:: YELLOW_6 semantic presentation begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; func :::"the_universe_of"::: "X" -> ($#m1_hidden :::"set"::: ) equals :: YELLOW_6:def 1 (Set ($#k1_classes1 :::"Tarski-Class"::: ) (Set "(" ($#k5_classes1 :::"the_transitive-closure_of"::: ) "X" ")" )); end; :: deftheorem defines :::"the_universe_of"::: YELLOW_6:def 1 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_yellow_6 :::"the_universe_of"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_classes1 :::"Tarski-Class"::: ) (Set "(" ($#k5_classes1 :::"the_transitive-closure_of"::: ) (Set (Var "X")) ")" )))); registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_yellow_6 :::"the_universe_of"::: ) "X") -> ($#v1_ordinal1 :::"epsilon-transitive"::: ) ($#v2_classes1 :::"Tarski"::: ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k1_yellow_6 :::"the_universe_of"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_classes2 :::"universal"::: ) ; end; theorem :: YELLOW_6:1 (Bool "for" (Set (Var "T")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r2_hidden :::"in"::: ) (Set (Var "T"))) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_tarski :::"c="::: ) (Set (Var "T")))) "holds" (Bool (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) ($#r2_hidden :::"in"::: ) (Set (Var "T"))))) ; begin begin theorem :: YELLOW_6:2 (Bool "for" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "J")) "being" ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "Y")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "Y")) "holds" (Bool (Set (Set "(" ($#k12_pralg_1 :::"Carrier"::: ) (Set (Var "J")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set (Var "J")) ($#k10_pralg_1 :::"."::: ) (Set (Var "i")) ")" )))))) ; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v3_funct_1 :::"constant"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_pralg_1 :::"1-sorted-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; notationlet "J" be ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#m1_hidden :::"Function":::); synonym :::"yielding_non-empty_carriers"::: "J" for :::"non-Empty"::: ; end; definitionlet "J" be ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#m1_hidden :::"Function":::); redefine attr "J" is :::"non-Empty"::: means :: YELLOW_6:def 2 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "J"))) "holds" (Bool (Set (Var "i")) "is" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) )); end; :: deftheorem defines :::"yielding_non-empty_carriers"::: YELLOW_6:def 2 : (Bool "for" (Set (Var "J")) "being" ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "J")) "is" ($#v4_waybel_3 :::"yielding_non-empty_carriers"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "J"))))) "holds" (Bool (Set (Var "i")) "is" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) )) ")" )); registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "L" be ($#l1_struct_0 :::"1-sorted"::: ) ; cluster (Set "X" ($#k2_funcop_1 :::"-->"::: ) "L") -> ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ; end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("I") ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#v4_waybel_3 :::"yielding_non-empty_carriers"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) "J" ")" )) -> ($#v4_funct_1 :::"functional"::: ) ; end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; let "J" be ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#v4_waybel_3 :::"yielding_non-empty_carriers"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set ($#k12_pralg_1 :::"Carrier"::: ) "J") -> bbbadV2_RELAT_1() ; end; begin registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "A" be ($#v12_waybel_0 :::"lower"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); cluster (Set "A" ($#k3_subset_1 :::"`"::: ) ) -> ($#v13_waybel_0 :::"upper"::: ) ; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "A" be ($#v13_waybel_0 :::"upper"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); cluster (Set "A" ($#k3_subset_1 :::"`"::: ) ) -> ($#v12_waybel_0 :::"lower"::: ) ; end; definitionlet "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; redefine attr "N" is :::"directed"::: means :: YELLOW_6:def 3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "N" (Bool "ex" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" "N" "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "z"))) & (Bool (Set (Var "y")) ($#r1_orders_2 :::"<="::: ) (Set (Var "z"))) ")" ))); end; :: deftheorem defines :::"directed"::: YELLOW_6:def 3 : (Bool "for" (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool "(" (Bool (Set (Var "N")) "is" ($#v7_waybel_0 :::"directed"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) (Bool "ex" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "z"))) & (Bool (Set (Var "y")) ($#r1_orders_2 :::"<="::: ) (Set (Var "z"))) ")" ))) ")" )); registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k3_yellow_1 :::"BoolePoset"::: ) "X") -> ($#v7_waybel_0 :::"directed"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_orders_2 :::"strict"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v7_waybel_0 :::"directed"::: ) for ($#l1_orders_2 :::"RelStr"::: ) ; end; definitionlet "M" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "M")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "N"))); let "m" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "M")); :: original: :::"."::: redefine func "f" :::"."::: "m" -> ($#m1_subset_1 :::"Element":::) "of" "N"; end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("I") ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"yielding_non-empty_carriers"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "J" be ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"yielding_non-empty_carriers"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set ($#k5_yellow_1 :::"product"::: ) "J") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; registrationlet "Y1", "Y2" be ($#v7_waybel_0 :::"directed"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; cluster (Set ($#k3_yellow_3 :::"[:"::: ) "Y1" "," "Y2" ($#k3_yellow_3 :::":]"::: ) ) -> ($#v7_waybel_0 :::"directed"::: ) ; end; theorem :: YELLOW_6:3 (Bool "for" (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" (Set (Var "R")) ($#k7_lattice3 :::"~"::: ) ")" )))) ; definitionlet "S" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); attr "N" is :::"constant"::: means :: YELLOW_6:def 4 (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" "N") "is" ($#v3_funct_1 :::"constant"::: ) ); end; :: deftheorem defines :::"constant"::: YELLOW_6:def 4 : (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "N")) "is" ($#v1_yellow_6 :::"constant"::: ) ) "iff" (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) "is" ($#v3_funct_1 :::"constant"::: ) ) ")" ))); definitionlet "R" be ($#l1_orders_2 :::"RelStr"::: ) ; let "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "p" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "T")); func :::"ConstantNet"::: "(" "R" "," "p" ")" -> ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" "T" means :: YELLOW_6:def 5 (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" "R") "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" "R") "#)" )) & (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" it) ($#r2_funct_2 :::"="::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#k8_funcop_1 :::"-->"::: ) "p")) ")" ); end; :: deftheorem defines :::"ConstantNet"::: YELLOW_6:def 5 : (Bool "for" (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "b4")) "being" ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k3_yellow_6 :::"ConstantNet"::: ) "(" (Set (Var "R")) "," (Set (Var "p")) ")" )) "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 "R"))) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" (Set (Var "R"))) "#)" )) & (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "b4"))) ($#r2_funct_2 :::"="::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b4"))) ($#k8_funcop_1 :::"-->"::: ) (Set (Var "p")))) ")" ) ")" ))))); registrationlet "R" be ($#l1_orders_2 :::"RelStr"::: ) ; let "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "p" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "T")); cluster (Set ($#k3_yellow_6 :::"ConstantNet"::: ) "(" "R" "," "p" ")" ) -> ($#v6_waybel_0 :::"strict"::: ) ($#v1_yellow_6 :::"constant"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) ; let "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "p" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "T")); cluster (Set ($#k3_yellow_6 :::"ConstantNet"::: ) "(" "R" "," "p" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_waybel_0 :::"strict"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_waybel_0 :::"directed"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "p" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "T")); cluster (Set ($#k3_yellow_6 :::"ConstantNet"::: ) "(" "R" "," "p" ")" ) -> ($#v6_waybel_0 :::"strict"::: ) ($#v7_waybel_0 :::"directed"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#l1_orders_2 :::"RelStr"::: ) ; let "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "p" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "T")); cluster (Set ($#k3_yellow_6 :::"ConstantNet"::: ) "(" "R" "," "p" ")" ) -> ($#v4_orders_2 :::"transitive"::: ) ($#v6_waybel_0 :::"strict"::: ) ; end; theorem :: YELLOW_6:4 (Bool "for" (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k3_yellow_6 :::"ConstantNet"::: ) "(" (Set (Var "R")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))))))) ; theorem :: YELLOW_6:5 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k3_yellow_6 :::"ConstantNet"::: ) "(" (Set (Var "R")) "," (Set (Var "p")) ")" ")" ) "holds" (Bool (Set (Set "(" ($#k3_yellow_6 :::"ConstantNet"::: ) "(" (Set (Var "R")) "," (Set (Var "p")) ")" ")" ) ($#k2_waybel_0 :::"."::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Var "p"))))))) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "T")); cluster (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" "N") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; begin theorem :: YELLOW_6:6 (Bool "for" (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) "holds" (Bool (Set (Var "R")) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "R")))) ; theorem :: YELLOW_6:7 (Bool "for" (Set (Var "R")) "being" ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "being" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Var "T")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "R")))))) ; definitionlet "S" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); mode :::"SubNetStr"::: "of" "N" -> ($#l1_waybel_0 :::"NetStr"::: ) "over" "S" means :: YELLOW_6:def 6 (Bool "(" (Bool it "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" "N") & (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" "N") ($#k2_partfun1 :::"|"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it))) ")" ); end; :: deftheorem defines :::"SubNetStr"::: YELLOW_6:def 6 : (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "," (Set (Var "b3")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "N"))) "iff" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "N"))) & (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) ($#k2_partfun1 :::"|"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))))) ")" ) ")" ))); theorem :: YELLOW_6:8 (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) "holds" (Bool (Set (Var "N")) "is" ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "N"))))) ; theorem :: YELLOW_6:9 (Bool "for" (Set (Var "Q")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "R")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "Q")) (Bool "for" (Set (Var "S")) "being" ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "R")) (Bool "for" (Set (Var "T")) "being" ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Var "T")) "is" ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "R"))))))) ; definitionlet "S" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); let "M" be ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Const "N")); attr "M" is :::"full"::: means :: YELLOW_6:def 7 (Bool "M" "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" "N"); end; :: deftheorem defines :::"full"::: YELLOW_6:def 7 : (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) (Bool "for" (Set (Var "M")) "being" ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "N")) "holds" (Bool "(" (Bool (Set (Var "M")) "is" ($#v2_yellow_6 :::"full"::: ) ) "iff" (Bool (Set (Var "M")) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "N"))) ")" )))); registrationlet "S" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); cluster ($#v6_waybel_0 :::"strict"::: ) ($#v2_yellow_6 :::"full"::: ) for ($#m1_yellow_6 :::"SubNetStr"::: ) "of" "N"; end; registrationlet "S" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_waybel_0 :::"strict"::: ) ($#v2_yellow_6 :::"full"::: ) for ($#m1_yellow_6 :::"SubNetStr"::: ) "of" "N"; end; theorem :: YELLOW_6:10 (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) (Bool "for" (Set (Var "M")) "being" ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "N")) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "M"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "N"))))))) ; theorem :: YELLOW_6:11 (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) (Bool "for" (Set (Var "M")) "being" ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "N")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "i"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "j"))) & (Bool (Set (Var "i")) ($#r1_orders_2 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "y")))))))) ; theorem :: YELLOW_6:12 (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_yellow_6 :::"full"::: ) ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "N")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "i"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "j"))) & (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "y")))) "holds" (Bool (Set (Var "i")) ($#r1_orders_2 :::"<="::: ) (Set (Var "j")))))))) ; begin registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#v6_waybel_0 :::"strict"::: ) ($#v7_waybel_0 :::"directed"::: ) ($#v1_yellow_6 :::"constant"::: ) for ($#l1_waybel_0 :::"NetStr"::: ) "over" "T"; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#v1_yellow_6 :::"constant"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "T")); cluster (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" "N") -> ($#v3_funct_1 :::"constant"::: ) ; end; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "T")); assume (Bool "(" (Bool (Set (Const "N")) "is" ($#v1_yellow_6 :::"constant"::: ) ) & (Bool (Bool "not" (Set (Const "N")) "is" ($#v2_struct_0 :::"empty"::: ) )) ")" ) ; func :::"the_value_of"::: "N" -> ($#m1_subset_1 :::"Element":::) "of" "T" equals :: YELLOW_6:def 8 (Set ($#k3_funct_1 :::"the_value_of"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" "N")); end; :: deftheorem defines :::"the_value_of"::: YELLOW_6:def 8 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "T")) "st" (Bool (Bool (Set (Var "N")) "is" ($#v1_yellow_6 :::"constant"::: ) ) & (Bool (Bool "not" (Set (Var "N")) "is" ($#v2_struct_0 :::"empty"::: ) ))) "holds" (Bool (Set ($#k4_yellow_6 :::"the_value_of"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set ($#k3_funct_1 :::"the_value_of"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))))))); theorem :: YELLOW_6:13 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_orders_2 :::"RelStr"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k4_yellow_6 :::"the_value_of"::: ) (Set "(" ($#k3_yellow_6 :::"ConstantNet"::: ) "(" (Set (Var "R")) "," (Set (Var "p")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Var "p")))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#l1_waybel_0 :::"net":::) "of" (Set (Const "T")); mode :::"subnet"::: "of" "N" -> ($#l1_waybel_0 :::"net":::) "of" "T" means :: YELLOW_6:def 9 (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" it "," "N" "st" (Bool "(" (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" it) ($#r2_funct_2 :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" "N") ($#k1_partfun1 :::"*"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element":::) "of" "N" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element":::) "of" it "st" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" it "st" (Bool (Bool (Set (Var "n")) ($#r1_orders_2 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set (Var "m")) ($#r1_orders_2 :::"<="::: ) (Set (Set (Var "f")) ($#k2_yellow_6 :::"."::: ) (Set (Var "p")))))) ")" ) ")" )); end; :: deftheorem defines :::"subnet"::: YELLOW_6:def 9 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "," (Set (Var "b3")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "b3")) "," (Set (Var "N")) "st" (Bool "(" (Bool (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "b3"))) ($#r2_funct_2 :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b3")) "st" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b3")) "st" (Bool (Bool (Set (Var "n")) ($#r1_orders_2 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set (Var "m")) ($#r1_orders_2 :::"<="::: ) (Set (Set (Var "f")) ($#k2_yellow_6 :::"."::: ) (Set (Var "p")))))) ")" ) ")" )) ")" ))); theorem :: YELLOW_6:14 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "holds" (Bool (Set (Var "N")) "is" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N"))))) ; theorem :: YELLOW_6:15 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N1")) "," (Set (Var "N2")) "," (Set (Var "N3")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "N1")) "is" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N2"))) & (Bool (Set (Var "N2")) "is" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N3")))) "holds" (Bool (Set (Var "N1")) "is" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N3"))))) ; theorem :: YELLOW_6:16 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#v1_yellow_6 :::"constant"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) "holds" (Bool (Set (Set (Var "N")) ($#k2_waybel_0 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_yellow_6 :::"the_value_of"::: ) (Set (Var "N"))))))) ; theorem :: YELLOW_6:17 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "X"))) & (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "X")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "Y")))))) ; theorem :: YELLOW_6:18 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "M")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "M")) ($#r2_waybel_0 :::"is_often_in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "N")) ($#r2_waybel_0 :::"is_often_in"::: ) (Set (Var "X"))))))) ; theorem :: YELLOW_6:19 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "N")) ($#r2_waybel_0 :::"is_often_in"::: ) (Set (Var "X")))))) ; theorem :: YELLOW_6:20 