:: YELLOW_2 semantic presentation begin theorem :: YELLOW_2:1 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x")))) "iff" (Bool (Set (Var "X")) ($#r2_lattice3 :::"is_<=_than"::: ) (Set (Var "x"))) ")" )))) ; theorem :: YELLOW_2:2 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k6_waybel_0 :::"uparrow"::: ) (Set (Var "x")))) "iff" (Bool (Set (Var "x")) ($#r1_lattice3 :::"is_<=_than"::: ) (Set (Var "X"))) ")" )))) ; theorem :: YELLOW_2:3 (Bool "for" (Set (Var "L")) "being" ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) & (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "Y")) "," (Set (Var "L")))) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y"))) "," (Set (Var "L"))) & (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")) ")" ) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ")" ) ($#k13_lattice3 :::""\/""::: ) (Set "(" ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "Y")) "," (Set (Var "L")) ")" ")" ))) ")" ))) ; theorem :: YELLOW_2:4 (Bool "for" (Set (Var "L")) "being" ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) & (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "Y")) "," (Set (Var "L")))) "holds" (Bool "(" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y"))) "," (Set (Var "L"))) & (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")) ")" ) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ")" ) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "Y")) "," (Set (Var "L")) ")" ")" ))) ")" ))) ; begin theorem :: YELLOW_2:5 (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "R")) ($#k2_wellord1 :::"|_2"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "R")) ($#k2_wellord1 :::"|_2"::: ) (Set (Var "Y")))))) ; theorem :: YELLOW_2:6 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))))) "holds" (Bool (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "S"))) ($#r1_relset_1 :::"c="::: ) (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "T")))))) ; theorem :: YELLOW_2:7 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "X")) "is" ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")))) "implies" (Bool (Set (Var "X")) "is" ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L"))) ")" & "(" (Bool (Bool (Set (Var "X")) "is" ($#v2_waybel_0 :::"filtered"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")))) "implies" (Bool (Set (Var "X")) "is" ($#v2_waybel_0 :::"filtered"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L"))) ")" ")" )))) ; theorem :: YELLOW_2:8 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))))) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T"))) & (Bool "(" "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "Y")))) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "X")) "is" ($#v2_waybel_0 :::"filtered"::: ) )) "implies" (Bool (Set (Var "Y")) "is" ($#v2_waybel_0 :::"filtered"::: ) ) ")" & "(" (Bool (Bool (Set (Var "X")) "is" ($#v1_waybel_0 :::"directed"::: ) )) "implies" (Bool (Set (Var "Y")) "is" ($#v1_waybel_0 :::"directed"::: ) ) ")" ")" ) ")" ) ")" )))) ; begin definitionlet "J" be ($#m1_hidden :::"set"::: ) ; let "L" be ($#l1_orders_2 :::"RelStr"::: ) ; let "f", "g" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); pred "f" :::"<="::: "g" means :: YELLOW_2:def 1 (Bool "for" (Set (Var "j")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) "J")) "holds" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set "f" ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set "g" ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) & (Bool (Set (Var "a")) ($#r1_orders_2 :::"<="::: ) (Set (Var "b"))) ")" ))); end; :: deftheorem defines :::"<="::: YELLOW_2:def 1 : (Bool "for" (Set (Var "J")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_yellow_2 :::"<="::: ) (Set (Var "g"))) "iff" (Bool "for" (Set (Var "j")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "j")) ($#r2_hidden :::"in"::: ) (Set (Var "J")))) "holds" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")))) & (Bool (Set (Var "a")) ($#r1_orders_2 :::"<="::: ) (Set (Var "b"))) ")" ))) ")" )))); notationlet "J" be ($#m1_hidden :::"set"::: ) ; let "L" be ($#l1_orders_2 :::"RelStr"::: ) ; let "f", "g" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); synonym "g" :::">="::: "f" for "f" :::"<="::: "g"; end; theorem :: YELLOW_2:9 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "M")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_yellow_2 :::"<="::: ) (Set (Var "g"))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_orders_2 :::"<="::: ) (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))))) ")" ))) ; begin definitionlet "L", "M" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "L")) "," (Set (Const "M")); func :::"Image"::: "f" -> ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" "M" equals :: YELLOW_2:def 2 (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) "f" ")" )); end; :: deftheorem defines :::"Image"::: YELLOW_2:def 2 : (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "M")) "holds" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")) ")" ))))); theorem :: YELLOW_2:10 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "M")) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k1_yellow_2 :::"Image"::: ) (Set (Var "f")) ")" ) (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "y"))))))) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); cluster (Set ($#k5_yellow_0 :::"subrelstr"::: ) "X") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; registrationlet "L", "M" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "L")) "," (Set (Const "M")); cluster (Set ($#k1_yellow_2 :::"Image"::: ) "f") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_yellow_0 :::"full"::: ) ; end; begin theorem :: YELLOW_2:11 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set ($#k3_struct_0 :::"id"::: ) (Set (Var "L"))) "is" ($#v5_orders_3 :::"monotone"::: ) )) ; theorem :: YELLOW_2:12 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "," (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "M")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "M")) "," (Set (Var "N")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) ) & (Bool (Set (Var "g")) "is" ($#v5_orders_3 :::"monotone"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v5_orders_3 :::"monotone"::: ) )))) ; theorem :: YELLOW_2:13 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "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 "L")) "," (Set (Var "M")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) ) & (Bool (Set (Var "x")) ($#r1_lattice3 :::"is_<=_than"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_lattice3 :::"is_<=_than"::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "X")))))))) ; theorem :: YELLOW_2:14 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "M")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) ) & (Bool (Set (Var "X")) ($#r2_lattice3 :::"is_<=_than"::: ) (Set (Var "x")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "X"))) ($#r2_lattice3 :::"is_<=_than"::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")))))))) ; theorem :: YELLOW_2:15 (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")) (Bool "for" (Set (Var "X")) "being" ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "X"))) "is" ($#v1_waybel_0 :::"directed"::: ) )))) ; theorem :: YELLOW_2:16 (Bool "for" (Set (Var "L")) "being" ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) ))) ; theorem :: YELLOW_2:17 (Bool "for" (Set (Var "L")) "being" ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v21_waybel_0 :::"filtered-infs-preserving"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) ))) ; begin theorem :: YELLOW_2:18 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v11_quantal1 :::"idempotent"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))))))) ; theorem :: YELLOW_2:19 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v11_quantal1 :::"idempotent"::: ) )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) : (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")))) "}" ))) ; theorem :: YELLOW_2:20 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v11_quantal1 :::"idempotent"::: ) ) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Var "X")))))) ; theorem :: YELLOW_2:21 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set ($#k3_struct_0 :::"id"::: ) (Set (Var "L"))) "is" ($#v11_quantal1 :::"idempotent"::: ) )) ; begin theorem :: YELLOW_2:22 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" )) & (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r3_orders_2 :::"<="::: ) (Set (Var "a"))) ")" )))) ; theorem :: YELLOW_2:23 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool "(" "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) ")" ) "iff" (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "Y")) "," (Set (Var "L")))) ")" )) ; theorem :: YELLOW_2:24 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) ")" )) "holds" (Bool (Set (Var "L")) "is" ($#v3_lattice3 :::"complete"::: ) )) ; theorem :: YELLOW_2:25 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) ")" )) "holds" (Bool (Set (Var "L")) "is" ($#v3_lattice3 :::"complete"::: ) )) ; theorem :: YELLOW_2:26 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "A")) "," (Set (Var "L"))) ")" )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) & (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ")" ) "," (Set (Var "L")) ")" )) ")" ))) ; theorem :: YELLOW_2:27 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "A")) "," (Set (Var "L"))) ")" )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) & (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ")" ) "," (Set (Var "L")) ")" )) ")" ))) ; theorem :: YELLOW_2:28 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "A")) "," (Set (Var "L"))) ")" )) "holds" (Bool (Set (Var "L")) "is" ($#v3_lattice3 :::"complete"::: ) )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#v25_waybel_0 :::"/\-complete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; theorem :: YELLOW_2:29 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) )) "holds" (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")))) "}" )) "holds" (Bool (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "M"))) "is" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::))))) ; theorem :: YELLOW_2:30 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_yellow_0 :::"full"::: ) ($#v7_yellow_0 :::"infs-inheriting"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")) "holds" (Bool (Set (Var "S")) "is" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::)))) ; theorem :: YELLOW_2:31 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_yellow_0 :::"full"::: ) ($#v8_yellow_0 :::"sups-inheriting"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "L")) "holds" (Bool (Set (Var "S")) "is" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::)))) ; theorem :: YELLOW_2:32 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "M")) "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 "L")) "," (Set (Var "M")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) )) "holds" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "f"))) "is" ($#v8_yellow_0 :::"sups-inheriting"::: ) )))) ; theorem :: YELLOW_2:33 