:: NFCONT_1 semantic presentation begin theorem :: NFCONT_1:1 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "seq1")) "," (Set (Var "seq2")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "seq1")) ($#k3_normsp_1 :::"-"::: ) (Set (Var "seq2"))) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "seq1")) ($#k2_normsp_1 :::"+"::: ) (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "seq2")) ")" ))))) ; theorem :: NFCONT_1:2 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "holds" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "seq"))) ($#r2_funct_2 :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k5_normsp_1 :::"*"::: ) (Set (Var "seq")))))) ; definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "x0" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "S")); mode :::"Neighbourhood"::: "of" "x0" -> ($#m1_subset_1 :::"Subset":::) "of" "S" means :: NFCONT_1:def 1 (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Point":::) "of" "S" : (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "y")) ($#k5_algstr_0 :::"-"::: ) "x0" ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) "}" ($#r1_tarski :::"c="::: ) it) ")" )); end; :: deftheorem defines :::"Neighbourhood"::: NFCONT_1:def 1 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (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_nfcont_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0"))) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) & (Bool "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) : (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "y")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) "}" ($#r1_tarski :::"c="::: ) (Set (Var "b3"))) ")" )) ")" )))); theorem :: NFCONT_1:3 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g")))) "holds" (Bool "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) : (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "y")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) "}" "is" ($#m1_nfcont_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")))))) ; theorem :: NFCONT_1:4 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "N")) "being" ($#m1_nfcont_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "holds" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))))) ; definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); attr "X" is :::"compact"::: means :: NFCONT_1:def 2 (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"sequence":::) "of" "S" "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) "X")) "holds" (Bool "ex" (Set (Var "s2")) "being" ($#m1_subset_1 :::"sequence":::) "of" "S" "st" (Bool "(" (Bool (Set (Var "s2")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "s1"))) & (Bool (Set (Var "s2")) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s2"))) ($#r2_hidden :::"in"::: ) "X") ")" ))); end; :: deftheorem defines :::"compact"::: NFCONT_1:def 2 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_nfcont_1 :::"compact"::: ) ) "iff" (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool "ex" (Set (Var "s2")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "s2")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "s1"))) & (Bool (Set (Var "s2")) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s2"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" ))) ")" ))); definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); attr "X" is :::"closed"::: means :: NFCONT_1:def 3 (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"sequence":::) "of" "S" "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) "X") & (Bool (Set (Var "s1")) "is" ($#v3_normsp_1 :::"convergent"::: ) )) "holds" (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s1"))) ($#r2_hidden :::"in"::: ) "X")); end; :: deftheorem defines :::"closed"::: NFCONT_1:def 3 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v2_nfcont_1 :::"closed"::: ) ) "iff" (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "s1")) "is" ($#v3_normsp_1 :::"convergent"::: ) )) "holds" (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s1"))) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" ))); definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); attr "X" is :::"open"::: means :: NFCONT_1:def 4 (Bool (Set "X" ($#k3_subset_1 :::"`"::: ) ) "is" ($#v2_nfcont_1 :::"closed"::: ) ); end; :: deftheorem defines :::"open"::: NFCONT_1:def 4 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v3_nfcont_1 :::"open"::: ) ) "iff" (Bool (Set (Set (Var "X")) ($#k3_subset_1 :::"`"::: ) ) "is" ($#v2_nfcont_1 :::"closed"::: ) ) ")" ))); definitionlet "S", "T" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "S")) "," (Set (Const "T")); let "x0" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "S")); pred "f" :::"is_continuous_in"::: "x0" means :: NFCONT_1:def 5 (Bool "(" (Bool "x0" ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"sequence":::) "of" "S" "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool (Set (Var "s1")) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) "x0")) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set "f" ($#k7_partfun1 :::"/."::: ) "x0") ($#r1_hidden :::"="::: ) (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"is_continuous_in"::: NFCONT_1:def 5 : (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "s1")) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ) ")" )))); definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x0" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "S")); pred "f" :::"is_continuous_in"::: "x0" means :: NFCONT_1:def 6 (Bool "(" (Bool "x0" ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"sequence":::) "of" "S" "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool (Set (Var "s1")) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) "x0")) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set "f" ($#k7_partfun1 :::"/."::: ) "x0") ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ); end; :: deftheorem defines :::"is_continuous_in"::: NFCONT_1:def 6 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "s1")) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ) ")" )))); theorem :: NFCONT_1:5 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_normsp_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h")))))))) ; theorem :: NFCONT_1:6 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq")))) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "seq")) ($#k1_normsp_1 :::"."::: ) (Set (Var "n"))))) ")" )))) ; theorem :: NFCONT_1:7 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_1:8 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_1:9 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "N1")) "being" ($#m1_nfcont_1 :::"Neighbourhood"::: ) "of" (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) (Bool "ex" (Set (Var "N")) "being" ($#m1_nfcont_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "for" (Set (Var "x1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1"))) ($#r2_hidden :::"in"::: ) (Set (Var "N1"))))) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_1:10 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "N1")) "being" ($#m1_nfcont_1 :::"Neighbourhood"::: ) "of" (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) (Bool "ex" (Set (Var "N")) "being" ($#m1_nfcont_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "N"))) ($#r1_tarski :::"c="::: ) (Set (Var "N1")))) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_1:11 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "ex" (Set (Var "N")) "being" ($#m1_nfcont_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "N"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )))) "holds" (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))))) ; theorem :: NFCONT_1:12 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "h1")) "," (Set (Var "h2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "h1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "h2")) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "h1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "h2")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_funct_2 :::"="::: ) (Set (Set "(" (Set (Var "h1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k2_normsp_1 :::"+"::: ) (Set "(" (Set (Var "h2")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))) & (Bool (Set (Set "(" (Set (Var "h1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "h2")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_funct_2 :::"="::: ) (Set (Set "(" (Set (Var "h1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k3_normsp_1 :::"-"::: ) (Set "(" (Set (Var "h2")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))) ")" )))) ; theorem :: NFCONT_1:13 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "h")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq"))) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "r")) ($#k5_normsp_1 :::"*"::: ) (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))))))) ; theorem :: NFCONT_1:14 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool "(" (Bool (Set ($#k4_normsp_0 :::"||."::: ) (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ) ($#k4_normsp_0 :::".