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) "holds" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))))) ; begin definitionlet "S" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; func "N" :::"""::: "X" -> ($#v6_waybel_0 :::"strict"::: ) ($#m1_yellow_6 :::"SubNetStr"::: ) "of" "N" means :: YELLOW_6:def 10 (Bool "(" (Bool it "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" "N") & (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" "N") ($#k8_relset_1 :::"""::: ) "X")) ")" ); end; :: deftheorem defines :::"""::: YELLOW_6:def 10 : (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b4")) "being" ($#v6_waybel_0 :::"strict"::: ) ($#m1_yellow_6 :::"SubNetStr"::: ) "of" (Set (Var "N")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "N")) ($#k5_yellow_6 :::"""::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set (Var "b4")) "is" ($#v4_yellow_0 :::"full"::: ) ($#m1_yellow_0 :::"SubRelStr"::: ) "of" (Set (Var "N"))) & (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N"))) ($#k8_relset_1 :::"""::: ) (Set (Var "X")))) ")" ) ")" ))))); registrationlet "S" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#v4_orders_2 :::"transitive"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "N" ($#k5_yellow_6 :::"""::: ) "X") -> ($#v4_orders_2 :::"transitive"::: ) ($#v6_waybel_0 :::"strict"::: ) ($#v2_yellow_6 :::"full"::: ) ; end; theorem :: YELLOW_6:21 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "N")) ($#r2_waybel_0 :::"is_often_in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Bool "not" (Set (Set (Var "N")) ($#k5_yellow_6 :::"""::: ) (Set (Var "X"))) "is" ($#v2_struct_0 :::"empty"::: ) )) & (Bool (Set (Set (Var "N")) ($#k5_yellow_6 :::"""::: ) (Set (Var "X"))) "is" ($#v7_waybel_0 :::"directed"::: ) ) ")" )))) ; theorem :: YELLOW_6:22 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "N")) ($#r2_waybel_0 :::"is_often_in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "N")) ($#k5_yellow_6 :::"""::: ) (Set (Var "X"))) "is" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")))))) ; theorem :: YELLOW_6:23 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "M")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")) "st" (Bool (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set (Var "N")) ($#k5_yellow_6 :::"""::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "M")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "X"))))))) ; begin definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; func :::"NetUniv"::: "X" -> ($#m1_hidden :::"set"::: ) means :: YELLOW_6:def 11 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "N")) "being" ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"net":::) "of" "X" "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "N"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_yellow_6 :::"the_universe_of"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X"))) ")" )) ")" )); end; :: deftheorem defines :::"NetUniv"::: YELLOW_6:def 11 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "X")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "ex" (Set (Var "N")) "being" ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "N"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_yellow_6 :::"the_universe_of"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))) ")" )) ")" )) ")" ))); registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster (Set ($#k6_yellow_6 :::"NetUniv"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; mode :::"net_set"::: "of" "X" "," "T" -> ($#m1_hidden :::"ManySortedSet":::) "of" "X" means :: YELLOW_6:def 12 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) it))) "holds" (Bool (Set (Var "i")) "is" ($#l1_waybel_0 :::"net":::) "of" "T")); end; :: deftheorem defines :::"net_set"::: YELLOW_6:def 12 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set (Var "X")) "," (Set (Var "T"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Var "i")) "is" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")))) ")" )))); theorem :: YELLOW_6:24 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set (Var "X")) "," (Set (Var "T"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) "is" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")))) ")" )))) ; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "J" be ($#m3_yellow_6 :::"net_set"::: ) "of" (Set (Const "X")) "," (Set (Const "T")); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); :: original: :::"."::: redefine func "J" :::"."::: "i" -> ($#l1_waybel_0 :::"net":::) "of" "T"; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; cluster -> ($#v1_yellow_1 :::"RelStr-yielding"::: ) for ($#m3_yellow_6 :::"net_set"::: ) "of" "X" "," "T"; end; registrationlet "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "Y" be ($#l1_waybel_0 :::"net":::) "of" (Set (Const "T")); cluster -> ($#v4_waybel_3 :::"yielding_non-empty_carriers"::: ) for ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y") "," "T"; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "Y" be ($#l1_waybel_0 :::"net":::) "of" (Set (Const "T")); let "J" be ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "Y"))) "," (Set (Const "T")); cluster (Set ($#k5_yellow_1 :::"product"::: ) "J") -> ($#v4_orders_2 :::"transitive"::: ) ($#v7_waybel_0 :::"directed"::: ) ; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; cluster -> ($#v4_waybel_3 :::"yielding_non-empty_carriers"::: ) for ($#m3_yellow_6 :::"net_set"::: ) "of" "X" "," "T"; end; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; let "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "X" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("X") ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#v1_yellow_1 :::"RelStr-yielding"::: ) ($#v4_waybel_3 :::"yielding_non-empty_carriers"::: ) for ($#m3_yellow_6 :::"net_set"::: ) "of" "X" "," "T"; end; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "Y" be ($#l1_waybel_0 :::"net":::) "of" (Set (Const "T")); let "J" be ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "Y"))) "," (Set (Const "T")); func :::"Iterated"::: "J" -> ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"net":::) "of" "T" means :: YELLOW_6:def 13 (Bool "(" (Bool (Set ($#g1_orders_2 :::"RelStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) "," (Set "the" ($#u1_orders_2 :::"InternalRel"::: ) "of" it) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#k3_yellow_3 :::"[:"::: ) "Y" "," (Set "(" ($#k5_yellow_1 :::"product"::: ) "J" ")" ) ($#k3_yellow_3 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" "Y" (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "Y")) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) "J" ")" )))) "holds" (Bool (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" "J" ($#k7_yellow_6 :::"."::: ) (Set (Var "i")) ")" )) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )))) ")" ) ")" ); end; :: deftheorem defines :::"Iterated"::: YELLOW_6:def 13 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "J")) "being" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "," (Set (Var "T")) (Bool "for" (Set (Var "b4")) "being" ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k8_yellow_6 :::"Iterated"::: ) (Set (Var "J")))) "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 ($#k3_yellow_3 :::"[:"::: ) (Set (Var "Y")) "," (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Var "J")) ")" ) ($#k3_yellow_3 :::":]"::: ) )) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "Y")) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y")))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Var "J")) ")" )))) "holds" (Bool (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "b4"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "i")) "," (Set (Var "f")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" (Set (Var "J")) ($#k7_yellow_6 :::"."::: ) (Set (Var "i")) ")" )) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" )))) ")" ) ")" ) ")" ))))); theorem :: YELLOW_6:25 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "J")) "being" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "Y")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "Y")) "holds" (Bool (Set (Set (Var "J")) ($#k7_yellow_6 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T")))) ")" )) "holds" (Bool (Set ($#k8_yellow_6 :::"Iterated"::: ) (Set (Var "J"))) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T"))))))) ; theorem :: YELLOW_6:26 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "J")) "being" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "N"))) "," (Set (Var "T")) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k8_yellow_6 :::"Iterated"::: ) (Set (Var "J")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "N"))) "," (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k12_pralg_1 :::"Carrier"::: ) (Set (Var "J")) ")" ) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))))) ; theorem :: YELLOW_6:27 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "J")) "being" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "N"))) "," (Set (Var "T")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k5_yellow_1 :::"product"::: ) (Set (Var "J")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k8_yellow_6 :::"Iterated"::: ) (Set (Var "J")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k7_yellow_3 :::"["::: ) (Set (Var "i")) "," (Set (Var "f")) ($#k7_yellow_3 :::"]"::: ) ))) "holds" (Bool (Set (Set "(" ($#k8_yellow_6 :::"Iterated"::: ) (Set (Var "J")) ")" ) ($#k2_waybel_0 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" (Set (Var "J")) ($#k7_yellow_6 :::"."::: ) (Set (Var "i")) ")" )) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "f")) ($#k4_waybel_3 :::"."::: ) (Set (Var "i")) ")" ))))))))) ; theorem :: YELLOW_6:28 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "Y")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "J")) "being" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "Y"))) "," (Set (Var "T")) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" ($#k8_yellow_6 :::"Iterated"::: ) (Set (Var "J")) ")" ))) ($#r1_tarski :::"c="::: ) (Set ($#k3_tarski :::"union"::: ) "{" (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" (Set (Var "J")) ($#k7_yellow_6 :::"."::: ) (Set (Var "i")) ")" )) ")" ) where i "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "Y")) : (Bool verum) "}" ))))) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "p" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); func :::"OpenNeighborhoods"::: "p" -> ($#l1_orders_2 :::"RelStr"::: ) equals :: YELLOW_6:def 14 (Set (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) "{" (Set (Var "V")) where V "is" ($#m1_subset_1 :::"Subset":::) "of" "T" : (Bool "(" (Bool "p" ($#r2_hidden :::"in"::: ) (Set (Var "V"))) & (Bool (Set (Var "V")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ) "}" ")" ) ($#k7_lattice3 :::"~"::: ) ); end; :: deftheorem defines :::"OpenNeighborhoods"::: YELLOW_6:def 14 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k9_yellow_6 :::"OpenNeighborhoods"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_yellow_1 :::"InclPoset"::: ) "{" (Set (Var "V")) where V "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "V"))) & (Bool (Set (Var "V")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ) "}" ")" ) ($#k7_lattice3 :::"~"::: ) )))); registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "p" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); cluster (Set ($#k9_yellow_6 :::"OpenNeighborhoods"::: ) "p") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; theorem :: YELLOW_6:29 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k9_yellow_6 :::"OpenNeighborhoods"::: ) (Set (Var "p")) ")" ) (Bool "ex" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "W")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ))))) ; theorem :: YELLOW_6:30 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k9_yellow_6 :::"OpenNeighborhoods"::: ) (Set (Var "p")) ")" ))) "iff" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "x"))) & (Bool (Set (Var "x")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ) ")" )))) ; theorem :: YELLOW_6:31 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k9_yellow_6 :::"OpenNeighborhoods"::: ) (Set (Var "p")) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_orders_2 :::"<="::: ) (Set (Var "y"))) "iff" (Bool (Set (Var "y")) ($#r1_tarski :::"c="::: ) (Set (Var "x"))) ")" )))) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "p" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); cluster (Set ($#k9_yellow_6 :::"OpenNeighborhoods"::: ) "p") -> ($#v4_orders_2 :::"transitive"::: ) ($#v7_waybel_0 :::"directed"::: ) ; end; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "N" be ($#l1_waybel_0 :::"net":::) "of" (Set (Const "T")); func :::"Lim"::: "N" -> ($#m1_subset_1 :::"Subset":::) "of" "T" means :: YELLOW_6:def 15 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "for" (Set (Var "V")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "p")) "holds" (Bool "N" ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "V")))) ")" )); end; :: deftheorem defines :::"Lim"::: YELLOW_6:def 15 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "for" (Set (Var "V")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "p")) "holds" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "V")))) ")" )) ")" )))); theorem :: YELLOW_6:32 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "Y")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")) "holds" (Bool (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N"))) ($#r1_tarski :::"c="::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "Y"))))))) ; theorem :: YELLOW_6:33 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#v1_yellow_6 :::"constant"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k4_yellow_6 :::"the_value_of"::: ) (Set (Var "N"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))))) ; theorem :: YELLOW_6:34 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N"))))) "holds" (Bool "for" (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) (Bool "ex" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "N")) ($#k2_waybel_0 :::"."::: ) (Set (Var "c")) ")" ) where c "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) : (Bool (Set (Var "d")) ($#r1_orders_2 :::"<="::: ) (Set (Var "c"))) "}" ) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "S")))) ")" )))))) ; theorem :: YELLOW_6:35 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v8_pre_topc :::"Hausdorff"::: ) ) "iff" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N"))))) "holds" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q"))))) ")" )) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v8_pre_topc :::"Hausdorff"::: ) ($#l1_pre_topc :::"TopSpace":::); let "N" be ($#l1_waybel_0 :::"net":::) "of" (Set (Const "T")); cluster (Set ($#k10_yellow_6 :::"Lim"::: ) "N") -> ($#v1_zfmisc_1 :::"trivial"::: ) ; end; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "N" be ($#l1_waybel_0 :::"net":::) "of" (Set (Const "T")); attr "N" is :::"convergent"::: means :: YELLOW_6:def 16 (Bool (Set ($#k10_yellow_6 :::"Lim"::: ) "N") ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )); end; :: deftheorem defines :::"convergent"::: YELLOW_6:def 16 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "N")) "is" ($#v3_yellow_6 :::"convergent"::: ) ) "iff" (Bool (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) ")" ))); registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#v7_waybel_0 :::"directed"::: ) ($#v1_yellow_6 :::"constant"::: ) -> ($#v3_yellow_6 :::"convergent"::: ) for ($#l1_waybel_0 :::"NetStr"::: ) "over" "T"; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_orders_2 :::"transitive"::: ) ($#v6_waybel_0 :::"strict"::: ) ($#v7_waybel_0 :::"directed"::: ) ($#v3_yellow_6 :::"convergent"::: ) for ($#l1_waybel_0 :::"NetStr"::: ) "over" "T"; end; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v8_pre_topc :::"Hausdorff"::: ) ($#l1_pre_topc :::"TopSpace":::); let "N" be ($#v3_yellow_6 :::"convergent"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Const "T")); func :::"lim"::: "N" -> ($#m1_subset_1 :::"Element":::) "of" "T" means :: YELLOW_6:def 17 (Bool it ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) "N")); end; :: deftheorem defines :::"lim"::: YELLOW_6:def 17 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v8_pre_topc :::"Hausdorff"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#v3_yellow_6 :::"convergent"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k11_yellow_6 :::"lim"::: ) (Set (Var "N")))) "iff" (Bool (Set (Var "b3")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))) ")" )))); theorem :: YELLOW_6:36 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v8_pre_topc :::"Hausdorff"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#v1_yellow_6 :::"constant"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k11_yellow_6 :::"lim"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set ($#k4_yellow_6 :::"the_value_of"::: ) (Set (Var "N")))))) ; theorem :: YELLOW_6:37 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))))) "holds" (Bool "ex" (Set (Var "Y")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")) "st" (Bool "for" (Set (Var "Z")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "Y")) "holds" (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "Z")))))))))) ; theorem :: YELLOW_6:38 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T"))))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))))) "holds" (Bool "ex" (Set (Var "Y")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")) "st" (Bool "(" (Bool (Set (Var "Y")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T")))) & (Bool "(" "for" (Set (Var "Z")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "Y")) "holds" (Bool (Bool "not" (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "Z"))))) ")" ) ")" ))))) ; theorem :: YELLOW_6:39 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N"))))) "holds" (Bool "for" (Set (Var "J")) "being" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "N"))) "," (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) "holds" (Bool (Set (Set (Var "N")) ($#k2_waybel_0 :::"."::: ) (Set (Var "i"))) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set "(" (Set (Var "J")) ($#k7_yellow_6 :::"."::: ) (Set (Var "i")) ")" ))) ")" )) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set "(" ($#k8_yellow_6 :::"Iterated"::: ) (Set (Var "J")) ")" ))))))) ; begin definitionlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; mode :::"Convergence-Class"::: "of" "S" -> ($#m1_hidden :::"set"::: ) means :: YELLOW_6:def 18 (Bool it ($#r1_tarski :::"c="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k6_yellow_6 :::"NetUniv"::: ) "S" ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; :: deftheorem defines :::"Convergence-Class"::: YELLOW_6:def 18 