:: WAYBEL_2 semantic presentation begin registrationlet "X", "Y" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "Y")); let "Z" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "X")); cluster (Set "f" ($#k7_relat_1 :::".:"::: ) "Z") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v16_waybel_0 :::"connected"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registrationlet "C" be ($#l1_orders_2 :::"Chain":::); cluster (Set ($#k2_struct_0 :::"[#]"::: ) "C") -> ($#v1_waybel_0 :::"directed"::: ) ; end; theorem :: WAYBEL_2:1 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "D"))) "," (Set (Var "L")))))) ; theorem :: WAYBEL_2:2 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"sup-Semilattice":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k2_yellow_4 :::""\/""::: ) (Set (Var "D"))) "," (Set (Var "L")))))) ; theorem :: WAYBEL_2:3 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"sup-Semilattice":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "A")) ($#r2_lattice3 :::"is_<=_than"::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "A")) ($#k2_yellow_4 :::""\/""::: ) (Set (Var "B")) ")" ))))) ; theorem :: WAYBEL_2:4 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"sup-Semilattice":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "A")) ($#k2_yellow_4 :::""\/""::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "A")) ")" ) ($#k13_lattice3 :::""\/""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "B")) ")" ))))) ; theorem :: WAYBEL_2:5 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool "{" (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ) ")" ) ")" ) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")))) "}" ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" ) where X "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) : (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")))) ")" )) "}" ))) ; theorem :: WAYBEL_2:6 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool (Set ($#k3_tarski :::"union"::: ) "{" (Set (Var "X")) where X "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) : (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")))) ")" )) "}" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")) ")" ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ))))) ; theorem :: WAYBEL_2:7 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set (Var "X")) where X "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) : (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")))) ")" )) "}" ) "," (Set (Var "L"))))) ; theorem :: WAYBEL_2:8 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) "{" (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" ) where X "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) : (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")))) ")" )) "}" "," (Set (Var "L"))))) ; theorem :: WAYBEL_2:9 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" "{" (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" ) where X "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) : (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")))) ")" )) "}" "," (Set (Var "L")) ")" ) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" ($#k3_tarski :::"union"::: ) "{" (Set (Var "X")) where X "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) : (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")))) ")" )) "}" ")" ) "," (Set (Var "L")) ")" )))) ; theorem :: WAYBEL_2:10 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" "{" (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" ) where X "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) : (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")))) ")" )) "}" "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set "(" ($#k3_tarski :::"union"::: ) "{" (Set (Var "X")) where X "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) : (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")))) ")" )) "}" ")" ) "," (Set (Var "L")) ")" )))) ; registrationlet "S", "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k3_yellow_3 :::"[:"::: ) "S" "," "T" ($#k3_yellow_3 :::":]"::: ) ) -> ($#v24_waybel_0 :::"up-complete"::: ) ; end; theorem :: WAYBEL_2:11 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "S")) "," (Set (Var "T")) ($#k3_yellow_3 :::":]"::: ) ) "is" ($#v24_waybel_0 :::"up-complete"::: ) )) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v24_waybel_0 :::"up-complete"::: ) ) & (Bool (Set (Var "T")) "is" ($#v24_waybel_0 :::"up-complete"::: ) ) ")" )) ; theorem :: WAYBEL_2:12 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool (Set ($#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")) ")" ) ")" ) ($#k7_yellow_3 :::"]"::: ) )))) ; theorem :: WAYBEL_2:13 (Bool "for" (Set (Var "S1")) "," (Set (Var "S2")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S1")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S1")) "," (Set (Var "S2")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k3_waybel_0 :::"downarrow"::: ) (Set (Var "D")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k3_waybel_0 :::"downarrow"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "D")) ")" )))))) ; theorem :: WAYBEL_2:14 (Bool "for" (Set (Var "S1")) "," (Set (Var "S2")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "D")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S1")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S1")) "," (Set (Var "S2")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v5_orders_3 :::"monotone"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k4_waybel_0 :::"uparrow"::: ) (Set (Var "D")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k4_waybel_0 :::"uparrow"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "D")) ")" )))))) ; registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) -> (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v2_waybel_1 :::"distributive"::: ) ($#v10_waybel_1 :::"complemented"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v1_orders_2 :::"strict"::: ) bbbadV2_ORDERS_2() ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#~v1_yellow_3 "non" ($#v1_yellow_3 :::"void"::: ) ) ($#v1_lattice3 :::"with_suprema"::: ) ($#v2_lattice3 :::"with_infima"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; theorem :: WAYBEL_2:15 (Bool "for" (Set (Var "H")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "H")) (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "H")) "holds" (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" )))))) ; theorem :: WAYBEL_2:16 (Bool "for" (Set (Var "H")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "H")) (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "H")) "holds" (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ($#k2_yellow_4 :::""\/""::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k13_lattice3 :::""\/""::: ) (Set "(" ($#k2_yellow_0 :::"inf"::: ) (Set (Var "X")) ")" )))))) ; theorem :: WAYBEL_2:17 (Bool "for" (Set (Var "H")) "being" ($#v2_waybel_1 :::"distributive"::: ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"LATTICE":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "H")) (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "H")) "holds" (Bool (Set (Set (Var "a")) ($#k4_waybel_1 :::""/\""::: ) ) ($#r4_waybel_0 :::"preserves_sup_of"::: ) (Set (Var "X")))))) ; begin scheme :: WAYBEL_2:sch 1 ExNet{ F1() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) , F2() -> ($#l1_waybel_0 :::"prenet":::) "of" (Set F1 "(" ")" ), F3( ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) } : (Bool "ex" (Set (Var "M")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set F1 "(" ")" ) "st" (Bool "(" (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "M"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "M"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set F2 "(" ")" )) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set F2 "(" ")" )) "#)" )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "M")) "holds" (Bool (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "M"))) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set "(" (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set F2 "(" ")" )) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ")" )) ")" ) ")" )) proof end; theorem :: WAYBEL_2:18 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "N")) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_waybel_0 :::"netmap"::: ) "(" (Set (Var "N")) "," (Set (Var "L")) ")" ")" )) "is" ($#v1_waybel_0 :::"directed"::: ) ))) ; theorem :: WAYBEL_2:19 (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 "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "D")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "holds" (Bool (Set ($#g1_waybel_0 :::"NetStr"::: ) "(#" (Set (Var "D")) "," (Set "(" (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L"))) ($#k1_toler_1 :::"|_2"::: ) (Set (Var "D")) ")" ) "," (Set (Var "n")) "#)" ) "is" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")))))) ; theorem :: WAYBEL_2:20 (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 "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "D")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "n")) ($#r1_funct_2 :::"="::: ) (Set ($#k6_partfun1 :::"id"::: ) (Set (Var "D")))) & (Bool (Set (Var "N")) ($#r1_hidden :::"="::: ) (Set ($#g1_waybel_0 :::"NetStr"::: ) "(#" (Set (Var "D")) "," (Set "(" (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "L"))) ($#k1_toler_1 :::"|_2"::: ) (Set (Var "D")) ")" ) "," (Set (Var "n")) "#)" ))) "holds" (Bool (Set (Var "N")) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) ))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "N" be ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "L")); func :::"sup"::: "N" -> ($#m1_subset_1 :::"Element":::) "of" "L" equals :: WAYBEL_2:def 1 (Set ($#k4_yellow_2 :::"Sup"::: ) ); end; :: deftheorem defines :::"sup"::: WAYBEL_2:def 1 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "L")) "holds" (Bool (Set ($#k1_waybel_2 :::"sup"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set ($#k4_yellow_2 :::"Sup"::: ) )))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "J" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); func :::"FinSups"::: "f" -> ($#l1_waybel_0 :::"prenet":::) "of" "L" means :: WAYBEL_2:def 2 (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k5_finsub_1 :::"Fin"::: ) "J" ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) "J") "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" "f" ($#k7_relset_1 :::".:"::: ) (Set (Var "x")) ")" ))) & (Bool it ($#r1_hidden :::"="::: ) (Set ($#g1_waybel_0 :::"NetStr"::: ) "(#" (Set "(" ($#k5_finsub_1 :::"Fin"::: ) "J" ")" ) "," (Set "(" ($#k1_yellow_1 :::"RelIncl"::: ) (Set "(" ($#k5_finsub_1 :::"Fin"::: ) "J" ")" ) ")" ) "," (Set (Var "g")) "#)" )) ")" ))); end; :: deftheorem defines :::"FinSups"::: WAYBEL_2:def 2 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "J")) "being" ($#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"))) (Bool "for" (Set (Var "b4")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k2_waybel_2 :::"FinSups"::: ) (Set (Var "f")))) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "J")) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "J"))) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "x")) ")" ))) & (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#g1_waybel_0 :::"NetStr"::: ) "(#" (Set "(" ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "J")) ")" ) "," (Set "(" ($#k1_yellow_1 :::"RelIncl"::: ) (Set "(" ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "J")) ")" ) ")" ) "," (Set (Var "g")) "#)" )) ")" ))) ")" ))))); theorem :: WAYBEL_2:21 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "J")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k2_waybel_2 :::"FinSups"::: ) (Set (Var "f")) ")" )) "iff" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "J")))) ")" )))) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "J" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); cluster (Set ($#k2_waybel_2 :::"FinSups"::: ) "f") -> ($#v8_waybel_0 :::"monotone"::: ) ; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); let "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "L")); func "x" :::""/\""::: "N" -> ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" "L" means :: WAYBEL_2:def 3 (Bool "(" (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" it) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "N") "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" "N") "#)" )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" it (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool "(" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" "N") ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" it) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set "x" ($#k11_lattice3 :::""/\""::: ) (Set (Var "y")))) ")" )) ")" ) ")" ); end; :: deftheorem defines :::""/\""::: WAYBEL_2:def 3 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "L")) (Bool "for" (Set (Var "b4")) "being" ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k3_waybel_2 :::""/\""::: ) (Set (Var "N")))) "iff" (Bool "(" (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b4"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "b4"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "N"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "N"))) "#)" )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b4")) (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "b4"))) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y")))) ")" )) ")" ) ")" ) ")" ))))); theorem :: WAYBEL_2:22 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N"))) "iff" (Bool (Set (Var "y")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" (Set (Var "x")) ($#k3_waybel_2 :::""/\""::: ) (Set (Var "N")) ")" )) ")" ))))) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); let "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "L")); cluster (Set "x" ($#k3_waybel_2 :::""/\""::: ) "N") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_waybel_0 :::"strict"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); let "N" be ($#l1_waybel_0 :::"prenet":::) "of" (Set (Const "L")); cluster (Set "x" ($#k3_waybel_2 :::""/\""::: ) "N") -> ($#v6_waybel_0 :::"strict"::: ) ($#v7_waybel_0 :::"directed"::: ) ; end; theorem :: WAYBEL_2:23 (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 "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "L")) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" (Set (Var "x")) ($#k3_waybel_2 :::""/\""::: ) (Set (Var "F")) ")" ))) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_yellow_4 :::""/\""::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "F"))) ")" )))))) ; theorem :: WAYBEL_2:24 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "J")) "being" ($#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 "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "x"))) "," (Set (Var "L"))) ")" )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_waybel_0 :::"netmap"::: ) "(" (Set "(" ($#k2_waybel_2 :::"FinSups"::: ) (Set (Var "f")) ")" ) "," (Set (Var "L")) ")" ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k11_waybel_0 :::"finsups"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")) ")" )))))) ; theorem :: WAYBEL_2:25 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "J")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_waybel_0 :::"netmap"::: ) "(" (Set "(" ($#k2_waybel_2 :::"FinSups"::: ) (Set (Var "f")) ")" ) "," (Set (Var "L")) ")" ")" )))))) ; theorem :: WAYBEL_2:26 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "J")) "being" ($#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 ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) "," (Set (Var "L"))) & (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_waybel_0 :::"netmap"::: ) "(" (Set "(" ($#k2_waybel_2 :::"FinSups"::: ) (Set (Var "f")) ")" ) "," (Set (Var "L")) ")" ")" )) "," (Set (Var "L"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_finsub_1 :::"Fin"::: ) (Set (Var "J"))) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "x"))) "," (Set (Var "L"))) ")" )) "holds" (Bool (Set ($#k4_yellow_2 :::"Sup"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_2 :::"sup"::: ) (Set "(" ($#k2_waybel_2 :::"FinSups"::: ) (Set (Var "f")) ")" )))))) ; theorem :: WAYBEL_2:27 (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 "N")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "N")) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) )) "holds" (Bool (Set (Set (Var "x")) ($#k3_waybel_2 :::""/\""::: ) (Set (Var "N"))) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) )))) ; theorem :: WAYBEL_2:28 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "E"))))) "holds" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "E")) ")" )))) ")" )) "holds" (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "D")) ")" )))))) ; theorem :: WAYBEL_2:29 (Bool "for" (Set (Var "L")) "being" ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_yellow_4 :::""/\""::: ) (Set (Var "X")) ")" )))) ")" )) "holds" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "L")))) "holds" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_yellow_4 :::""/\""::: ) (Set "(" ($#k11_waybel_0 :::"finsups"::: ) (Set (Var "X")) ")" ) ")" )))))) ; theorem :: WAYBEL_2:30 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "st" (Bool (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")) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k11_waybel_0 :::"finsups"::: ) (Set (Var "X")) ")" ) ")" )))) ")" )) "holds" (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "X")) ")" )))))) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"inf_op"::: "L" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) "L" "," "L" ($#k3_yellow_3 :::":]"::: ) ) "," "L" means :: WAYBEL_2:def 4 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y"))))); end; :: deftheorem defines :::"inf_op"::: WAYBEL_2:def 4 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "," (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_waybel_2 :::"inf_op"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "b2")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y"))))) ")" ))); theorem :: WAYBEL_2:31 (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 ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool (Set (Set "(" ($#k4_waybel_2 :::"inf_op"::: ) (Set (Var "L")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k8_yellow_3 :::"`1"::: ) ")" ) ($#k11_lattice3 :::""/\""::: ) (Set "(" (Set (Var "x")) ($#k9_yellow_3 :::"`2"::: ) ")" ))))) ; registrationlet "L" be ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k4_waybel_2 :::"inf_op"::: ) "L") -> ($#v5_orders_3 :::"monotone"::: ) ; end; theorem :: WAYBEL_2:32 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set "(" ($#k4_waybel_2 :::"inf_op"::: ) (Set (Var "S")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set ($#k6_yellow_3 :::"[:"::: ) (Set (Var "D1")) "," (Set (Var "D2")) ($#k6_yellow_3 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "D1")) ($#k3_yellow_4 :::""/\""::: ) (Set (Var "D2")))))) ; theorem :: WAYBEL_2:33 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k4_waybel_2 :::"inf_op"::: ) (Set (Var "L")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "D")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set "(" ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")) ")" ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ) ")" ))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; func :::"sup_op"::: "L" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) "L" "," "L" ($#k3_yellow_3 :::":]"::: ) ) "," "L" means :: WAYBEL_2:def 5 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool (Set it ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "y"))))); end; :: deftheorem defines :::"sup_op"::: WAYBEL_2:def 5 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "," (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_waybel_2 :::"sup_op"::: ) (Set (Var "L")))) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "b2")) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k10_lattice3 :::""\/""::: ) (Set (Var "y"))))) ")" ))); theorem :: WAYBEL_2:34 (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 ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_waybel_2 :::"sup_op"::: ) (Set (Var "L")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k8_yellow_3 :::"`1"::: ) ")" ) ($#k10_lattice3 :::""\/""::: ) (Set "(" (Set (Var "x")) ($#k9_yellow_3 :::"`2"::: ) ")" ))))) ; registrationlet "L" be ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k5_waybel_2 :::"sup_op"::: ) "L") -> ($#v5_orders_3 :::"monotone"::: ) ; end; theorem :: WAYBEL_2:35 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set "(" ($#k5_waybel_2 :::"sup_op"::: ) (Set (Var "S")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set ($#k6_yellow_3 :::"[:"::: ) (Set (Var "D1")) "," (Set (Var "D2")) ($#k6_yellow_3 :::":]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "D1")) ($#k1_yellow_4 :::""\/""::: ) (Set (Var "D2")))))) ; theorem :: WAYBEL_2:36 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_waybel_0 :::"filtered"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k3_yellow_3 :::"[:"::: ) (Set (Var "L")) "," (Set (Var "L")) ($#k3_yellow_3 :::":]"::: ) ) "holds" (Bool (Set ($#k2_yellow_0 :::"inf"::: ) (Set "(" (Set "(" ($#k5_waybel_2 :::"sup_op"::: ) (Set (Var "L")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "D")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::"inf"::: ) (Set "(" (Set "(" ($#k4_yellow_3 :::"proj1"::: ) (Set (Var "D")) ")" ) ($#k2_yellow_4 :::""\/""::: ) (Set "(" ($#k5_yellow_3 :::"proj2"::: ) (Set (Var "D")) ")" ) ")" ))))) ; begin definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; attr "R" is :::"satisfying_MC"::: means :: WAYBEL_2:def 6 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "R" (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" "R" "holds" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_yellow_4 :::""/\""::: ) (Set (Var "D")) ")" ))))); end; :: deftheorem defines :::"satisfying_MC"::: WAYBEL_2:def 6 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v1_waybel_2 :::"satisfying_MC"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_yellow_4 :::""/\""::: ) (Set (Var "D")) ")" ))))) ")" )); definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; attr "R" is :::"meet-continuous"::: means :: WAYBEL_2:def 7 (Bool "(" (Bool "R" "is" ($#v24_waybel_0 :::"up-complete"::: ) ) & (Bool "R" "is" ($#v1_waybel_2 :::"satisfying_MC"::: ) ) ")" ); end; :: deftheorem defines :::"meet-continuous"::: WAYBEL_2:def 7 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) "iff" (Bool "(" (Bool (Set (Var "R")) "is" ($#v24_waybel_0 :::"up-complete"::: ) ) & (Bool (Set (Var "R")) "is" ($#v1_waybel_2 :::"satisfying_MC"::: ) ) ")" ) ")" )); registration cluster (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) -> (Num 1) ($#v13_struct_0 :::"-element"::: ) ($#v3_orders_2 :::"reflexive"::: ) ($#v1_waybel_2 :::"satisfying_MC"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v2_waybel_2 :::"meet-continuous"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#v1_waybel_2 :::"satisfying_MC"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#v1_waybel_2 :::"satisfying_MC"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v2_waybel_2 :::"meet-continuous"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; theorem :: WAYBEL_2:37 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" "{" (Set "(" (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set (Var "y")) ")" ) where y "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) : (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "}" "," (Set (Var "S")) ")" ))) ")" )) "holds" (Bool (Set (Var "S")) "is" ($#v1_waybel_2 :::"satisfying_MC"::: ) )) ; theorem :: WAYBEL_2:38 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "st" (Bool (Bool (Set ($#k2_yellow_2 :::"SupMap"::: ) (Set (Var "L"))) "is" ($#v19_waybel_0 :::"meet-preserving"::: ) )) "holds" (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I1")) ")" ) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "I1")) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "I2")) ")" ))))) ; registrationlet "L" be ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"sup-Semilattice":::); cluster (Set ($#k2_yellow_2 :::"SupMap"::: ) "L") -> ($#v20_waybel_0 :::"join-preserving"::: ) ; end; theorem :: WAYBEL_2:39 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "st" (Bool (Bool "(" "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I1")) ")" ) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "I1")) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "I2")) ")" ))) ")" )) "holds" (Bool (Set ($#k2_yellow_2 :::"SupMap"::: ) (Set (Var "L"))) "is" ($#v19_waybel_0 :::"meet-preserving"::: ) )) ; theorem :: WAYBEL_2:40 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "st" (Bool (Bool "(" "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I1")) ")" ) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "I1")) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "I2")) ")" ))) ")" )) "holds" (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D1")) ")" ) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "D1")) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "D2")) ")" ))))) ; theorem :: WAYBEL_2:41 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set (Var "L")) "is" ($#v1_waybel_2 :::"satisfying_MC"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "N")) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) )) "holds" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set "(" ($#k1_waybel_2 :::"sup"::: ) (Set (Var "N")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_yellow_4 :::""/\""::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_waybel_0 :::"netmap"::: ) "(" (Set (Var "N")) "," (Set (Var "L")) ")" ")" ) ")" ) ")" )))))) ; theorem :: WAYBEL_2:42 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "N")) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) )) "holds" (Bool (Set (Set (Var "x")) ($#k11_lattice3 :::""/\""::: ) (Set "(" ($#k1_waybel_2 :::"sup"::: ) (Set (Var "N")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_yellow_4 :::""/\""::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_waybel_0 :::"netmap"::: ) "(" (Set (Var "N")) "," (Set (Var "L")) ")" ")" ) ")" ) ")" )))) ")" )) "holds" (Bool (Set (Var "L")) "is" ($#v1_waybel_2 :::"satisfying_MC"::: ) )) ; theorem :: WAYBEL_2:43 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool (Set ($#k4_waybel_2 :::"inf_op"::: ) (Set (Var "L"))) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) "holds" (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D1")) ")" ) ($#k11_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "D1")) ($#k3_yellow_4 :::""/\""::: ) (Set (Var "D2")) ")" ))))) ; theorem :: WAYBEL_2:44 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D1")) ")" ) ($#k11_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "D1")) ($#k3_yellow_4 :::""/\""::: ) (Set (Var "D2")) ")" ))) ")" )) "holds" (Bool (Set (Var "L")) "is" ($#v1_waybel_2 :::"satisfying_MC"::: ) )) ; theorem :: WAYBEL_2:45 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_lattice3 :::"with_infima"::: ) ($#v1_waybel_2 :::"satisfying_MC"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D"))))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "D")) ")" )))))) ; theorem :: WAYBEL_2:46 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "E"))))) "holds" (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "E")) ")" )))) ")" )) "holds" (Bool (Set ($#k4_waybel_2 :::"inf_op"::: ) (Set (Var "L"))) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) )) ; theorem :: WAYBEL_2:47 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"RelStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "N")) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) )) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_waybel_2 :::"sup"::: ) (Set (Var "N")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_waybel_0 :::"netmap"::: ) "(" (Set (Var "N")) "," (Set (Var "L")) ")" ")" ) ")" ) ")" )))) ")" )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "J")) "being" ($#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"))) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k4_yellow_2 :::"Sup"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_2 :::"sup"::: ) (Set "(" (Set (Var "x")) ($#k3_waybel_2 :::""/\""::: ) (Set "(" ($#k2_waybel_2 :::"FinSups"::: ) (Set (Var "f")) ")" ) ")" ))))))) ; theorem :: WAYBEL_2:48 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "st" (Bool (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "J")) "being" ($#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"))) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k4_yellow_2 :::"Sup"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_2 :::"sup"::: ) (Set "(" (Set (Var "x")) ($#k3_waybel_2 :::""/\""::: ) (Set "(" ($#k2_waybel_2 :::"FinSups"::: ) (Set (Var "f")) ")" ) ")" ))))) ")" )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "N")) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) )) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_waybel_2 :::"sup"::: ) (Set (Var "N")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_waybel_0 :::"netmap"::: ) "(" (Set (Var "N")) "," (Set (Var "L")) ")" ")" ) ")" ) ")" )))))) ; theorem :: WAYBEL_2:49 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) "iff" (Bool "(" (Bool (Set ($#k2_yellow_2 :::"SupMap"::: ) (Set (Var "L"))) "is" ($#v19_waybel_0 :::"meet-preserving"::: ) ) & (Bool (Set ($#k2_yellow_2 :::"SupMap"::: ) (Set (Var "L"))) "is" ($#v20_waybel_0 :::"join-preserving"::: ) ) ")" ) ")" )) ; registrationlet "L" be ($#v2_waybel_2 :::"meet-continuous"::: ) ($#l1_orders_2 :::"LATTICE":::); cluster (Set ($#k2_yellow_2 :::"SupMap"::: ) "L") -> ($#v19_waybel_0 :::"meet-preserving"::: ) ($#v20_waybel_0 :::"join-preserving"::: ) ; end; theorem :: WAYBEL_2:50 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) "iff" (Bool "for" (Set (Var "I1")) "," (Set (Var "I2")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I1")) ")" ) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "I2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "I1")) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "I2")) ")" )))) ")" )) ; theorem :: WAYBEL_2:51 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) "iff" (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D1")) ")" ) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D2")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set (Var "D1")) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "D2")) ")" )))) ")" )) ; theorem :: WAYBEL_2:52 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"LATTICE":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_waybel_0 :::"directed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r3_orders_2 :::"<="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set (Var "D"))))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set (Var "D")) ")" ))))) ")" )) ; theorem :: WAYBEL_2:53 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) "iff" (Bool (Set ($#k4_waybel_2 :::"inf_op"::: ) (Set (Var "L"))) "is" ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ) ")" )) ; registrationlet "L" be ($#v2_waybel_2 :::"meet-continuous"::: ) ($#l1_orders_2 :::"Semilattice":::); cluster (Set ($#k4_waybel_2 :::"inf_op"::: ) "L") -> ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ; end; theorem :: WAYBEL_2:54 (Bool "for" (Set (Var "L")) "being" ($#v24_waybel_0 :::"up-complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"prenet":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "N")) "is" ($#v10_waybel_0 :::"eventually-directed"::: ) )) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k1_waybel_2 :::"sup"::: ) (Set (Var "N")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k4_yellow_4 :::""/\""::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_waybel_0 :::"netmap"::: ) "(" (Set (Var "N")) "," (Set (Var "L")) ")" ")" ) ")" ) ")" ))))) ")" )) ; theorem :: WAYBEL_2:55 (Bool "for" (Set (Var "L")) "being" ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Semilattice":::) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "J")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "J")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "holds" (Bool (Set (Set (Var "x")) ($#k12_lattice3 :::""/\""::: ) (Set "(" ($#k4_yellow_2 :::"Sup"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_waybel_2 :::"sup"::: ) (Set "(" (Set (Var "x")) ($#k3_waybel_2 :::""/\""::: ) (Set "(" ($#k2_waybel_2 :::"FinSups"::: ) (Set (Var "f")) ")" ) ")" )))))) ")" )) ; registrationlet "L" be ($#v2_waybel_2 :::"meet-continuous"::: ) ($#l1_orders_2 :::"Semilattice":::); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "L")); cluster (Set "x" ($#k4_waybel_1 :::""/\""::: ) ) -> ($#v22_waybel_0 :::"directed-sups-preserving"::: ) ; end; theorem :: WAYBEL_2:56 (Bool "for" (Set (Var "H")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Poset":::) "holds" (Bool "(" (Bool (Set (Var "H")) "is" ($#v9_waybel_1 :::"Heyting"::: ) ) "iff" (Bool "(" (Bool (Set (Var "H")) "is" ($#v2_waybel_2 :::"meet-continuous"::: ) ) & (Bool (Set (Var "H")) "is" ($#v2_waybel_1 :::"distributive"::: ) ) ")" ) ")" )) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v9_waybel_1 :::"Heyting"::: ) ($#v3_lattice3 :::"complete"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_waybel_1 :::"distributive"::: ) ($#v2_waybel_2 :::"meet-continuous"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v2_waybel_1 :::"distributive"::: ) ($#v3_lattice3 :::"complete"::: ) ($#v2_waybel_2 :::"meet-continuous"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v9_waybel_1 :::"Heyting"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end;