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "M")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) )) "holds" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "f"))) "is" ($#v7_yellow_0 :::"infs-inheriting"::: ) )))) ; theorem :: YELLOW_2:34 (Bool "for" (Set (Var "L")) "," (Set (Var "M")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "M")) "st" (Bool (Bool "(" (Bool (Set (Var "f")) "is" ($#v18_waybel_0 :::"sups-preserving"::: ) ) "or" (Bool (Set (Var "f")) "is" ($#v17_waybel_0 :::"infs-preserving"::: ) ) ")" )) "holds" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "f"))) "is" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::)))) ; theorem :: YELLOW_2:35 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set (Var "L")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v11_quantal1 :::"idempotent"::: ) ) & (Bool (Set (Var "f")) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "f"))) "is" ($#v4_waybel_0 :::"directed-sups-inheriting"::: ) ) & (Bool (Set ($#k1_yellow_2 :::"Image"::: ) (Set (Var "f"))) "is" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::)) ")" ))) ; begin theorem :: YELLOW_2:36 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "L")) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "X")) "is" ($#v12_waybel_0 :::"lower"::: ) ) ")" )) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))) "is" ($#v12_waybel_0 :::"lower"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L"))))) ; theorem :: YELLOW_2:37 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "L")) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "X")) "is" ($#v13_waybel_0 :::"upper"::: ) ) ")" )) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))) "is" ($#v13_waybel_0 :::"upper"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L"))))) ; theorem :: YELLOW_2:38 (Bool "for" (Set (Var "L")) "being" ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "L")) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v12_waybel_0 :::"lower"::: ) ) & (Bool (Set (Var "X")) "is" ($#v1_waybel_0 :::"directed"::: ) ) ")" ) ")" )) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))) "is" ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L"))))) ; theorem :: YELLOW_2:39 (Bool "for" (Set (Var "L")) "being" ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "L")) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v13_waybel_0 :::"upper"::: ) ) & (Bool (Set (Var "X")) "is" ($#v2_waybel_0 :::"filtered"::: ) ) ")" ) ")" )) "holds" (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))) "is" ($#v2_waybel_0 :::"filtered"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L"))))) ; theorem :: YELLOW_2:40 (Bool "for" (Set (Var "L")) "being" ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "I")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "J"))) "is" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L"))))) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k7_waybel_0 :::"Ids"::: ) "L") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: YELLOW_2:41 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) ")" )) "iff" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L"))) ")" ))) ; theorem :: YELLOW_2:42 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")))))) ; theorem :: YELLOW_2:43 (Bool "for" (Set (Var "L")) "being" ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) ")" ) "holds" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "y")))))) ; registrationlet "L" be ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"Poset":::); cluster (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) "L" ")" )) -> ($#v2_lattice3 :::"with_infima"::: ) ; end; theorem :: YELLOW_2:44 (Bool "for" (Set (Var "L")) "being" ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) ")" ) (Bool "ex" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) "{" (Set (Var "z")) where z "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "x"))) "or" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "y"))) "or" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "x"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "y"))) & (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k13_lattice3 :::""\/""::: ) (Set (Var "b")))) ")" )) ")" ) "}" ) & (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k7_domain_1 :::"}"::: ) ) "," (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ))) & (Bool (Set (Set (Var "x")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set ($#k3_waybel_0 :::"downarrow"::: ) (Set (Var "Z")))) ")" )))) ; registrationlet "L" be ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::); cluster (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) "L" ")" )) -> ($#v1_lattice3 :::"with_suprema"::: ) ; end; theorem :: YELLOW_2:45 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"sup-Semilattice":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) "holds" (Bool (Set ($#k1_setfam_1 :::"meet"::: ) (Set (Var "X"))) "is" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L"))))) ; theorem :: YELLOW_2:46 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"sup-Semilattice":::) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) ")" ) "holds" (Bool "(" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Var "A")) "," (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ))) & (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k1_setfam_1 :::"meet"::: ) (Set (Var "A")))) ")" ))) ; theorem :: YELLOW_2:47 (Bool "for" (Set (Var "L")) "being" ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::) "holds" (Bool "(" (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ))) & (Bool (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set ($#k1_xboole_0 :::"{}"::: ) ) "," (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "L")))) ")" )) ; theorem :: YELLOW_2:48 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"sup-Semilattice":::) "holds" (Bool (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" )) "is" ($#v3_lattice3 :::"complete"::: ) )) ; registrationlet "L" be ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"sup-Semilattice":::); cluster (Set ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) "L" ")" )) -> ($#v3_lattice3 :::"complete"::: ) ; end; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::); func :::"SupMap"::: "L" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) "L" ")" ) ")" ) "," "L" means :: YELLOW_2:def 3 (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" "L" "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I"))))); end; :: deftheorem defines :::"SupMap"::: YELLOW_2:def 3 : (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 "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) ")" ) "," (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_2 :::"SupMap"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I"))))) ")" ))); theorem :: YELLOW_2:49 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_yellow_2 :::"SupMap"::: ) (Set (Var "L")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k2_yellow_2 :::"SupMap"::: ) (Set (Var "L")) ")" )) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L"))) ")" )) ; theorem :: YELLOW_2:50 (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 :::"Poset":::) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k2_yellow_2 :::"SupMap"::: ) (Set (Var "L")) ")" ))) "iff" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L"))) ")" ))) ; theorem :: YELLOW_2:51 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Poset":::) "holds" (Bool (Set ($#k2_yellow_2 :::"SupMap"::: ) (Set (Var "L"))) "is" ($#v5_orders_3 :::"monotone"::: ) )) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Poset":::); cluster (Set ($#k2_yellow_2 :::"SupMap"::: ) "L") -> ($#v5_orders_3 :::"monotone"::: ) ; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::); func :::"IdsMap"::: "L" -> ($#m1_subset_1 :::"Function":::) "of" "L" "," (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) "L" ")" ) ")" ) means :: YELLOW_2:def 4 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x"))))); end; :: deftheorem defines :::"IdsMap"::: YELLOW_2: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 (Var "L")) "," (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k7_waybel_0 :::"Ids"::: ) (Set (Var "L")) ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_yellow_2 :::"IdsMap"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_waybel_0 :::"downarrow"::: ) (Set (Var "x"))))) ")" ))); theorem :: YELLOW_2:52 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) "holds" (Bool (Set ($#k3_yellow_2 :::"IdsMap"::: ) (Set (Var "L"))) "is" ($#v5_orders_3 :::"monotone"::: ) )) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::); cluster (Set ($#k3_yellow_2 :::"IdsMap"::: ) "L") -> ($#v5_orders_3 :::"monotone"::: ) ; end; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "F" be ($#m1_hidden :::"Relation":::); func :::"\\/"::: "(" "F" "," "L" ")" -> ($#m1_subset_1 :::"Element":::) "of" "L" equals :: YELLOW_2:def 5 (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) "F" ")" ) "," "L" ")" ); func :::"//\"::: "(" "F" "," "L" ")" -> ($#m1_subset_1 :::"Element":::) "of" "L" equals :: YELLOW_2:def 6 (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) "F" ")" ) "," "L" ")" ); end; :: deftheorem defines :::"\\/"::: YELLOW_2:def 5 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool (Set ($#k4_yellow_2 :::"\\/"::: ) "(" (Set (Var "F")) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F")) ")" ) "," (Set (Var "L")) ")" )))); :: deftheorem defines :::"//\"::: YELLOW_2:def 6 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool (Set ($#k5_yellow_2 :::"//\"::: ) "(" (Set (Var "F")) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F")) ")" ) "," (Set (Var "L")) ")" )))); notationlet "J" be ($#m1_hidden :::"set"::: ) ; let "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "F" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); synonym :::"Sup"::: "F" for :::"\\/"::: "(" "L" "," "J" ")" ; synonym :::"Inf"::: "F" for :::"//\"::: "(" "L" "," "J" ")" ; end; theorem :: YELLOW_2:53 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "J")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "holds" (Bool "(" (Bool (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "j"))) ($#r3_orders_2 :::"<="::: ) (Set ($#k4_yellow_2 :::"Sup"::: ) )) & (Bool (Set ($#k5_yellow_2 :::"Inf"::: ) ) ($#r3_orders_2 :::"<="::: ) (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")))) ")" ))))) ; theorem :: YELLOW_2:54 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "st" (Bool (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "J")) "holds" (Bool (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "j"))) ($#r3_orders_2 :::"<="::: ) (Set (Var "a"))) ")" )) "holds" (Bool (Set ($#k4_yellow_2 :::"Sup"::: ) ) ($#r3_orders_2 :::"<="::: ) (Set (Var "a"))))))) ; theorem :: YELLOW_2:55 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "J")) "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 "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "st" (Bool (Bool "(" "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "J")) "holds" (Bool (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "j")))) ")" )) "holds" (Bool (Set (Var "a")) ($#r3_orders_2 :::"<="::: ) (Set ($#k5_yellow_2 :::"Inf"::: ) )))))) ;