($#k1_domain_1 :::"]"::: ) ) where x, y "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool (Set (Set (Var "k")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r3_orders_2 :::"<="::: ) (Set (Set (Var "k")) ($#k3_funct_2 :::"."::: ) (Set (Var "y")))) "}" ) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "LR")) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ) "," (Set (Var "x")) ")" )) ")" )) "holds" (Bool (Set (Var "g")) "is" ($#m1_waybel16 :::"CLHomomorphism"::: ) "of" (Set (Var "L")) "," (Set (Var "LR"))) ")" ) ")" ))))) ; theorem :: WAYBEL20:38 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "R")) "is" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L")))) & (Bool "ex" (Set (Var "LR")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "LR"))) ($#r1_hidden :::"="::: ) (Set ($#k8_eqrel_1 :::"Class"::: ) (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "LR")) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ) "," (Set (Var "x")) ")" )) ")" )) "holds" (Bool (Set (Var "g")) "is" ($#m1_waybel16 :::"CLHomomorphism"::: ) "of" (Set (Var "L")) "," (Set (Var "LR"))) ")" ) ")" ))) "holds" (Bool (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "R"))) "is" ($#m1_yellow_0 :::"CLSubFrame":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) )))) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#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"::: ) bbbadV1_PARTFUN1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")) bbbadV1_FUNCT_2((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")) ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ($#v8_waybel_1 :::"kernel"::: ) for ($#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") ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "k" be ($#v8_waybel_1 :::"kernel"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "L")) "," (Set (Const "L")); func :::"kernel_congruence"::: "k" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) "L" "," "L" ($#k3_yellow_3 :::":]"::: ) ) equals :: WAYBEL20:def 4 (Set (Set ($#k1_waybel20 :::"[:"::: ) "k" "," "k" ($#k1_waybel20 :::":]"::: ) ) ($#k8_relset_1 :::"""::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ")" )); end; :: deftheorem defines :::"kernel_congruence"::: WAYBEL20:def 4 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "k")) "being" ($#v8_waybel_1 :::"kernel"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool (Set ($#k4_waybel20 :::"kernel_congruence"::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_waybel20 :::"[:"::: ) (Set (Var "k")) "," (Set (Var "k")) ($#k1_waybel20 :::":]"::: ) ) ($#k8_relset_1 :::"""::: ) (Set "(" ($#k6_partfun1 :::"id"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ")" ))))); theorem :: WAYBEL20:39 (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 "k")) "being" ($#v8_waybel_1 :::"kernel"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool (Set ($#k4_waybel20 :::"kernel_congruence"::: ) (Set (Var "k"))) "is" ($#m1_subset_1 :::"Equivalence_Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L")))))) ; theorem :: WAYBEL20:40 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "k")) "being" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ($#v8_waybel_1 :::"kernel"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool (Set ($#k4_waybel20 :::"kernel_congruence"::: ) (Set (Var "k"))) "is" ($#v1_waybel20 :::"CLCongruence"::: ) ))) ; definitionlet "L" be ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::); let "R" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Const "L")) "," (Set (Const "L")) ($#k3_yellow_3 :::":]"::: ) ); assume (Bool (Set (Const "R")) "is" ($#v1_waybel20 :::"CLCongruence"::: ) ) ; func "L" :::"./."::: "R" -> ($#v1_orders_2 :::"strict"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) means :: WAYBEL20:def 5 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k8_eqrel_1 :::"Class"::: ) (Set "(" ($#k2_waybel20 :::"EqRel"::: ) "R" ")" ))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" it "holds" (Bool "(" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "x")) "," "L" ")" ) ($#r3_orders_2 :::"<="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "y")) "," "L" ")" )) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"./."::: WAYBEL20:def 5 : (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "R")) "is" ($#v1_waybel20 :::"CLCongruence"::: ) )) "holds" (Bool "for" (Set (Var "b3")) "being" ($#v1_orders_2 :::"strict"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k5_waybel20 :::"./."::: ) (Set (Var "R")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k8_eqrel_1 :::"Class"::: ) (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b3")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "x")) "," (Set (Var "L")) ")" ) ($#r3_orders_2 :::"<="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "y")) "," (Set (Var "L")) ")" )) ")" ) ")" ) ")" ) ")" )))); theorem :: WAYBEL20:41 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "R")) "is" ($#v1_waybel20 :::"CLCongruence"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "L")) ($#k5_waybel20 :::"./."::: ) (Set (Var "R")) ")" )) "iff" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k6_eqrel_1 :::"Class"::: ) "(" (Set "(" ($#k2_waybel20 :::"EqRel"::: ) (Set (Var "R")) ")" ) "," (Set (Var "y")) ")" ))) ")" )))) ; theorem :: WAYBEL20:42 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "R")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "st" (Bool (Bool (Set (Var "R")) "is" ($#v1_waybel20 :::"CLCongruence"::: ) )) "holds" (Bool (Set (Var "R")) ($#r1_hidden :::"="::: ) (Set ($#k4_waybel20 :::"kernel_congruence"::: ) (Set "(" ($#k3_waybel20 :::"kernel_op"::: ) (Set (Var "R")) ")" ))))) ; theorem :: WAYBEL20:43 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "k")) "being" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ($#v8_waybel_1 :::"kernel"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "holds" (Bool (Set (Var "k")) ($#r2_funct_2 :::"="::: ) (Set ($#k3_waybel20 :::"kernel_op"::: ) (Set "(" ($#k4_waybel20 :::"kernel_congruence"::: ) (Set (Var "k")) ")" ))))) ; theorem :: WAYBEL20:44 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "p")) "being" ($#v6_waybel_1 :::"projection"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "p"))) "is" ($#v3_waybel_3 :::"continuous"::: ) ($#l1_orders_2 :::"LATTICE":::)) & (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "p"))) "is" ($#v7_yellow_0 :::"infs-inheriting"::: ) ) ")" ))) ;