:: YELLOW19 semantic presentation begin theorem :: YELLOW19:1 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "not" (Bool (Set (Var "A")) "is" ($#v1_xboole_0 :::"empty"::: ) ))))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); func :::"NeighborhoodSystem"::: "x" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) "T" ")" ) ")" ) equals :: YELLOW19:def 1 "{" (Set (Var "A")) where A "is" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" "x" : (Bool verum) "}" ; end; :: deftheorem defines :::"NeighborhoodSystem"::: YELLOW19:def 1 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_yellow19 :::"NeighborhoodSystem"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "x")) : (Bool verum) "}" ))); theorem :: YELLOW19:2 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set ($#k1_yellow19 :::"NeighborhoodSystem"::: ) (Set (Var "x")))) "iff" (Bool (Set (Var "A")) "is" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "x"))) ")" )))) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "T")); cluster (Set ($#k1_yellow19 :::"NeighborhoodSystem"::: ) "x") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_subset_1 :::"proper"::: ) ($#v2_waybel_0 :::"filtered"::: ) ($#v13_waybel_0 :::"upper"::: ) ; end; theorem :: YELLOW19:3 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "F")) "being" ($#v13_waybel_0 :::"upper"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) "iff" (Bool (Set ($#k1_yellow19 :::"NeighborhoodSystem"::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set (Var "F"))) ")" )))) ; theorem :: YELLOW19:4 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set ($#k1_yellow19 :::"NeighborhoodSystem"::: ) (Set (Var "x"))) "," (Set (Var "T"))))) ; theorem :: YELLOW19:5 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T")))) "holds" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))))) ")" ))) ; definitionlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); mode :::"Subset"::: "of" "S" "," "N" -> ($#m1_subset_1 :::"Subset":::) "of" "S" means :: YELLOW19:def 2 (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" "N" "st" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" "N" ($#k4_waybel_9 :::"|"::: ) (Set (Var "i")) ")" ))))); end; :: deftheorem defines :::"Subset"::: YELLOW19:def 2 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_yellow19 :::"Subset"::: ) "of" (Set (Var "S")) "," (Set (Var "N"))) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) "st" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" (Set (Var "N")) ($#k4_waybel_9 :::"|"::: ) (Set (Var "i")) ")" ))))) ")" )))); theorem :: YELLOW19:6 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" (Set (Var "N")) ($#k4_waybel_9 :::"|"::: ) (Set (Var "i")) ")" ))) "is" ($#m1_yellow19 :::"Subset"::: ) "of" (Set (Var "S")) "," (Set (Var "N")))))) ; registrationlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); cluster -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_yellow19 :::"Subset"::: ) "of" "S" "," "N"; end; theorem :: YELLOW19:7 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set "the" ($#u1_waybel_0 :::"mapping"::: ) "of" (Set "(" (Set (Var "N")) ($#k5_waybel_9 :::"|"::: ) (Set (Var "i")) ")" )))) "iff" (Bool "ex" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "N")) "st" (Bool "(" (Bool (Set (Var "i")) ($#r1_orders_2 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "N")) ($#k2_waybel_0 :::"."::: ) (Set (Var "j")))) ")" )) ")" ))))) ; theorem :: YELLOW19:8 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "A")) "being" ($#m1_yellow19 :::"Subset"::: ) "of" (Set (Var "S")) "," (Set (Var "N")) "holds" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A")))))) ; theorem :: YELLOW19:9 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "F")) "holds" (Bool (Set (Var "A")) "is" ($#m1_yellow19 :::"Subset"::: ) "of" (Set (Var "S")) "," (Set (Var "N"))) ")" )) "holds" (Bool "ex" (Set (Var "B")) "being" ($#m1_yellow19 :::"Subset"::: ) "of" (Set (Var "S")) "," (Set (Var "N")) "st" (Bool (Set (Var "B")) ($#r1_tarski :::"c="::: ) (Set ($#k1_setfam_1 :::"meet"::: ) (Set (Var "F")))))))) ; definitionlet "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")); func :::"a_filter"::: "N" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) "T" ")" ) ")" ) equals :: YELLOW19:def 3 "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" "T" : (Bool "N" ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))) "}" ; end; :: deftheorem defines :::"a_filter"::: YELLOW19:def 3 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "T")) "holds" (Bool (Set ($#k2_yellow19 :::"a_filter"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "A")) where A "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) : (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))) "}" ))); theorem :: YELLOW19:10 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set ($#k2_yellow19 :::"a_filter"::: ) (Set (Var "N")))) "iff" (Bool "(" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T"))) ")" ) ")" )))) ; 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 ($#k2_yellow19 :::"a_filter"::: ) "N") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v13_waybel_0 :::"upper"::: ) ; end; registrationlet "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")); cluster (Set ($#k2_yellow19 :::"a_filter"::: ) "N") -> ($#v1_subset_1 :::"proper"::: ) ($#v2_waybel_0 :::"filtered"::: ) ; end; theorem :: YELLOW19:11 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r3_waybel_9 :::"is_a_cluster_point_of"::: ) (Set (Var "N"))) "iff" (Bool (Set (Var "x")) ($#r1_waybel_7 :::"is_a_cluster_point_of"::: ) (Set ($#k2_yellow19 :::"a_filter"::: ) (Set (Var "N"))) "," (Set (Var "T"))) ")" )))) ; theorem :: YELLOW19:12 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))) "iff" (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set ($#k2_yellow19 :::"a_filter"::: ) (Set (Var "N"))) "," (Set (Var "T"))) ")" )))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "O" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); let "F" be ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Const "O")) ")" ); func :::"a_net"::: "F" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" "L" means :: YELLOW19:def 4 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set (Var "a")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ) where a "is" ($#m1_subset_1 :::"Element":::) "of" "L", f "is" ($#m2_subset_1 :::"Element"::: ) "of" "F" : (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "f"))) "}" ) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element":::) "of" it "holds" (Bool "(" (Bool (Set (Var "i")) ($#r1_orders_2 :::"<="::: ) (Set (Var "j"))) "iff" (Bool (Set (Set (Var "j")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "i")) ($#k2_xtuple_0 :::"`2"::: ) )) ")" ) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" it "holds" (Bool (Set it ($#k2_waybel_0 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k1_xtuple_0 :::"`1"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"a_net"::: YELLOW19:def 4 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "O")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Var "O")) ")" ) (Bool "for" (Set (Var "b4")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_waybel_0 :::"strict"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k3_yellow19 :::"a_net"::: ) (Set (Var "F")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b4"))) ($#r1_hidden :::"="::: ) "{" (Set ($#k4_tarski :::"["::: ) (Set (Var "a")) "," (Set (Var "f")) ($#k4_tarski :::"]"::: ) ) where a "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")), f "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Var "F")) : (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "f"))) "}" ) & (Bool "(" "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b4")) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r1_orders_2 :::"<="::: ) (Set (Var "j"))) "iff" (Bool (Set (Set (Var "j")) ($#k2_xtuple_0 :::"`2"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "i")) ($#k2_xtuple_0 :::"`2"::: ) )) ")" ) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b4")) "holds" (Bool (Set (Set (Var "b4")) ($#k2_waybel_0 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k1_xtuple_0 :::"`1"::: ) )) ")" ) ")" ) ")" ))))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "O" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); let "F" be ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Const "O")) ")" ); cluster (Set ($#k3_yellow19 :::"a_net"::: ) "F") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_orders_2 :::"reflexive"::: ) ($#v4_orders_2 :::"transitive"::: ) ($#v6_waybel_0 :::"strict"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "O" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); let "F" be ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set (Const "O")) ")" ); cluster (Set ($#k3_yellow19 :::"a_net"::: ) "F") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_waybel_0 :::"strict"::: ) ($#v7_waybel_0 :::"directed"::: ) ; end; theorem :: YELLOW19:13 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) "holds" (Bool (Set (Set (Var "F")) ($#k7_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow19 :::"a_filter"::: ) (Set "(" ($#k3_yellow19 :::"a_net"::: ) (Set (Var "F")) ")" ))))) ; theorem :: YELLOW19:14 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) "holds" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k2_yellow19 :::"a_filter"::: ) (Set "(" ($#k3_yellow19 :::"a_net"::: ) (Set (Var "F")) ")" ))))) ; theorem :: YELLOW19:15 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) "iff" (Bool (Set ($#k3_yellow19 :::"a_net"::: ) (Set (Var "F"))) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))) ")" )))) ; theorem :: YELLOW19:16 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r3_waybel_9 :::"is_a_cluster_point_of"::: ) (Set ($#k3_yellow19 :::"a_net"::: ) (Set (Var "F")))) "iff" (Bool (Set (Var "x")) ($#r1_waybel_7 :::"is_a_cluster_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) ")" )))) ; theorem :: YELLOW19:17 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set "(" ($#k3_yellow19 :::"a_net"::: ) (Set (Var "F")) ")" ))) "iff" (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) ")" )))) ; theorem :: YELLOW19:18 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))))) "holds" (Bool "for" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k1_yellow19 :::"NeighborhoodSystem"::: ) (Set (Var "x"))))) "holds" (Bool (Set ($#k3_yellow19 :::"a_net"::: ) (Set (Var "F"))) ($#r2_waybel_0 :::"is_often_in"::: ) (Set (Var "A"))))))) ; theorem :: YELLOW19:19 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A")))) "holds" (Bool "for" (Set (Var "S")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")) "holds" (Bool (Set (Var "S")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))))))) ; theorem :: YELLOW19:20 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "G"))) & (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T")))) "holds" (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "G")) "," (Set (Var "T"))))) ; theorem :: YELLOW19:21 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")))) "iff" (Bool "ex" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))) & (Bool (Set (Var "x")) ($#r3_waybel_9 :::"is_a_cluster_point_of"::: ) (Set (Var "N"))) ")" )) ")" )))) ; theorem :: YELLOW19:22 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")))) "iff" (Bool "ex" (Set (Var "N")) "being" ($#v3_yellow_6 :::"convergent"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N")))) ")" )) ")" )))) ; theorem :: YELLOW19:23 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A")))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "x")) ($#r3_waybel_9 :::"is_a_cluster_point_of"::: ) (Set (Var "N")))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) ")" ))) ; theorem :: YELLOW19:24 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "N")) "being" ($#v3_yellow_6 :::"convergent"::: ) ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A")))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_yellow_6 :::"Lim"::: ) (Set (Var "N"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) ")" ))) ; theorem :: YELLOW19:25 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")))) "iff" (Bool "ex" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) "st" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) & (Bool (Set (Var "x")) ($#r1_waybel_7 :::"is_a_cluster_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) ")" )) ")" )))) ; theorem :: YELLOW19:26 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")))) "iff" (Bool "ex" (Set (Var "F")) "being" ($#v3_waybel_7 :::"ultra"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) "st" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) & (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))) ")" )) ")" )))) ; theorem :: YELLOW19:27 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "x")) ($#r1_waybel_7 :::"is_a_cluster_point_of"::: ) (Set (Var "F")) "," (Set (Var "T")))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) ")" ))) ; theorem :: YELLOW19:28 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "F")) "being" ($#v3_waybel_7 :::"ultra"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T")))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))))) ")" ))) ; theorem :: YELLOW19:29 (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 "s")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "s")) ($#r3_waybel_9 :::"is_a_cluster_point_of"::: ) (Set (Var "N"))) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_yellow19 :::"Subset"::: ) "of" (Set (Var "T")) "," (Set (Var "N")) "holds" (Bool (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))))) ")" )))) ; theorem :: YELLOW19:30 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v2_tops_2 :::"closed"::: ) )) "holds" (Bool (Set ($#k2_cantor_1 :::"FinMeetCl"::: ) (Set (Var "F"))) "is" ($#v2_tops_2 :::"closed"::: ) ))) ; theorem :: YELLOW19:31 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_compts_1 :::"compact"::: ) ) "iff" (Bool "for" (Set (Var "F")) "being" ($#v3_waybel_7 :::"ultra"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Set (Var "x")) ($#r2_waybel_7 :::"is_a_convergence_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))))) ")" )) ; theorem :: YELLOW19:32 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_compts_1 :::"compact"::: ) ) "iff" (Bool "for" (Set (Var "F")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "T")) ")" ) ")" ) (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Set (Var "x")) ($#r1_waybel_7 :::"is_a_cluster_point_of"::: ) (Set (Var "F")) "," (Set (Var "T"))))) ")" )) ; theorem :: YELLOW19:33 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_compts_1 :::"compact"::: ) ) "iff" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Set (Var "x")) ($#r3_waybel_9 :::"is_a_cluster_point_of"::: ) (Set (Var "N"))))) ")" )) ; theorem :: YELLOW19:34 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_compts_1 :::"compact"::: ) ) "iff" (Bool "for" (Set (Var "N")) "being" ($#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 "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Set (Var "x")) ($#r3_waybel_9 :::"is_a_cluster_point_of"::: ) (Set (Var "N"))))) ")" )) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#v4_orders_2 :::"transitive"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "L")); cluster ($#v2_yellow_6 :::"full"::: ) -> ($#v4_orders_2 :::"transitive"::: ) ($#v2_yellow_6 :::"full"::: ) for ($#m1_yellow_6 :::"SubNetStr"::: ) "of" "N"; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_waybel_0 :::"directed"::: ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "L")); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_waybel_0 :::"strict"::: ) ($#v7_waybel_0 :::"directed"::: ) ($#v2_yellow_6 :::"full"::: ) for ($#m1_yellow_6 :::"SubNetStr"::: ) "of" "N"; end; theorem :: YELLOW19:35 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_compts_1 :::"compact"::: ) ) "iff" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) (Bool "ex" (Set (Var "S")) "being" ($#m2_yellow_6 :::"subnet"::: ) "of" (Set (Var "N")) "st" (Bool (Set (Var "S")) "is" ($#v3_yellow_6 :::"convergent"::: ) ))) ")" )) ; definitionlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "N" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Const "S")); attr "N" is :::"Cauchy"::: means :: YELLOW19:def 5 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "S" "holds" (Bool "(" (Bool "N" ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))) "or" (Bool "N" ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Set (Var "A")) ($#k3_subset_1 :::"`"::: ) )) ")" )); end; :: deftheorem defines :::"Cauchy"::: YELLOW19:def 5 : (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "N")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_waybel_0 :::"NetStr"::: ) "over" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "N")) "is" ($#v1_yellow19 :::"Cauchy"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Var "A"))) "or" (Bool (Set (Var "N")) ($#r1_waybel_0 :::"is_eventually_in"::: ) (Set (Set (Var "A")) ($#k3_subset_1 :::"`"::: ) )) ")" )) ")" ))); registrationlet "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "F" be ($#v3_waybel_7 :::"ultra"::: ) ($#m1_subset_1 :::"Filter":::) "of" (Set "(" ($#k3_yellow_1 :::"BoolePoset"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Const "S")) ")" ) ")" ); cluster (Set ($#k3_yellow19 :::"a_net"::: ) "F") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_waybel_0 :::"strict"::: ) ($#v1_yellow19 :::"Cauchy"::: ) ; end; theorem :: YELLOW19:36 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_compts_1 :::"compact"::: ) ) "iff" (Bool "for" (Set (Var "N")) "being" ($#l1_waybel_0 :::"net":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "N")) "is" ($#v1_yellow19 :::"Cauchy"::: ) )) "holds" (Bool (Set (Var "N")) "is" ($#v3_yellow_6 :::"convergent"::: ) )) ")" )) ;