:: NFCONT_3 semantic presentation begin theorem :: NFCONT_3:1 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set (Var "seq")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h")))))))) ; theorem :: NFCONT_3:2 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "h1")) "," (Set (Var "h2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "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")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))) ")" )))) ; theorem :: NFCONT_3:3 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "h"))) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "r")) ($#k5_normsp_1 :::"*"::: ) (Set (Var "h"))))))) ; theorem :: NFCONT_3:4 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (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")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")) ")" ))))))) ; theorem :: NFCONT_3:5 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "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")) ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "seq")))) ")" )))) ; begin definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "x0" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; pred "f" :::"is_continuous_in"::: "x0" means :: NFCONT_3:def 1 (Bool "(" (Bool "x0" ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#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_3:def 1 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#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")) ")" ))) ")" ) ")" ) ")" ) ")" )))); theorem :: NFCONT_3:6 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))))))) ; theorem :: NFCONT_3:7 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"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" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "s1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"<>"::: ) (Set (Var "x0"))) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#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")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" ) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_3:8 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"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"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x0")) ")" )) ($#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_3:9 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#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_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "for" (Set (Var "x1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "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"))) ($#r2_hidden :::"in"::: ) (Set (Var "N1"))))) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_3:10 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"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" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "N"))) ($#r1_tarski :::"c="::: ) (Set (Var "N1")))) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_3:11 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_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_3 :::"is_continuous_in"::: ) (Set (Var "x0")))))) ; theorem :: NFCONT_3:12 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Var "f1")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" )))) ; theorem :: NFCONT_3:13 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))))))) ; theorem :: NFCONT_3:14 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" )))) ; theorem :: NFCONT_3:15 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "f1")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))))))) ; definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); attr "f" is :::"continuous"::: means :: NFCONT_3:def 2 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f"))) "holds" (Bool "f" ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0")))); end; :: deftheorem defines :::"continuous"::: NFCONT_3:def 2 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0")))) ")" ))); theorem :: NFCONT_3:16 (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" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "s1")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "s1"))) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "s1")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"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 "(" ($#k2_seq_2 :::"lim"::: ) (Set (Var "s1")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "s1")) ")" ))) ")" )) ")" )))) ; theorem :: NFCONT_3:17 (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" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x0")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#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 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )))) ")" )))) ; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v3_funct_1 :::"constant"::: ) -> ($#v1_nfcont_3 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_nfcont_3 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#v1_nfcont_3 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v1_nfcont_3 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S"); end; theorem :: NFCONT_3:18 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) )))) ; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v1_nfcont_3 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "X" be ($#v1_zfmisc_1 :::"trivial"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v1_nfcont_3 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S"); end; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f1", "f2" be ($#v1_nfcont_3 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); cluster (Set "f1" ($#k1_vfunct_1 :::"+"::: ) "f2") -> ($#v1_nfcont_3 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S"); cluster (Set "f1" ($#k2_vfunct_1 :::"-"::: ) "f2") -> ($#v1_nfcont_3 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S"); end; theorem :: NFCONT_3:19 (Bool "for" (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" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) ")" )))) ; theorem :: NFCONT_3:20 (Bool "for" (Set (Var "S")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) ")" )))) ; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#v1_nfcont_3 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "r" be ($#m1_subset_1 :::"Real":::); cluster (Set "r" ($#k4_vfunct_1 :::"(#)"::: ) "f") -> ($#v1_nfcont_3 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S"); end; theorem :: NFCONT_3:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ))))) ; theorem :: NFCONT_3:22 (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" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) )) "holds" (Bool "(" (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) ")" )))) ; theorem :: NFCONT_3:23 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "x1")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "x2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ))) ")" ) & (Bool "ex" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Set (Var "f")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) "is" ($#v1_nfcont_3 :::"continuous"::: ) ))) ; theorem :: NFCONT_3:24 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))) "is" ($#v1_rcomp_1 :::"compact"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" )) "is" ($#v1_nfcont_3 :::"continuous"::: ) )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) "is" ($#v1_nfcont_1 :::"compact"::: ) ))) ; theorem :: NFCONT_3:25 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_rcomp_1 :::"compact"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y"))) "is" ($#v1_nfcont_1 :::"compact"::: ) )))) ; begin definitionlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); attr "f" is :::"Lipschitzian"::: means :: NFCONT_3:def 3 (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f"))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x2")) ")" ) ")" ))) ")" ) ")" )); end; :: deftheorem defines :::"Lipschitzian"::: NFCONT_3:def 3 : (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) "iff" (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#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 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x2")) ")" ) ")" ))) ")" ) ")" )) ")" ))); theorem :: NFCONT_3:26 (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" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) "iff" (Bool "ex" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" )))) "holds" (Bool (Set ($#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 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k4_real_1 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k6_xcmplx_0 :::"-"::: ) (Set (Var "x2")) ")" ) ")" ))) ")" ) ")" )) ")" )))) ; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v2_nfcont_3 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#v2_nfcont_3 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v2_nfcont_3 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S"); end; theorem :: NFCONT_3:27 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) )))) ; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f1", "f2" be ($#v2_nfcont_3 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); cluster (Set "f1" ($#k1_vfunct_1 :::"+"::: ) "f2") -> ($#v2_nfcont_3 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S"); cluster (Set "f1" ($#k2_vfunct_1 :::"-"::: ) "f2") -> ($#v2_nfcont_3 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S"); end; theorem :: NFCONT_3:28 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) )))) ; theorem :: NFCONT_3:29 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) )))) ; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#v2_nfcont_3 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); let "p" be ($#m1_subset_1 :::"Real":::); cluster (Set "p" ($#k4_vfunct_1 :::"(#)"::: ) "f") -> ($#v2_nfcont_3 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S"); end; theorem :: NFCONT_3:30 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ))))) ; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#v2_nfcont_3 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))); cluster (Set ($#k2_normsp_0 :::"||."::: ) "f" ($#k2_normsp_0 :::".||"::: ) ) -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: NFCONT_3:31 (Bool "for" (Set (Var "X")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) )) "holds" (Bool "(" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" )) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) & (Bool (Set (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) & (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) ")" )))) ; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v3_funct_1 :::"constant"::: ) -> ($#v2_nfcont_3 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "S" be ($#l1_normsp_1 :::"RealNormSpace":::); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v2_nfcont_3 :::"Lipschitzian"::: ) -> ($#v1_nfcont_3 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: NFCONT_3:32 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) )))) "holds" (Bool (Set (Var "f")) "is" ($#v1_nfcont_3 :::"continuous"::: ) ))) ; theorem :: NFCONT_3:33 (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" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool "(" "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x0")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "r")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "p")))) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) ))))) ;