:: NFCONT_2 semantic presentation begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "S", "T" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "S")) "," (Set (Const "T")); pred "f" :::"is_uniformly_continuous_on"::: "X" means :: NFCONT_2:def 1 (Bool "(" (Bool "X" ($#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" "S" "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"::: NFCONT_2:def 1 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"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" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x1")) ($#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 "S" be ($#l1_normsp_1 :::"RealNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ); pred "f" :::"is_uniformly_continuous_on"::: "X" means :: NFCONT_2: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" ($#m1_subset_1 :::"Point":::) "of" "S" "st" (Bool (Bool (Set (Var "x1")) ($#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"::: NFCONT_2:def 2 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_nfcont_2 :::"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" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x1")) ($#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"))) ")" ) ")" )) ")" ) ")" ) ")" )))); theorem :: NFCONT_2:1 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))))) ; theorem :: NFCONT_2:2 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f1")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1"))))))) ; theorem :: NFCONT_2:3 (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f1")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X1")))) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1"))))))) ; theorem :: NFCONT_2:4 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "p")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X"))))))) ; theorem :: NFCONT_2:5 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NFCONT_2:6 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#r2_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NFCONT_2:7 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NFCONT_2:8 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r4_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NFCONT_2:9 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) ($#r5_nfcont_1 :::"is_Lipschitzian_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))) ; theorem :: NFCONT_2:10 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "Y")))))) ; theorem :: NFCONT_2:11 (Bool "for" (Set (Var "T")) "," (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"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 :: NFCONT_2:12 (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Y")) "is" ($#v1_nfcont_1 :::"compact"::: ) ) & (Bool (Set (Var "f")) ($#r2_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "Y")))) "holds" (Bool "ex" (Set (Var "x1")) "," (Set (Var "x2")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Var "x2")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y")) ")" ))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y")) ")" ))) ")" ))))) ; theorem :: NFCONT_2:13 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Var "f")) ($#r1_nfcont_2 :::"is_uniformly_continuous_on"::: ) (Set (Var "X")))))) ; begin definitionlet "M" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_normsp_1 :::"NORMSTR"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "M")) "," (Set (Const "M")); attr "f" is :::"contraction"::: means :: NFCONT_2:def 3 (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")) ")" ) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )); end; :: deftheorem defines :::"contraction"::: NFCONT_2:def 3 : (Bool "for" (Set (Var "M")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_normsp_1 :::"NORMSTR"::: ) (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_nfcont_2 :::"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")) ")" ) ($#k1_normsp_0 :::".||"::: ) ))) ")" ) ")" )) ")" ))); registrationlet "M" be ($#l1_normsp_1 :::"RealNormSpace":::); 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"::: ) ($#v1_nfcont_2 :::"contraction"::: ) for ($#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 ($#l1_normsp_1 :::"RealNormSpace":::); mode Contraction of "M" is ($#v1_nfcont_2 :::"contraction"::: ) ($#m1_subset_1 :::"Function":::) "of" "M" "," "M"; end; theorem :: NFCONT_2:14 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealBanachSpace":::) (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")) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" )))) ; theorem :: NFCONT_2:15 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealBanachSpace":::) (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 ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set ($#k1_abian :::"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"))) ")" ) ")" )))) ; theorem :: NFCONT_2:16 (Bool "for" (Set (Var "K")) "," (Set (Var "L")) "," (Set (Var "e")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "K"))) & (Bool (Set (Var "K")) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "e")))) "holds" (Bool "ex" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "L")) ($#k3_xcmplx_0 :::"*"::: ) (Set "(" (Set (Var "K")) ($#k3_power :::"to_power"::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "e"))))) ;