:: NFCONT_4 semantic presentation begin definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "x0" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; pred "f" :::"is_continuous_in"::: "x0" means :: NFCONT_4:def 1 (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "n" ")" ) "st" (Bool "(" (Bool "f" ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) "x0") ")" )); end; :: deftheorem defines :::"is_continuous_in"::: NFCONT_4:def 1 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" )) ")" )))); theorem :: NFCONT_4:1 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool (Set (Var "h")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ))))) ; theorem :: NFCONT_4:2 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))))))) ; theorem :: NFCONT_4:3 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"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")) ($#k10_binop_2 :::"-"::: ) (Set (Var "x0")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_4:4 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) (Bool "for" (Set (Var "w")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set (Var "w")))) "holds" (Bool "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "y")) ($#k8_euclid :::"-"::: ) (Set (Var "z")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" ($#r1_hidden :::"="::: ) "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) : (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "y")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" ))))) ; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "Z" be ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); func :::"|.":::"f":::".|"::: -> ($#m1_subset_1 :::"PartFunc":::) "of" "Z" "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: NFCONT_4:def 2 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k12_euclid :::"|."::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"|."::: NFCONT_4:def 2 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) )) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) )) ")" ) ")" ) ")" ))))); definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "Z" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); func :::"-"::: "f" -> ($#m1_subset_1 :::"PartFunc":::) "of" "Z" "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) means :: NFCONT_4:def 3 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k6_euclid :::"-"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"-"::: NFCONT_4:def 3 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "b4")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "Z")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k6_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" ) ")" ) ")" )))); theorem :: NFCONT_4:5 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "W")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_hidden :::"="::: ) (Set (Var "g1"))) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set (Var "g2")))) "holds" (Bool (Set (Set (Var "f1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g1")) ($#k7_integr15 :::"+"::: ) (Set (Var "g2")))))))) ; theorem :: NFCONT_4:6 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "W")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) ($#r1_hidden :::"="::: ) (Set (Var "g1")))) "holds" (Bool (Set (Set (Var "a")) ($#k4_vfunct_1 :::"(#)"::: ) (Set (Var "f1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "g1"))))))))) ; theorem :: NFCONT_4:7 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "W")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set "(" ($#k7_binop_2 :::"-"::: ) (Num 1) ")" ) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f1"))) ($#r2_relset_1 :::"="::: ) (Set ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f1"))))))) ; theorem :: NFCONT_4:8 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "W")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_hidden :::"="::: ) (Set (Var "g1")))) "holds" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f1"))) ($#r1_hidden :::"="::: ) (Set ($#k2_nfcont_4 :::"-"::: ) (Set (Var "g1")))))))) ; theorem :: NFCONT_4:9 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "W")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_hidden :::"="::: ) (Set (Var "g1")))) "holds" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f1")) ($#k3_normsp_0 :::".||"::: ) ) ($#r2_relset_1 :::"="::: ) (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "g1")) ($#k1_nfcont_4 :::".|"::: ) )))))) ; theorem :: NFCONT_4:10 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "W")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g1")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "W")) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_hidden :::"="::: ) (Set (Var "g1"))) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set (Var "g2")))) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "g1")) ($#k8_integr15 :::"-"::: ) (Set (Var "g2")))))))) ; theorem :: NFCONT_4:11 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "N1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "y")) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" ($#r1_hidden :::"="::: ) (Set (Var "N1"))) ")" ))) "holds" (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_4:12 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"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" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool "{" (Set (Var "y")) where y "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "y")) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" ($#r1_hidden :::"="::: ) (Set (Var "N1"))) ")" ))) "holds" (Bool "ex" (Set (Var "N")) "being" ($#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_4:13 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool "ex" (Set (Var "N")) "being" ($#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_4 :::"is_continuous_in"::: ) (Set (Var "x0")))))) ; theorem :: NFCONT_4:14 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "x0")) ($#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_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2"))) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))))) ; theorem :: NFCONT_4:15 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "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")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2"))) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))))) ; theorem :: NFCONT_4:16 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f"))) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))))))) ; theorem :: NFCONT_4:17 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0")))))) ; theorem :: NFCONT_4:18 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f"))) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))))) ; theorem :: NFCONT_4:19 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set (Var "S")) "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_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")))) & (Bool (Set (Var "f2")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "z")))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))))))))) ; theorem :: NFCONT_4:20 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "S")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set (Var "S")) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set ($#k1_numbers :::"REAL"::: ) ) "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")) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))))) ; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x0" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Const "n"))); pred "f" :::"is_continuous_in"::: "x0" means :: NFCONT_4:def 4 (Bool "ex" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "n" ")" )(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "n" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool "x0" ($#r1_hidden :::"="::: ) (Set (Var "y0"))) & (Bool "f" ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "y0"))) ")" ))); end; :: deftheorem defines :::"is_continuous_in"::: NFCONT_4:def 4 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "ex" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" )(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool "(" (Bool (Set (Var "x0")) ($#r1_hidden :::"="::: ) (Set (Var "y0"))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "y0"))) ")" ))) ")" )))); theorem :: NFCONT_4:21 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) (Bool "for" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "h"))) & (Bool (Set (Var "x0")) ($#r1_hidden :::"="::: ) (Set (Var "y0")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool (Set (Var "h")) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "y0"))) ")" )))))) ; theorem :: NFCONT_4:22 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "f1")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r2_nfcont_4 :::"is_continuous_in"::: ) (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))))) ; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); attr "f" is :::"continuous"::: means :: NFCONT_4:def 5 (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" ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))); end; :: deftheorem defines :::"continuous"::: NFCONT_4:def 5 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "x0")) "being" ($#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_4 :::"is_continuous_in"::: ) (Set (Var "x0")))) ")" ))); theorem :: NFCONT_4:23 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool "(" (Bool (Set (Var "g")) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) "iff" (Bool (Set (Var "f")) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) ")" )))) ; theorem :: NFCONT_4:24 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "x0")) "being" ($#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 ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k10_binop_2 :::"-"::: ) (Set (Var "x0")) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )))) ")" )))) ; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v3_funct_1 :::"constant"::: ) -> ($#v1_nfcont_4 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_euclid :::"REAL"::: ) "n") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV7_VALUED_2() bbbadV8_VALUED_2() bbbadV9_VALUED_2() ($#v1_nfcont_4 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#v1_nfcont_4 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v1_nfcont_4 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ); end; theorem :: NFCONT_4:25 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) )))) ; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v1_nfcont_4 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "X" be ($#v1_zfmisc_1 :::"trivial"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v1_nfcont_4 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ); end; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f1", "f2" be ($#v1_nfcont_4 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); cluster (Set "f1" ($#k7_integr15 :::"+"::: ) "f2") -> ($#v1_nfcont_4 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ); end; theorem :: NFCONT_4:26 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "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_4 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) ")" )))) ; theorem :: NFCONT_4:27 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "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_4 :::"continuous"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) ")" )))) ; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#v1_nfcont_4 :::"continuous"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "r" be ($#m1_subset_1 :::"Real":::); cluster (Set "r" ($#k9_integr15 :::"(#)"::: ) "f") -> ($#v1_nfcont_4 :::"continuous"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ); end; theorem :: NFCONT_4:28 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) ))))) ; theorem :: NFCONT_4:29 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "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_4 :::"continuous"::: ) )) "holds" (Bool "(" (Bool (Set (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_fcont_1 :::"continuous"::: ) ) & (Bool (Set (Set "(" ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) ")" )))) ; theorem :: NFCONT_4:30 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "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")) ($#k9_binop_2 :::"+"::: ) (Set (Var "x2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k7_euclid :::"+"::: ) (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_4 :::"is_continuous_in"::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) "is" ($#v1_nfcont_4 :::"continuous"::: ) ))) ; theorem :: NFCONT_4:31 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "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" ($#v1_nfcont_4 :::"continuous"::: ) ) & (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "Y")) "is" ($#v1_nfcont_1 :::"compact"::: ) )))) ; theorem :: NFCONT_4:32 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "Z")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "Y")))) & (Bool (Set (Var "Y")) "is" ($#v1_rcomp_1 :::"compact"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) )) "holds" (Bool (Set (Var "Z")) "is" ($#v1_nfcont_1 :::"compact"::: ) ))))) ; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); attr "f" is :::"Lipschitzian"::: means :: NFCONT_4:def 6 (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "n" ")" ) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) "f") & (Bool (Set (Var "g")) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) ")" )); end; :: deftheorem defines :::"Lipschitzian"::: NFCONT_4:def 6 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) ) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "f"))) & (Bool (Set (Var "g")) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) ")" )) ")" ))); theorem :: NFCONT_4:33 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_nfcont_4 :::"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 ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k10_binop_2 :::"-"::: ) (Set (Var "x2")) ")" ) ")" ))) ")" ) ")" )) ")" ))) ; theorem :: NFCONT_4:34 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "h")))) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) ) "iff" (Bool (Set (Var "h")) "is" ($#v2_nfcont_3 :::"Lipschitzian"::: ) ) ")" )))) ; theorem :: NFCONT_4:35 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_4 :::"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 ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x2")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "r")) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k18_complex1 :::"abs"::: ) (Set "(" (Set (Var "x1")) ($#k10_binop_2 :::"-"::: ) (Set (Var "x2")) ")" ) ")" ))) ")" ) ")" )) ")" )))) ; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_funct_1 :::"Function-like"::: ) -> ($#v2_nfcont_4 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster ($#v1_xboole_0 :::"empty"::: ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_euclid :::"REAL"::: ) "n") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV7_VALUED_2() bbbadV8_VALUED_2() bbbadV9_VALUED_2() for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#v2_nfcont_4 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v2_nfcont_4 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ); end; theorem :: NFCONT_4:36 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "," (Set (Var "X1")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) ) & (Bool (Set (Var "X1")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) )))) ; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f1", "f2" be ($#v2_nfcont_4 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); cluster (Set "f1" ($#k7_integr15 :::"+"::: ) "f2") -> ($#v2_nfcont_4 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ); cluster (Set "f1" ($#k8_integr15 :::"-"::: ) "f2") -> ($#v2_nfcont_4 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ); end; theorem :: NFCONT_4:37 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) )))) ; theorem :: NFCONT_4:38 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X1"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "X1")) ")" )) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) )))) ; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#v2_nfcont_4 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "p" be ($#m1_subset_1 :::"Real":::); cluster (Set "p" ($#k9_integr15 :::"(#)"::: ) "f") -> ($#v2_nfcont_4 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ); end; theorem :: NFCONT_4:39 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) ) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) ))))) ; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#v2_nfcont_4 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); cluster (Set ($#k1_nfcont_4 :::"|."::: ) "f" ($#k1_nfcont_4 :::".|"::: ) ) -> ($#v2_fcont_1 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); end; theorem :: NFCONT_4:40 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) )) "holds" (Bool "(" (Bool (Set ($#k2_nfcont_4 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" )) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) ) & (Bool (Set (Set ($#k1_nfcont_4 :::"|."::: ) (Set (Var "f")) ($#k1_nfcont_4 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_fcont_1 :::"Lipschitzian"::: ) ) & (Bool (Set (Set "(" ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_nfcont_4 :::"Lipschitzian"::: ) ) ")" )))) ; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v3_funct_1 :::"constant"::: ) -> ($#v2_nfcont_4 :::"Lipschitzian"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v2_nfcont_4 :::"Lipschitzian"::: ) -> ($#v1_nfcont_4 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: NFCONT_4:41 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x0")) ($#k9_euclid :::"*"::: ) (Set (Var "r")) ")" ) ($#k7_euclid :::"+"::: ) (Set (Var "p")))) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) ))))) ; theorem :: NFCONT_4:42 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r2_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: NFCONT_4:43 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "h")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "h"))) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ) ")" )))) ; theorem :: NFCONT_4:44 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v1_nfcont_4 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "h"))) "is" ($#v1_fcont_1 :::"continuous"::: ) )) ")" ))) ; theorem :: NFCONT_4:45 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: NFCONT_4:46 (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "h"))) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0")))) ")" )))) ; theorem :: NFCONT_4:47 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v1_nfcont_3 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n"))))) "holds" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "h"))) "is" ($#v1_nfcont_3 :::"continuous"::: ) )) ")" ))) ;