:: KURATO_2 semantic presentation begin definitionlet "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; mode SetSequence of "T" is ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T"); end; begin registrationlet "f" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )); cluster (Set ($#k3_topreal1 :::"L~"::: ) "f") -> ($#v4_pre_topc :::"closed"::: ) ; end; theorem :: KURATO_2:1 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "x")) "," (Set (Var "r")) ")" ) "is" ($#v3_pre_topc :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ))))) ; theorem :: KURATO_2:2 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p9")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "p9"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" ))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p9")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))))))) ; theorem :: KURATO_2:3 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "p9")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "p9"))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p9")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p")) "," (Set (Var "r")) ")" )))))) ; theorem :: KURATO_2:4 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "X"))))) "holds" (Bool "ex" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "of" (Set (Var "n")) "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "seq")) "is" ($#v3_toprns_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_toprns_1 :::"lim"::: ) (Set (Var "seq"))) ($#r1_hidden :::"="::: ) (Set (Var "r"))) ")" ))))) ; registrationlet "M" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::); cluster (Set ($#k3_pcomps_1 :::"TopSpaceMetr"::: ) "M") -> ($#v1_frechet :::"first-countable"::: ) ; end; theorem :: KURATO_2:5 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_frechet :::"first-countable"::: ) ) "iff" (Bool (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) "#)" ) "is" ($#v1_frechet :::"first-countable"::: ) ) ")" )) ; registrationlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k15_euclid :::"TOP-REAL"::: ) "n") -> ($#v1_frechet :::"first-countable"::: ) ; end; theorem :: KURATO_2:6 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p9")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "p9")))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")))) "iff" (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p9")) "," (Set (Var "r")) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "A")))) ")" ))))) ; theorem :: KURATO_2:7 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x9")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "x9")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool "not" (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "x9")) "," (Set (Var "r")) ")" ))))))) ; theorem :: KURATO_2:8 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "S")) "is" ($#v9_rltopsp1 :::"bounded"::: ) )) "iff" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool (Set ($#k4_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_xxreal_0 :::">"::: ) (Set (Var "r"))) ")" ))) ")" ))) ; theorem :: KURATO_2:9 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "x")) "," (Set (Var "a")) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "y")) "," (Set (Var "b")) ")" ))) "holds" (Bool (Set ($#k4_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b"))))))) ; theorem :: KURATO_2:10 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "x")) "," (Set (Var "a")) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "z")) "," (Set (Var "c")) ")" )) & (Bool (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "z")) "," (Set (Var "c")) ")" ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "y")) "," (Set (Var "b")) ")" ))) "holds" (Bool (Set ($#k4_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set "(" (Set (Var "a")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "b")) ")" ) ($#k3_real_1 :::"+"::: ) (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "c")) ")" )))))) ; theorem :: KURATO_2:11 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "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 "X")) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "Y")) (Bool "for" (Set (Var "V")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set (Var "X")) "," (Set (Var "Y")) ($#k2_borsuk_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "V")) "is" ($#m2_connsp_2 :::"a_neighborhood"::: ) "of" (Set ($#k3_borsuk_1 :::"[:"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) "," (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "y")) ($#k6_domain_1 :::"}"::: ) ) ($#k3_borsuk_1 :::":]"::: ) )) "iff" (Bool (Set (Var "V")) "is" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set ($#k4_borsuk_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k4_borsuk_1 :::"]"::: ) )) ")" ))))) ; begin theorem :: KURATO_2:12 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "G")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "F"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set (Var "G")) ($#r2_funct_2 :::"="::: ) (Set (Var "F")))))) ; theorem :: KURATO_2:13 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "R")) "being" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "S")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "R")) ($#k3_funct_2 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k3_funct_2 :::"."::: ) (Set (Var "m")))) ")" )))))) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "S" be ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "T"))); func :::"Lim_inf"::: "S" -> ($#m1_subset_1 :::"Subset":::) "of" "T" means :: KURATO_2:def 1 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" "T" "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "for" (Set (Var "G")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "p")) (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool (Set "S" ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "G")))))) ")" )); end; :: deftheorem defines :::"Lim_inf"::: KURATO_2:def 1 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "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 ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "for" (Set (Var "G")) "being" ($#m1_connsp_2 :::"a_neighborhood"::: ) "of" (Set (Var "p")) (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set (Var "S")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "G")))))) ")" )) ")" )))); theorem :: KURATO_2:14 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" )) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "p9")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "p9")))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "m")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "k")))) "holds" (Bool (Set (Set (Var "S")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "p9")) "," (Set (Var "r")) ")" ))))) ")" ))))) ; theorem :: KURATO_2:15 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "S")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "S")))))) ; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "S" be ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "T"))); cluster (Set ($#k1_kurato_2 :::"Lim_inf"::: ) "S") -> ($#v4_pre_topc :::"closed"::: ) ; end; theorem :: KURATO_2:16 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "R")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "S")))) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "R")))))) ; theorem :: KURATO_2:17 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")))) ")" )) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "B")))))) ; theorem :: KURATO_2:18 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "C")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ))) ")" )) "holds" (Bool (Set (Set "(" ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "A")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "B")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "C")))))) ; theorem :: KURATO_2:19 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "C")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ))) ")" )) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "C"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "A")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "B")) ")" ))))) ; theorem :: KURATO_2:20 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "G")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ))) ")" )) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "F")))))) ; theorem :: KURATO_2:21 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" )) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "of" (Set (Var "n")) "st" (Bool "(" (Bool (Set (Var "s")) "is" ($#v3_toprns_1 :::"convergent"::: ) ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "S")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")))) ")" ) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k6_toprns_1 :::"lim"::: ) (Set (Var "s")))) ")" ))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "S"))))))) ; theorem :: KURATO_2:22 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Var "P"))) ")" )) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "s"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "P"))))))) ; theorem :: KURATO_2:23 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))))))) ; theorem :: KURATO_2:24 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "A")) "being" ($#v4_pre_topc :::"closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Var "A")))))) ; theorem :: KURATO_2:25 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" )) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "P")) "is" ($#v9_rltopsp1 :::"bounded"::: ) ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "S")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Var "P"))) ")" )) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "S"))) "is" ($#v9_rltopsp1 :::"bounded"::: ) )))) ; theorem :: KURATO_2:26 (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "P")) "is" ($#v9_rltopsp1 :::"bounded"::: ) ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "S")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Var "P"))) ")" )) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "S"))) "is" ($#v2_compts_1 :::"compact"::: ) ))) ; theorem :: KURATO_2:27 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" )) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "n")) ")" ) ($#k2_borsuk_1 :::":]"::: ) )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "C")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set "(" (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) ($#k3_borsuk_1 :::":]"::: ) )) ")" )) "holds" (Bool (Set ($#k3_borsuk_1 :::"[:"::: ) (Set "(" ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "A")) ")" ) "," (Set "(" ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "B")) ")" ) ($#k3_borsuk_1 :::":]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "C"))))))) ; theorem :: KURATO_2:28 (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k3_kurato_0 :::"lim_inf"::: ) (Set (Var "S"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "S"))))) ; theorem :: KURATO_2:29 (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Simple_closed_curve":::) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k2_tops_1 :::"Fr"::: ) (Set "(" (Set "(" ($#k2_jordan2c :::"UBD"::: ) (Set "(" ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ) ")" ) ")" ) ($#k3_subset_1 :::"`"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_topreal1 :::"L~"::: ) (Set "(" ($#k1_jordan9 :::"Cage"::: ) "(" (Set (Var "C")) "," (Set (Var "i")) ")" ")" ))))) ; begin definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "S" be ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "T"))); func :::"Lim_sup"::: "S" -> ($#m1_subset_1 :::"Subset":::) "of" "T" means :: KURATO_2:def 2 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "A")) "being" ($#m2_valued_0 :::"subsequence"::: ) "of" "S" "st" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "A"))))) ")" )); end; :: deftheorem defines :::"Lim_sup"::: KURATO_2:def 2 : (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "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 ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "S")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "A")) "being" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "S")) "st" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "A"))))) ")" )) ")" )))); theorem :: KURATO_2:30 (Bool "for" (Set (Var "N")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "N")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Var "N")) ")" ) (Bool "for" (Set (Var "x9")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k14_euclid :::"Euclid"::: ) (Set (Var "N")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "x9")))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_frechet2 :::"is_a_cluster_point_of"::: ) (Set (Var "F"))) "iff" (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "m"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_metric_1 :::"Ball"::: ) "(" (Set (Var "x9")) "," (Set (Var "r")) ")" )) ")" )))) ")" ))))) ; theorem :: KURATO_2:31 (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 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "A")))))) ; theorem :: KURATO_2:32 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")))) ")" ) & (Bool (Set (Var "C")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "A")))) "holds" (Bool "ex" (Set (Var "D")) "being" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "B")) "st" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "C")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "D")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))))))) ; theorem :: KURATO_2:33 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")))) ")" ) & (Bool (Set (Var "C")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "B")))) "holds" (Bool "ex" (Set (Var "D")) "being" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "A")) "st" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "D")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "C")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))))))) ; theorem :: KURATO_2:34 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")))) ")" )) "holds" (Bool (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "A"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "B"))))) ; theorem :: KURATO_2:35 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "C")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ))) ")" )) "holds" (Bool (Set (Set "(" ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "A")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "B")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "C"))))) ; theorem :: KURATO_2:36 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "C")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ))) ")" )) "holds" (Bool (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "C"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "A")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "B")) ")" )))) ; theorem :: KURATO_2:37 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "C")) "," (Set (Var "C1")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k2_borsuk_1 :::":]"::: ) )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "C")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set "(" (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) ($#k3_borsuk_1 :::":]"::: ) )) ")" ) & (Bool (Set (Var "C1")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "C")))) "holds" (Bool "ex" (Set (Var "A1")) "," (Set (Var "B1")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) "st" (Bool "(" (Bool (Set (Var "A1")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "A"))) & (Bool (Set (Var "B1")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "B"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "C1")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set "(" (Set (Var "A1")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "B1")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) ($#k3_borsuk_1 :::":]"::: ) )) ")" ) ")" )))) ; theorem :: KURATO_2:38 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_borsuk_1 :::"[:"::: ) (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "," (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) ($#k2_borsuk_1 :::":]"::: ) )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "C")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set "(" (Set (Var "A")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "B")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ) ($#k3_borsuk_1 :::":]"::: ) )) ")" )) "holds" (Bool (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "C"))) ($#r1_tarski :::"c="::: ) (Set ($#k3_borsuk_1 :::"[:"::: ) (Set "(" ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "A")) ")" ) "," (Set "(" ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "B")) ")" ) ($#k3_borsuk_1 :::":]"::: ) )))) ; theorem :: KURATO_2:39 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set ($#k1_kurato_2 :::"Lim_inf"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "F"))))))) ; theorem :: KURATO_2:40 (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "F")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "A"))) ")" )) "holds" (Bool (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")))))) ; theorem :: KURATO_2:41 (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"SetSequence":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" )) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "G")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "i")) ")" ))) ")" )) "holds" (Bool (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "G"))) ($#r1_hidden :::"="::: ) (Set ($#k2_kurato_2 :::"Lim_sup"::: ) (Set (Var "F"))))) ;