||"::: ) ) ($#r2_funct_2 :::"="::: ) (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "h")) ($#k3_normsp_0 :::".||"::: ) ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")))) & (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "h")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" )) ($#r2_funct_2 :::"="::: ) (Set (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "h")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")))) ")" )))) ; theorem :: NFCONT_1:15 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "f1")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" )))) ; theorem :: NFCONT_1:16 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))))) ; theorem :: NFCONT_1:17 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" )))) ; definitionlet "S", "T" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "S")) "," (Set (Const "T")); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_continuous_on"::: "X" means :: NFCONT_1:def 7 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "X") ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ); end; :: deftheorem defines :::"is_continuous_on"::: NFCONT_1:def 7 : (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ) ")" )))); definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_continuous_on"::: "X" means :: NFCONT_1:def 8 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "X") ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ); end; :: deftheorem defines :::"is_continuous_on"::: NFCONT_1:def 8 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ) ")" )))); theorem :: NFCONT_1:18 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "s1")) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s1"))) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1"))) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k6_normsp_1 :::"lim"::: ) (Set (Var "s1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_1:19 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" ))) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_1:20 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" ))) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_1:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" )))) ; theorem :: NFCONT_1:22 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" )))) ; theorem :: NFCONT_1:23 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X1")))))) ; theorem :: NFCONT_1:24 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: NFCONT_1:25 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f1")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" )))) ; theorem :: NFCONT_1:26 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f1")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")))) & (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")))) ")" )))) ; theorem :: NFCONT_1:27 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f"))) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NFCONT_1:28 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" )))) ; theorem :: NFCONT_1:29 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "x1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "x2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ))) ")" ) & (Bool "ex" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))))) ; theorem :: NFCONT_1:30 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) "is" ($#v1_nfcont_1 :::"compact"::: ) ))) ; theorem :: NFCONT_1:31 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) "is" ($#v1_rcomp_1 :::"compact"::: ) ))) ; theorem :: NFCONT_1:32 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y"))) "is" ($#v1_nfcont_1 :::"compact"::: ) )))) ; theorem :: NFCONT_1:33 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")) ")" ))) ")" )))) ; theorem :: NFCONT_1:34 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "ex" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ")" ))) & (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ")" ))) ")" )))) ; theorem :: NFCONT_1:35 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set ($#k3_normsp_0 :::"||."::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k3_normsp_0 :::".||"::: ) ))))) ; theorem :: NFCONT_1:36 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool "ex" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y")) ")" ))) & (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y")) ")" ))) ")" ))))) ; theorem :: NFCONT_1:37 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool "ex" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y")) ")" ))) ")" ))))) ; definitionlet "S", "T" be ($#l1_normsp_1 :::"RealNormSpace":::); let "X" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "S")) "," (Set (Const "T")); pred "f" :::"is_Lipschitzian_on"::: "X" means :: NFCONT_1:def 9 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )) ")" ); end; :: deftheorem defines :::"is_Lipschitzian_on"::: NFCONT_1:def 9 : (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )) ")" ) ")" )))); definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "X" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ); pred "f" :::"is_Lipschitzian_on"::: "X" means :: NFCONT_1:def 10 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )) ")" ); end; :: deftheorem defines :::"is_Lipschitzian_on"::: NFCONT_1:def 10 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r6_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )) ")" ) ")" )))); theorem :: NFCONT_1:38 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X1")))))) ; theorem :: NFCONT_1:39 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f1")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1"))))))) ; theorem :: NFCONT_1:40 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f1")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1"))))))) ; theorem :: NFCONT_1:41 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "p")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f"))) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X"))))))) ; theorem :: NFCONT_1:42 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X"))) & (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#r6_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X"))) ")" )))) ; theorem :: NFCONT_1:43 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" bbbadV3_FUNCT_1())) "holds" (Bool (Set (Var "f")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X")))))) ; theorem :: NFCONT_1:44 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool (Set ($#k1_partfun2 :::"id"::: ) (Set (Var "Y"))) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "Y"))))) ; theorem :: NFCONT_1:45 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NFCONT_1:46 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r6_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NFCONT_1:47 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "st" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) )))) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))))) ; theorem :: NFCONT_1:48 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" bbbadV3_FUNCT_1())) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NFCONT_1:49 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "S")) "st" (Bool (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) ")" )) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))))) ; theorem :: NFCONT_1:50 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set ($#k1_partfun2 :::"id"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))))) ; theorem :: NFCONT_1:51 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) ($#r2_relset_1 :::"="::: ) (Set ($#k1_partfun2 :::"id"::: ) (Set (Var "Y"))))) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "Y")))))) ; theorem :: NFCONT_1:52 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "S")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "x0")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "p")))) ")" )) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))))))) ; theorem :: NFCONT_1:53 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x0")) ($#k1_normsp_0 :::".||"::: ) )) ")" )) "holds" (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))))) ; theorem :: NFCONT_1:54 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "x0")) ($#k1_normsp_0 :::".||"::: ) )) ")" )) "holds" (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ;