:: WAYBEL35 semantic presentation begin begin registrationlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v5_relat_1 :::"-valued"::: ) ($#v1_waybel_4 :::"auxiliary(i)"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "L" be ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v5_relat_1 :::"-valued"::: ) ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#v2_waybel_4 :::"auxiliary(ii)"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "L" be ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v5_relat_1 :::"-valued"::: ) ($#v3_waybel_4 :::"auxiliary(iii)"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v5_relat_1 :::"-valued"::: ) ($#v4_waybel_4 :::"auxiliary(iv)"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "L")); attr "R" is :::"extra-order"::: means :: WAYBEL35:def 1 (Bool "(" (Bool "R" "is" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ) & (Bool "R" "is" ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ) & (Bool "R" "is" ($#v4_waybel_4 :::"auxiliary(iv)"::: ) ) ")" ); end; :: deftheorem defines :::"extra-order"::: WAYBEL35:def 1 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v1_waybel35 :::"extra-order"::: ) ) "iff" (Bool "(" (Bool (Set (Var "R")) "is" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ) & (Bool (Set (Var "R")) "is" ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ) & (Bool (Set (Var "R")) "is" ($#v4_waybel_4 :::"auxiliary(iv)"::: ) ) ")" ) ")" ))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v1_waybel35 :::"extra-order"::: ) -> ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#v4_waybel_4 :::"auxiliary(iv)"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); cluster ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#v4_waybel_4 :::"auxiliary(iv)"::: ) -> ($#v1_waybel35 :::"extra-order"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v3_waybel_4 :::"auxiliary(iii)"::: ) ($#v1_waybel35 :::"extra-order"::: ) -> ($#v5_waybel_4 :::"auxiliary"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); cluster ($#v5_waybel_4 :::"auxiliary"::: ) -> ($#v1_waybel35 :::"extra-order"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#v5_relat_1 :::"-valued"::: ) ($#v1_waybel35 :::"extra-order"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "L" be ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::); let "R" be ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "L")); func "R" :::"-LowerMap"::: -> ($#m1_subset_1 :::"Function":::) "of" "L" "," (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k4_lattice7 :::"LOWER"::: ) "L" ")" ) ")" ) means :: WAYBEL35:def 2 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set "R" ($#k5_waybel_4 :::"-below"::: ) (Set (Var "x"))))); end; :: deftheorem defines :::"-LowerMap"::: WAYBEL35:def 2 : (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "L")) "," (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) (Set "(" ($#k4_lattice7 :::"LOWER"::: ) (Set (Var "L")) ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "R")) ($#k1_waybel35 :::"-LowerMap"::: ) )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "R")) ($#k5_waybel_4 :::"-below"::: ) (Set (Var "x"))))) ")" )))); registrationlet "L" be ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::); let "R" be ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Const "L")); cluster (Set "R" ($#k1_waybel35 :::"-LowerMap"::: ) ) -> ($#v5_orders_3 :::"monotone"::: ) ; end; definitionlet "L" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); mode :::"strict_chain"::: "of" "R" -> ($#m1_subset_1 :::"Subset":::) "of" "L" means :: WAYBEL35:def 3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) it) & (Bool (Bool "not" (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R")) & (Bool (Bool "not" (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "y")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R")); end; :: deftheorem defines :::"strict_chain"::: WAYBEL35:def 3 : (Bool "for" (Set (Var "L")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R"))) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) & (Bool (Bool "not" (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) & (Bool (Bool "not" (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "y")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) ")" )))); theorem :: WAYBEL35:1 (Bool "for" (Set (Var "L")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#v1_zfmisc_1 :::"trivial"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "holds" (Bool (Set (Var "C")) "is" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")))))) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); cluster (Num 1) ($#v3_card_1 :::"-element"::: ) for ($#m1_waybel35 :::"strict_chain"::: ) "of" "R"; end; theorem :: WAYBEL35:2 (Bool "for" (Set (Var "L")) "being" ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "x")) ($#r2_orders_2 :::"<"::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))))))) ; theorem :: WAYBEL35:3 (Bool "for" (Set (Var "L")) "being" ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "y")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y")))))) ; theorem :: WAYBEL35:4 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "R")) "being" ($#v4_waybel_4 :::"auxiliary(iv)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k7_domain_1 :::"{"::: ) (Set "(" ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L")) ")" ) "," (Set (Var "x")) ($#k7_domain_1 :::"}"::: ) ) "is" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")))))) ; theorem :: WAYBEL35:5 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#v4_waybel_4 :::"auxiliary(iv)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "C")) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) )) "is" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")))))) ; definitionlet "L" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); let "C" be ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Const "R")); attr "C" is :::"maximal"::: means :: WAYBEL35:def 4 (Bool "for" (Set (Var "D")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" "R" "st" (Bool (Bool "C" ($#r1_lattice7 :::"c="::: ) (Set (Var "D")))) "holds" (Bool "C" ($#r1_hidden :::"="::: ) (Set (Var "D")))); end; :: deftheorem defines :::"maximal"::: WAYBEL35:def 4 : (Bool "for" (Set (Var "L")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v2_waybel35 :::"maximal"::: ) ) "iff" (Bool "for" (Set (Var "D")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "C")) ($#r1_lattice7 :::"c="::: ) (Set (Var "D")))) "holds" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Var "D")))) ")" )))); definitionlet "L" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); let "C" be ($#m1_hidden :::"set"::: ) ; func :::"Strict_Chains"::: "(" "R" "," "C" ")" -> ($#m1_hidden :::"set"::: ) means :: WAYBEL35:def 5 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_waybel35 :::"strict_chain"::: ) "of" "R") & (Bool "C" ($#r1_tarski :::"c="::: ) (Set (Var "x"))) ")" ) ")" )); end; :: deftheorem defines :::"Strict_Chains"::: WAYBEL35:def 5 : (Bool "for" (Set (Var "L")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) (Bool "for" (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k2_waybel35 :::"Strict_Chains"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" )) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b4"))) "iff" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R"))) & (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "x"))) ")" ) ")" )) ")" ))))); registrationlet "L" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); let "C" be ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Const "R")); cluster (Set ($#k2_waybel35 :::"Strict_Chains"::: ) "(" "R" "," "C" ")" ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; notationlet "R" be ($#m1_hidden :::"Relation":::); let "X" be ($#m1_hidden :::"set"::: ) ; synonym "X" :::"is_inductive_wrt"::: "R" for "X" :::"has_upper_Zorn_property_wrt"::: "R"; end; theorem :: WAYBEL35:6 (Bool "for" (Set (Var "L")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set ($#k2_waybel35 :::"Strict_Chains"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" ) ($#r4_orders_1 :::"is_inductive_wrt"::: ) (Set ($#k1_yellow_1 :::"RelIncl"::: ) (Set "(" ($#k2_waybel35 :::"Strict_Chains"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" ")" ))) & (Bool "ex" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "D")) ($#r6_orders_1 :::"is_maximal_in"::: ) (Set ($#k1_yellow_1 :::"RelIncl"::: ) (Set "(" ($#k2_waybel35 :::"Strict_Chains"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" ")" ))) & (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "D"))) ")" )) ")" )))) ; theorem :: WAYBEL35:7 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "C")) "st" (Bool (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) & (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "C")))) & (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "C")) ")" ) ")" )) ")" )))) ; theorem :: WAYBEL35:8 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "C")) "st" (Bool (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) & (Bool (Set (Var "C")) "is" ($#v2_waybel35 :::"maximal"::: ) )) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "C")))))))) ; theorem :: WAYBEL35:9 