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "S"))) "iff" (Bool (Set (Var "b2")) ($#r1_tarski :::"c="::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "S")) ")" ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#k2_zfmisc_1 :::":]"::: ) )) ")" ))); registrationlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster -> ($#v1_relat_1 :::"Relation-like"::: ) for ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" "S"; end; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); func :::"Convergence"::: "T" -> ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" "T" means :: YELLOW_6:def 19 (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" "T" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "(" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) "T")) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))) ")" ) ")" ))); end; :: deftheorem defines :::"Convergence"::: YELLOW_6:def 19 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "b2")) "being" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k12_yellow_6 :::"Convergence"::: ) (Set (Var "T")))) "iff" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "(" (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T")))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))) ")" ) ")" ))) ")" ))); definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "C" be ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Const "T")); attr "C" is :::"(CONSTANTS)"::: means :: YELLOW_6:def 20 (Bool "for" (Set (Var "N")) "being" ($#v1_yellow_6 :::"constant"::: ) ($#l1_waybel_0 :::"net":::) "of" "T" "st" (Bool (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) "T"))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set "(" ($#k4_yellow_6 :::"the_value_of"::: ) (Set (Var "N")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "C")); attr "C" is :::"(SUBNETS)"::: means :: YELLOW_6:def 21 (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" "T" (Bool "for" (Set (Var "Y")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")) "st" (Bool (Bool (Set (Var "Y")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) "T"))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" "T" "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "C")) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "Y")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "C")))); attr "C" is :::"(DIVERGENCE)"::: means :: YELLOW_6:def 22 (Bool "for" (Set (Var "X")) "being" ($#l1_waybel_0 :::"net":::) "of" "T" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" "T" "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) "T")) & (Bool (Bool "not" (Set ($#k4_tarski :::"["::: ) (Set (Var "X")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "C"))) "holds" (Bool "ex" (Set (Var "Y")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "Y")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) "T")) & (Bool "(" "for" (Set (Var "Z")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "Y")) "holds" (Bool (Bool "not" (Set ($#k4_tarski :::"["::: ) (Set (Var "Z")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "C")) ")" ) ")" )))); attr "C" is :::"(ITERATED_LIMITS)"::: means :: YELLOW_6:def 23 (Bool "for" (Set (Var "X")) "being" ($#l1_waybel_0 :::"net":::) "of" "T" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" "T" "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "X")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "C")) "holds" (Bool "for" (Set (Var "J")) "being" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," "T" "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "J")) ($#k7_yellow_6 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "X")) ($#k2_waybel_0 :::"."::: ) (Set (Var "i")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "C") ")" )) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k8_yellow_6 :::"Iterated"::: ) (Set (Var "J")) ")" ) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "C")))); end; :: deftheorem defines :::"(CONSTANTS)"::: YELLOW_6:def 20 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v4_yellow_6 :::"(CONSTANTS)"::: ) ) "iff" (Bool "for" (Set (Var "N")) "being" ($#v1_yellow_6 :::"constant"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "N")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T"))))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set "(" ($#k4_yellow_6 :::"the_value_of"::: ) (Set (Var "N")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) ")" ))); :: deftheorem defines :::"(SUBNETS)"::: YELLOW_6:def 21 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v5_yellow_6 :::"(SUBNETS)"::: ) ) "iff" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "Y")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")) "st" (Bool (Bool (Set (Var "Y")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T"))))) "holds" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "Y")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C")))))) ")" ))); :: deftheorem defines :::"(DIVERGENCE)"::: YELLOW_6:def 22 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v6_yellow_6 :::"(DIVERGENCE)"::: ) ) "iff" (Bool "for" (Set (Var "X")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "X")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T")))) & (Bool (Bool "not" (Set ($#k4_tarski :::"["::: ) (Set (Var "X")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C"))))) "holds" (Bool "ex" (Set (Var "Y")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "Y")) ($#r2_hidden :::"in"::: ) (Set ($#k6_yellow_6 :::"NetUniv"::: ) (Set (Var "T")))) & (Bool "(" "for" (Set (Var "Z")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "Y")) "holds" (Bool (Bool "not" (Set ($#k4_tarski :::"["::: ) (Set (Var "Z")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) ")" ) ")" )))) ")" ))); :: deftheorem defines :::"(ITERATED_LIMITS)"::: YELLOW_6:def 23 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v7_yellow_6 :::"(ITERATED_LIMITS)"::: ) ) "iff" (Bool "for" (Set (Var "X")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "X")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool "for" (Set (Var "J")) "being" ($#m3_yellow_6 :::"net_set"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "," (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" (Set (Var "J")) ($#k7_yellow_6 