:: NCFCONT2 semantic presentation begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "CNS1", "CNS2" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "CNS1")) "," (Set (Const "CNS2")); pred "f" :::"is_uniformly_continuous_on"::: "X" means :: NCFCONT2:def 1 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "r")) "being" ($#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"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" "CNS1" "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "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 (Var "r"))) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"is_uniformly_continuous_on"::: NCFCONT2:def 1 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_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 "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")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS1")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#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")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" )))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "RNS" be ($#l1_normsp_1 :::"RealNormSpace":::); let "CNS" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "CNS")) "," (Set (Const "RNS")); pred "f" :::"is_uniformly_continuous_on"::: "X" means :: NCFCONT2:def 2 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "r")) "being" ($#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"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" "CNS" "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "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 (Var "r"))) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"is_uniformly_continuous_on"::: NCFCONT2:def 2 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#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"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#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 "x2")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" ))))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "RNS" be ($#l1_normsp_1 :::"RealNormSpace":::); let "CNS" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "RNS")) "," (Set (Const "CNS")); pred "f" :::"is_uniformly_continuous_on"::: "X" means :: NCFCONT2:def 3 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "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")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" "RNS" "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"is_uniformly_continuous_on"::: NCFCONT2:def 3 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_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 "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")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#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")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" ))))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "CNS" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "CNS"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); pred "f" :::"is_uniformly_continuous_on"::: "X" means :: NCFCONT2:def 4 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "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")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" "CNS" "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"is_uniformly_continuous_on"::: NCFCONT2:def 4 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "CNS"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#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"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" )))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "CNS" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "CNS"))) "," (Set ($#k1_numbers :::"REAL"::: ) ); pred "f" :::"is_uniformly_continuous_on"::: "X" means :: NCFCONT2:def 5 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "r")) "being" ($#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"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" "CNS" "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"is_uniformly_continuous_on"::: NCFCONT2:def 5 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "CNS"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r5_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#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"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#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 "x2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" )))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "RNS" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "RNS"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); pred "f" :::"is_uniformly_continuous_on"::: "X" means :: NCFCONT2:def 6 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "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")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" "RNS" "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"is_uniformly_continuous_on"::: NCFCONT2:def 6 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "RNS"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r6_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#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"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x2")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" )))); theorem :: NCFCONT2:1 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "st" (Bool (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))))) ; theorem :: NCFCONT2:2 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1"))))))) ; theorem :: NCFCONT2:3 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1"))))))) ; theorem :: NCFCONT2:4 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "st" (Bool (Bool (Set (Var "f1")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2"))) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1"))))))) ; theorem :: NCFCONT2:5 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "f1")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")))))))) ; theorem :: NCFCONT2:6 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "f1")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2"))) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")))))))) ; theorem :: NCFCONT2:7 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "st" (Bool (Bool (Set (Var "f1")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1"))))))) ; theorem :: NCFCONT2:8 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "f1")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")))))))) ; theorem :: NCFCONT2:9 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "f1")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")))))))) ; theorem :: NCFCONT2:10 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "st" (Bool (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f"))) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:11 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "r")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f"))) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))))) ; theorem :: NCFCONT2:12 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f"))) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))))) ; theorem :: NCFCONT2:13 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "st" (Bool (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NCFCONT2:14 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:15 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:16 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "st" (Bool (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#r5_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NCFCONT2:17 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#r5_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:18 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#r2_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:19 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "st" (Bool (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r7_ncfcont1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NCFCONT2:20 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r8_ncfcont1 :::"is_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r9_ncfcont1 :::"is_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:22 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "CNS"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r10_ncfcont1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NCFCONT2:23 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "CNS"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r5_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r11_ncfcont1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NCFCONT2:24 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "RNS"))) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r6_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r12_ncfcont1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NCFCONT2:25 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "st" (Bool (Bool (Set (Var "f")) ($#r13_ncfcont1 :::"is_Lipschitzian_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NCFCONT2:26 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "f")) ($#r14_ncfcont1 :::"is_Lipschitzian_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:27 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "f")) ($#r15_ncfcont1 :::"is_Lipschitzian_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:28 (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "CNS1")) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v1_ncfcont1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r7_ncfcont1 :::"is_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "Y")))))) ; theorem :: NCFCONT2:29 (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v1_ncfcont1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r8_ncfcont1 :::"is_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "Y"))))))) ; theorem :: NCFCONT2:30 (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r9_ncfcont1 :::"is_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "Y"))))))) ; theorem :: NCFCONT2:31 (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "CNS1")) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_ncfcont1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y"))) "is" ($#v1_ncfcont1 :::"compact"::: ) )))) ; theorem :: NCFCONT2:32 (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_ncfcont1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y"))) "is" ($#v1_nfcont_1 :::"compact"::: ) ))))) ; theorem :: NCFCONT2:33 (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "RNS")) "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_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y"))) "is" ($#v1_ncfcont1 :::"compact"::: ) ))))) ; theorem :: NCFCONT2:34 (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "CNS"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "Y")) ($#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_ncfcont1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r5_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool "ex" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "CNS")) "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")) ")" ))) ")" ))))) ; theorem :: NCFCONT2:35 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS1")) "," (Set (Var "CNS2")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Var "f")) ($#r1_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NCFCONT2:36 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "CNS")) "," (Set (Var "RNS")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Var "f")) ($#r2_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NCFCONT2:37 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "RNS")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "CNS")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "RNS")) "," (Set (Var "CNS")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Var "f")) ($#r3_ncfcont2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; begin definitionlet "M" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_clvect_1 :::"CNORMSTR"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "M")) "," (Set (Const "M")); attr "f" is :::"contraction"::: means :: NCFCONT2:def 7 (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "L"))) & (Bool (Set (Var "L")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" "M" "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "L")) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )); end; :: deftheorem defines :::"contraction"::: NCFCONT2:def 7 : (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_clvect_1 :::"CNORMSTR"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "M")) "," (Set (Var "M")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_ncfcont2 :::"contraction"::: ) ) "iff" (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "L"))) & (Bool (Set (Var "L")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "M")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" ) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "L")) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )) ")" ))); registrationlet "M" be ($#l2_clvect_1 :::"ComplexBanachSpace":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "M") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "M") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v1_ncfcont2 :::"contraction"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "M") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "M"))))); end; definitionlet "M" be ($#l2_clvect_1 :::"ComplexBanachSpace":::); mode Contraction of "M" is ($#v1_ncfcont2 :::"contraction"::: ) ($#m1_subset_1 :::"Function":::) "of" "M" "," "M"; end; theorem :: NCFCONT2:38 (Bool "for" (Set (Var "X")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) "iff" (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) ")" ))) ; theorem :: NCFCONT2:39 (Bool "for" (Set (Var "X")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "y")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: NCFCONT2:40 (Bool "for" (Set (Var "X")) "being" ($#l2_clvect_1 :::"ComplexBanachSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) "is" ($#m1_subset_1 :::"Contraction":::) "of" (Set (Var "X")))) "holds" (Bool "ex" (Set (Var "xp")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "xp"))) ($#r1_hidden :::"="::: ) (Set (Var "xp"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x")))) "holds" (Bool (Set (Var "xp")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" )))) ; theorem :: NCFCONT2:41 (Bool "for" (Set (Var "X")) "being" ($#l2_clvect_1 :::"ComplexBanachSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "X")) "," (Set (Var "X")) "st" (Bool (Bool "ex" (Set (Var "n0")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set ($#k9_funct_7 :::"iter"::: ) "(" (Set (Var "f")) "," (Set (Var "n0")) ")" ) "is" ($#m1_subset_1 :::"Contraction":::) "of" (Set (Var "X"))))) "holds" (Bool "ex" (Set (Var "xp")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "xp"))) ($#r1_hidden :::"="::: ) (Set (Var "xp"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x")))) "holds" (Bool (Set (Var "xp")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" )))) ;