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "C")) "st" (Bool (Bool ($#r2_yellow_0 :::"ex_inf_of"::: ) (Set (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set "(" ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "C"))) "," (Set (Var "L"))) & (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "L"))) & (Bool (Set (Var "C")) "is" ($#v2_waybel35 :::"maximal"::: ) )) "holds" (Bool "(" (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "C")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set "(" ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "C")) ")" ) "," (Set (Var "L")) ")" )) & "(" (Bool (Bool (Bool "not" (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "C"))))) "implies" (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r2_orders_2 :::"<"::: ) (Set ($#k2_yellow_0 :::""/\""::: ) "(" (Set "(" (Set "(" ($#k6_waybel_0 :::"uparrow"::: ) (Set "(" ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "C")) ")" ) "," (Set (Var "L")) ")" )) ")" ")" ))))) ; theorem :: WAYBEL35:10 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_waybel35 :::"maximal"::: ) )) "holds" (Bool (Set ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "C"))) "is" ($#v3_lattice3 :::"complete"::: ) )))) ; theorem :: WAYBEL35:11 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#v4_waybel_4 :::"auxiliary(iv)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_waybel35 :::"maximal"::: ) )) "holds" (Bool (Set ($#k3_yellow_0 :::"Bottom"::: ) (Set (Var "L"))) ($#r2_hidden :::"in"::: ) (Set (Var "C")))))) ; theorem :: WAYBEL35:12 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_yellow_0 :::"upper-bounded"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_waybel35 :::"maximal"::: ) ) & (Bool (Set (Var "m")) ($#r4_waybel_1 :::"is_maximum_of"::: ) (Set (Var "C"))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "m")) "," (Set "(" ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")) ")" ) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R")))) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set "(" ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")) ")" ) "," (Set "(" ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")) ")" ) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set ($#k4_yellow_0 :::"Top"::: ) (Set (Var "L")))) ")" ))))) ; definitionlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; let "C" be ($#m1_hidden :::"set"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); pred "R" :::"satisfies_SIC_on"::: "C" means :: WAYBEL35:def 6 (Bool "for" (Set (Var "x")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "C") & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) "C") & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "z")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R") & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z")))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "C") & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R") & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "R") & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) ")" ))); end; :: deftheorem defines :::"satisfies_SIC_on"::: WAYBEL35:def 6 : (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "holds" (Bool "(" (Bool (Set (Var "R")) ($#r1_waybel35 :::"satisfies_SIC_on"::: ) (Set (Var "C"))) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "z")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z")))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) ")" ))) ")" )))); definitionlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); let "C" be ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Const "R")); attr "C" is :::"satisfying_SIC"::: means :: WAYBEL35:def 7 (Bool "R" ($#r1_waybel35 :::"satisfies_SIC_on"::: ) "C"); end; :: deftheorem defines :::"satisfying_SIC"::: WAYBEL35:def 7 : (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v3_waybel35 :::"satisfying_SIC"::: ) ) "iff" (Bool (Set (Var "R")) ($#r1_waybel35 :::"satisfies_SIC_on"::: ) (Set (Var "C"))) ")" )))); theorem :: WAYBEL35:13 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "R")) ($#r1_waybel35 :::"satisfies_SIC_on"::: ) (Set (Var "C")))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "z")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z")))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "y")) "," (Set (Var "z")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set (Var "x")) ($#r2_orders_2 :::"<"::: ) (Set (Var "y"))) ")" )))))) ; registrationlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); cluster ($#v1_zfmisc_1 :::"trivial"::: ) -> ($#v3_waybel35 :::"satisfying_SIC"::: ) for ($#m1_waybel35 :::"strict_chain"::: ) "of" "R"; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); cluster (Num 1) ($#v3_card_1 :::"-element"::: ) for ($#m1_waybel35 :::"strict_chain"::: ) "of" "R"; end; theorem :: WAYBEL35:14 (Bool "for" (Set (Var "L")) "being" ($#v1_yellow_0 :::"lower-bounded"::: ) ($#v1_lattice3 :::"with_suprema"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_waybel35 :::"maximal"::: ) ) & (Bool (Set (Var "R")) "is" ($#v8_waybel_4 :::"satisfying_SI"::: ) )) "holds" (Bool (Set (Var "R")) ($#r1_waybel35 :::"satisfies_SIC_on"::: ) (Set (Var "C")))))) ; definitionlet "R" be ($#m1_hidden :::"Relation":::); let "C", "y" be ($#m1_hidden :::"set"::: ) ; func :::"SetBelow"::: "(" "R" "," "C" "," "y" ")" -> ($#m1_hidden :::"set"::: ) equals :: WAYBEL35:def 8 (Set (Set "(" "R" ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) "y" ($#k1_tarski :::"}"::: ) ) ")" ) ($#k3_xboole_0 :::"/\"::: ) "C"); end; :: deftheorem defines :::"SetBelow"::: WAYBEL35:def 8 : (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) (Bool "for" (Set (Var "C")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k3_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "R")) ($#k8_relat_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "y")) ($#k1_tarski :::"}"::: ) ) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "C")))))); theorem :: WAYBEL35:15 (Bool "for" (Set (Var "R")) "being" ($#m1_hidden :::"Relation":::) (Bool "for" (Set (Var "C")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "y")) ")" )) "iff" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" ) ")" ))) ; definitionlet "L" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); let "C", "y" be ($#m1_hidden :::"set"::: ) ; :: original: :::"SetBelow"::: redefine func :::"SetBelow"::: "(" "R" "," "C" "," "y" ")" -> ($#m1_subset_1 :::"Subset":::) "of" "L"; end; theorem :: WAYBEL35:16 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "y")) ")" ) ($#r2_lattice3 :::"is_<=_than"::: ) (Set (Var "y"))))))) ; theorem :: WAYBEL35:17 (Bool "for" (Set (Var "L")) "being" ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "x")) ")" ) ($#r1_lattice7 :::"c="::: ) (Set ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "y")) ")" )))))) ; theorem :: WAYBEL35:18 (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "x")) "," (Set (Var "x")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "x")) ")" ) "," (Set (Var "L")))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "x")) ")" ")" ))))))) ; definitionlet "L" be ($#l1_orders_2 :::"RelStr"::: ) ; let "C" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); attr "C" is :::"sup-closed"::: means :: WAYBEL35:def 9 (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" "C" "st" (Bool (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," "L")) "holds" (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," "L" ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k5_yellow_0 :::"subrelstr"::: ) "C" ")" ) ")" ))); end; :: deftheorem defines :::"sup-closed"::: WAYBEL35:def 9 : (Bool "for" (Set (Var "L")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v4_waybel35 :::"sup-closed"::: ) ) "iff" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "C")) "st" (Bool (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set (Var "X")) "," (Set (Var "L")))) "holds" (Bool (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set (Var "L")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" (Set (Var "X")) "," (Set "(" ($#k5_yellow_0 :::"subrelstr"::: ) (Set (Var "C")) ")" ) ")" ))) ")" ))); theorem :: WAYBEL35:19 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel35 :::"extra-order"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#v3_waybel35 :::"satisfying_SIC"::: ) ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "p")) ($#r2_orders_2 :::"<"::: ) (Set (Var "q")))) "holds" (Bool "ex" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r2_orders_2 :::"<"::: ) (Set (Var "y"))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "y")) "," (Set (Var "q")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "y")) ")" ")" ))) ")" )))))) ; theorem :: WAYBEL35:20 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_yellow_0 :::"lower-bounded"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel35 :::"extra-order"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v4_waybel35 :::"sup-closed"::: ) ) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "c")) ")" ) "," (Set (Var "L"))) ")" ) & (Bool (Set (Var "R")) ($#r1_waybel35 :::"satisfies_SIC_on"::: ) (Set (Var "C")))) "holds" (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "c")) ")" ")" ))))))) ; theorem :: WAYBEL35:21 