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "X")) ($#k2_waybel_0 :::"."::: ) (Set (Var "i")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )) "holds" (Bool (Set ($#k4_tarski :::"["::: ) (Set "(" ($#k8_yellow_6 :::"Iterated"::: ) (Set (Var "J")) ")" ) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C")))))) ")" ))); registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k12_yellow_6 :::"Convergence"::: ) "T") -> ($#v4_yellow_6 :::"(CONSTANTS)"::: ) ($#v5_yellow_6 :::"(SUBNETS)"::: ) ($#v6_yellow_6 :::"(DIVERGENCE)"::: ) ($#v7_yellow_6 :::"(ITERATED_LIMITS)"::: ) ; end; definitionlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "C" be ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Const "S")); func :::"ConvergenceSpace"::: "C" -> ($#v1_pre_topc :::"strict"::: ) ($#l1_pre_topc :::"TopStruct"::: ) means :: YELLOW_6:def 24 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")) & (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" it) ($#r1_hidden :::"="::: ) "{" (Set (Var "V")) where V "is" ($#m1_subset_1 :::"Subset":::) "of" "S" : (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" "S" "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" "S" "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) "C")) "holds" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "V"))))) "}" ) ")" ); end; :: deftheorem defines :::"ConvergenceSpace"::: YELLOW_6:def 24 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#v1_pre_topc :::"strict"::: ) ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k13_yellow_6 :::"ConvergenceSpace"::: ) (Set (Var "C")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))) & (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "V")) where V "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) : (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "V")))) "holds" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "V"))))) "}" ) ")" ) ")" )))); registrationlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "C" be ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Const "S")); cluster (Set ($#k13_yellow_6 :::"ConvergenceSpace"::: ) "C") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ; end; registrationlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "C" be ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Const "S")); cluster (Set ($#k13_yellow_6 :::"ConvergenceSpace"::: ) "C") -> ($#v1_pre_topc :::"strict"::: ) ($#v2_pre_topc :::"TopSpace-like"::: ) ; end; theorem :: YELLOW_6:40 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "S")) "holds" (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set ($#k12_yellow_6 :::"Convergence"::: ) (Set "(" ($#k13_yellow_6 :::"ConvergenceSpace"::: ) (Set (Var "C")) ")" ))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "C" be ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Const "T")); attr "C" is :::"topological"::: means :: YELLOW_6:def 25 (Bool "(" (Bool "C" "is" ($#v4_yellow_6 :::"(CONSTANTS)"::: ) ) & (Bool "C" "is" ($#v5_yellow_6 :::"(SUBNETS)"::: ) ) & (Bool "C" "is" ($#v6_yellow_6 :::"(DIVERGENCE)"::: ) ) & (Bool "C" "is" ($#v7_yellow_6 :::"(ITERATED_LIMITS)"::: ) ) ")" ); end; :: deftheorem defines :::"topological"::: YELLOW_6:def 25 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v8_yellow_6 :::"topological"::: ) ) "iff" (Bool "(" (Bool (Set (Var "C")) "is" ($#v4_yellow_6 :::"(CONSTANTS)"::: ) ) & (Bool (Set (Var "C")) "is" ($#v5_yellow_6 :::"(SUBNETS)"::: ) ) & (Bool (Set (Var "C")) "is" ($#v6_yellow_6 :::"(DIVERGENCE)"::: ) ) & (Bool (Set (Var "C")) "is" ($#v7_yellow_6 :::"(ITERATED_LIMITS)"::: ) ) ")" ) ")" ))); registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v8_yellow_6 :::"topological"::: ) for ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" "T"; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#v8_yellow_6 :::"topological"::: ) -> ($#v4_yellow_6 :::"(CONSTANTS)"::: ) ($#v5_yellow_6 :::"(SUBNETS)"::: ) ($#v6_yellow_6 :::"(DIVERGENCE)"::: ) ($#v7_yellow_6 :::"(ITERATED_LIMITS)"::: ) for ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" "T"; cluster ($#v4_yellow_6 :::"(CONSTANTS)"::: ) ($#v5_yellow_6 :::"(SUBNETS)"::: ) ($#v6_yellow_6 :::"(DIVERGENCE)"::: ) ($#v7_yellow_6 :::"(ITERATED_LIMITS)"::: ) -> ($#v8_yellow_6 :::"topological"::: ) for ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" "T"; end; theorem :: YELLOW_6:41 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#v8_yellow_6 :::"topological"::: ) ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k13_yellow_6 :::"ConvergenceSpace"::: ) (Set (Var "C")) ")" ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v3_pre_topc :::"open"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "S"))))) ")" )))) ; theorem :: YELLOW_6:42 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#v8_yellow_6 :::"topological"::: ) ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k13_yellow_6 :::"ConvergenceSpace"::: ) (Set (Var "C")) ")" ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "N")) ($#r2_waybel_0 :::"is_often_in"::: ) (Set (Var "S")))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))))) ")" )))) ; theorem :: YELLOW_6:43 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#v8_yellow_6 :::"topological"::: ) ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k13_yellow_6 :::"ConvergenceSpace"::: ) (Set (Var "C")) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k13_yellow_6 :::"ConvergenceSpace"::: ) (Set (Var "C")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "S"))))) "holds" (Bool "ex" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set ($#k13_yellow_6 :::"ConvergenceSpace"::: ) (Set (Var "C"))) "st" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "N")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set (Var "N")))) ($#r1_tarski :::"c="::: ) (Set (Var "S"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))) ")" )))))) ; theorem :: YELLOW_6:44 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "C")) "being" ($#m4_yellow_6 :::"Convergence-Class"::: ) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set ($#k12_yellow_6 :::"Convergence"::: ) (Set "(" ($#k13_yellow_6 :::"ConvergenceSpace"::: ) (Set (Var "C")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "C"))) "iff" (Bool (Set (Var "C")) "is" ($#v8_yellow_6 :::"topological"::: ) ) ")" ))) ;