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v5_orders_2 :::"antisymmetric"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "st" (Bool (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool "(" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "c")) ")" ) "," (Set (Var "L"))) & (Bool (Set (Var "c")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "c")) ")" ")" ))) ")" ) ")" )) "holds" (Bool (Set (Var "R")) ($#r1_waybel35 :::"satisfies_SIC_on"::: ) (Set (Var "C")))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); let "C" be ($#m1_hidden :::"set"::: ) ; func :::"SupBelow"::: "(" "R" "," "C" ")" -> ($#m1_hidden :::"set"::: ) means :: WAYBEL35:def 10 (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k4_waybel35 :::"SetBelow"::: ) "(" "R" "," "C" "," (Set (Var "y")) ")" ")" ))) ")" )); end; :: deftheorem defines :::"SupBelow"::: WAYBEL35:def 10 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) (Bool "for" (Set (Var "C")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k5_waybel35 :::"SupBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" )) "iff" (Bool "for" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b4"))) "iff" (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::"sup"::: ) (Set "(" ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "y")) ")" ")" ))) ")" )) ")" ))))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "R" be ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "L"))); let "C" be ($#m1_hidden :::"set"::: ) ; :: original: :::"SupBelow"::: redefine func :::"SupBelow"::: "(" "R" "," "C" ")" -> ($#m1_subset_1 :::"Subset":::) "of" "L"; end; theorem :: WAYBEL35:22 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "st" (Bool (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "c")) ")" ) "," (Set (Var "L"))) ")" )) "holds" (Bool (Set ($#k6_waybel35 :::"SupBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" ) "is" ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")))))) ; theorem :: WAYBEL35:23 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel_4 :::"auxiliary(i)"::: ) ($#v2_waybel_4 :::"auxiliary(ii)"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "holds" (Bool ($#r1_yellow_0 :::"ex_sup_of"::: ) (Set ($#k4_waybel35 :::"SetBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) "," (Set (Var "c")) ")" ) "," (Set (Var "L"))) ")" )) "holds" (Bool (Set ($#k6_waybel35 :::"SupBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" ) "is" ($#v4_waybel35 :::"sup-closed"::: ) )))) ; theorem :: WAYBEL35:24 (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 "R")) "being" ($#v1_waybel35 :::"extra-order"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#v3_waybel35 :::"satisfying_SIC"::: ) ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k6_waybel35 :::"SupBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" ))) "holds" (Bool (Set (Var "d")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow_0 :::""\/""::: ) "(" "{" (Set (Var "b")) where b "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) : (Bool "(" (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set ($#k6_waybel35 :::"SupBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" )) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "b")) "," (Set (Var "d")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) ")" ) "}" "," (Set (Var "L")) ")" )))))) ; theorem :: WAYBEL35:25 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel35 :::"extra-order"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#v3_waybel35 :::"satisfying_SIC"::: ) ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) "holds" (Bool (Set (Var "R")) ($#r1_waybel35 :::"satisfies_SIC_on"::: ) (Set ($#k6_waybel35 :::"SupBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" ))))) ; theorem :: WAYBEL35:26 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_lattice3 :::"complete"::: ) ($#l1_orders_2 :::"Poset":::) (Bool "for" (Set (Var "R")) "being" ($#v1_waybel35 :::"extra-order"::: ) ($#m1_subset_1 :::"Relation":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "C")) "being" ($#v3_waybel35 :::"satisfying_SIC"::: ) ($#m1_waybel35 :::"strict_chain"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "a")) ($#r2_orders_2 :::"<"::: ) (Set (Var "b")))) "holds" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "d")) ($#r2_hidden :::"in"::: ) (Set ($#k6_waybel35 :::"SupBelow"::: ) "(" (Set (Var "R")) "," (Set (Var "C")) ")" )) & (Bool (Set (Var "a")) ($#r2_orders_2 :::"<"::: ) (Set (Var "d"))) & (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "d")) "," (Set (Var "b")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "R"))) ")" )))))) ;