:: NDIFF_4 semantic presentation begin theorem :: NDIFF_4:1 (Bool "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "holds" (Bool (Set (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set "(" ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f2")) ")" ))))) ; definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#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 ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; pred "f" :::"is_differentiable_in"::: "x" means :: NDIFF_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")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) "x") ")" )); end; :: deftheorem defines :::"is_differentiable_in"::: NDIFF_4:def 1 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x"))) "iff" (Bool "ex" (Set (Var "g")) "being" ($#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_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x"))) ")" )) ")" )))); theorem :: NDIFF_4:2 (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 "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")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x"))) "iff" (Bool (Set (Var "h")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x"))) ")" ))))) ; definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#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 ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"diff"::: "(" "f" "," "x" ")" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") means :: NDIFF_4:def 2 (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 it ($#r1_hidden :::"="::: ) (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "g")) "," "x" ")" )) ")" )); end; :: deftheorem defines :::"diff"::: NDIFF_4:def 2 : (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 "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" )) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "x")) ")" )) ")" )) ")" ))))); theorem :: NDIFF_4:3 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#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 "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "f")))) "holds" (Bool (Set ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "h")) "," (Set (Var "x")) ")" )))))) ; definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#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 ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_differentiable_on"::: "X" means :: NDIFF_4:def 3 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "X") ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x"))) ")" ) ")" ); end; :: deftheorem defines :::"is_differentiable_on"::: NDIFF_4:def 3 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_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 "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x"))) ")" ) ")" ) ")" )))); theorem :: NDIFF_4:4 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "X")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ))))) ; theorem :: NDIFF_4:5 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) "iff" (Bool "(" (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "f")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x"))) ")" ) ")" ) ")" )))) ; theorem :: NDIFF_4:6 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "Y")) "is" ($#v3_rcomp_1 :::"open"::: ) )))) ; definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "X" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Const "X"))) ; func "f" :::"`|"::: "X" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) means :: NDIFF_4:def 4 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "X") & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_ndiff_4 :::"diff"::: ) "(" "f" "," (Set (Var "x")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`|"::: NDIFF_4:def 4 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "X")))) "holds" (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_ndiff_4 :::"`|"::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" )) ")" ) ")" ) ")" ))))); theorem :: NDIFF_4:7 (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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "ex" (Set (Var "r")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) )))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_ndiff_4 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set (Var "n")))) ")" ) ")" )))) ; theorem :: NDIFF_4:8 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#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 "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "f")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "h")) "being" ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v1_fdiff_1 :::"-convergent"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "c")) "being" ($#v3_funct_1 :::"constant"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k1_rvsum_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k1_rvsum_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "N")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k1_ndiff_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "g")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k3_normsp_1 :::"-"::: ) (Set "(" (Set (Var "g")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_normsp_1 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k1_ndiff_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "g")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k3_normsp_1 :::"-"::: ) (Set "(" (Set (Var "g")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))))) ; theorem :: NDIFF_4:9 (Bool "for" (Set (Var "x0")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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 "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_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f"))) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_ndiff_4 :::"diff"::: ) "(" (Set "(" (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_euclid :::"*"::: ) (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ))) ")" )))) ; theorem :: NDIFF_4:10 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (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 "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_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f"))) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_ndiff_4 :::"diff"::: ) "(" (Set "(" ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_euclid :::"-"::: ) (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ))) ")" )))) ; theorem :: NDIFF_4:11 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (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 "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2"))) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_ndiff_4 :::"diff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k7_euclid :::"+"::: ) (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" )))) ; theorem :: NDIFF_4:12 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (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 "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2"))) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_ndiff_4 :::"diff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k8_euclid :::"-"::: ) (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" )))) ; theorem :: NDIFF_4:13 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f"))) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_ndiff_4 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_euclid :::"*"::: ) (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ")" ))) ")" ) ")" ))))) ; theorem :: NDIFF_4:14 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f"))) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k2_nfcont_4 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_ndiff_4 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_euclid :::"-"::: ) (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ")" ))) ")" ) ")" )))) ; theorem :: NDIFF_4:15 (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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Var "f1")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2"))) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k7_integr15 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_ndiff_4 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x")) ")" ")" ) ($#k7_euclid :::"+"::: ) (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x")) ")" ")" ))) ")" ) ")" )))) ; theorem :: NDIFF_4:16 (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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2")) ")" ))) & (Bool (Set (Var "f1")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2"))) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k8_integr15 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_ndiff_4 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x")) ")" ")" ) ($#k8_euclid :::"-"::: ) (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x")) ")" ")" ))) ")" ) ")" )))) ; theorem :: NDIFF_4:17 (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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Z"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_ndiff_4 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set (Var "n")))) ")" ) ")" )))) ; theorem :: NDIFF_4:18 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k9_euclid :::"*"::: ) (Set (Var "r")) ")" ) ($#k7_euclid :::"+"::: ) (Set (Var "p")))) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_ndiff_4 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "r"))) ")" ) ")" ))))) ; theorem :: NDIFF_4:19 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Var "f")) ($#r1_nfcont_4 :::"is_continuous_in"::: ) (Set (Var "x0")))))) ; theorem :: NDIFF_4:20 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (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 "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_nfcont_4 :::"continuous"::: ) )))) ; theorem :: NDIFF_4:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z"))))))) ; definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); attr "f" is :::"differentiable"::: means :: NDIFF_4:def 5 (Bool "f" ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")); end; :: deftheorem defines :::"differentiable"::: NDIFF_4:def 5 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_ndiff_4 :::"differentiable"::: ) ) "iff" (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ))); registrationlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k2_funcop_1 :::"-->"::: ) (Set "(" ($#k5_euclid :::"0*"::: ) "n" ")" )) -> ($#v1_ndiff_4 :::"differentiable"::: ) for ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ); end; registrationlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster ($#~v1_xboole_0 "non" ($#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"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) bbbadV7_VALUED_2() bbbadV8_VALUED_2() bbbadV9_VALUED_2() ($#v1_ndiff_4 :::"differentiable"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ))))); end; theorem :: NDIFF_4:22 (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 "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#v1_ndiff_4 :::"differentiable"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "Z")))))) ; theorem :: NDIFF_4:23 (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 "R")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "R")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#v1_ndiff_3 :::"RestFunc-like"::: ) ) "iff" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k17_complex1 :::"|."::: ) (Set (Var "z")) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d")))) "holds" (Bool (Set (Set "(" (Set ($#k17_complex1 :::"|."::: ) (Set (Var "z")) ($#k17_complex1 :::".|"::: ) ) ($#k8_binop_2 :::"""::: ) ")" ) ($#k11_binop_2 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "R")) ($#k7_partfun1 :::"/."::: ) (Set (Var "z")) ")" ) ($#k1_normsp_0 :::".||"::: ) )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" ))) ")" ))) ; theorem :: NDIFF_4:24 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "g")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "x0")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set "(" (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ")" )) ")" ))))) ; theorem :: NDIFF_4:25 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x0")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0")))) ")" )))) ; theorem :: NDIFF_4:26 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "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"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "f")) ($#r1_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k1_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set "(" (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x0")) ")" )) ")" ))))) ; theorem :: NDIFF_4:27 (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 "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_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0")))) ")" )))) ; theorem :: NDIFF_4:28 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (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 "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n"))) & (Bool (Set (Var "g")) ($#r2_ndiff_3 :::"is_differentiable_on"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))) ($#r2_ndiff_3 :::"is_differentiable_on"::: ) (Set (Var "X"))) & (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "g")) ($#k2_ndiff_3 :::"`|"::: ) (Set (Var "X")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ) ($#k2_ndiff_3 :::"`|"::: ) (Set (Var "X")))) ")" ))))) ; theorem :: NDIFF_4:29 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#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"))) & (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) ($#r2_ndiff_3 :::"is_differentiable_on"::: ) (Set (Var "X"))) & (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k2_ndiff_4 :::"`|"::: ) (Set (Var "X")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k2_ndiff_3 :::"`|"::: ) (Set (Var "X")))) ")" ))))) ; theorem :: NDIFF_4:30 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (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 "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r2_ndiff_3 :::"is_differentiable_on"::: ) (Set (Var "X"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))) ($#r2_ndiff_3 :::"is_differentiable_on"::: ) (Set (Var "X")))) ")" )))) ; theorem :: NDIFF_4:31 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (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 "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_ndiff_4 :::"is_differentiable_on"::: ) (Set (Var "X"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "n")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) ($#r2_ndiff_3 :::"is_differentiable_on"::: ) (Set (Var "X")))) ")" )))) ; theorem :: NDIFF_4:32 (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "st" (Bool (Bool (Set (Var "J")) ($#r1_funct_2 :::"="::: ) (Set ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ))) "holds" (Bool (Set (Var "J")) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: NDIFF_4:33 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k2_funct_1 :::"""::: ) ))) "holds" (Bool (Set (Var "I")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "x0"))))) ; theorem :: NDIFF_4:34 (Bool "for" (Set (Var "S")) "," (Set (Var "T")) "being" ($#l1_normsp_1 :::"RealNormSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "S")) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set (Var "T")) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ))) & (Bool (Set (Var "f1")) ($#r2_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))))) "holds" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))))))) ; theorem :: NDIFF_4:35 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (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 "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "J")) ($#r1_funct_2 :::"="::: ) (Set ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" )) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "x0")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "y0")) ($#k12_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "J"))))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool (Set (Var "g")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "y0"))) ")" ))))))) ; theorem :: NDIFF_4:36 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (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 "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k2_funct_1 :::"""::: ) )) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "x0")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "y0")) ($#k12_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "I"))) ($#r2_relset_1 :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool (Set (Var "g")) ($#r1_nfcont_3 :::"is_continuous_in"::: ) (Set (Var "y0"))) ")" ))))))) ; theorem :: NDIFF_4:37 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k2_funct_1 :::"""::: ) ))) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "I")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Num 1) ($#k12_finseq_1 :::"*>"::: ) )) ")" ))) ; definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Const "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Const "n")) ")" ); pred "f" :::"is_differentiable_in"::: "x" means :: NDIFF_4:def 6 (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) )(Bool "ex" (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") "st" (Bool "(" (Bool "f" ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool "x" ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool (Set (Var "g")) ($#r1_pdiff_7 :::"is_differentiable_in"::: ) (Set (Var "y"))) ")" ))); end; :: deftheorem defines :::"is_differentiable_in"::: NDIFF_4:def 6 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x"))) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) )(Bool "ex" (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool (Set (Var "g")) ($#r1_pdiff_7 :::"is_differentiable_in"::: ) (Set (Var "y"))) ")" ))) ")" )))); definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Const "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Const "n")) ")" ); func :::"diff"::: "(" "f" "," "x" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "n" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: NDIFF_4:def 7 (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) )(Bool "ex" (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") "st" (Bool "(" (Bool "f" ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool "x" ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool it ($#r1_funct_2 :::"="::: ) (Set ($#k1_pdiff_7 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "y")) ")" )) ")" ))); end; :: deftheorem defines :::"diff"::: NDIFF_4:def 7 : (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 "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k3_ndiff_4 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" )) "iff" (Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) )(Bool "ex" (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool "(" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool (Set (Var "b4")) ($#r1_funct_2 :::"="::: ) (Set ($#k1_pdiff_7 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "y")) ")" )) ")" ))) ")" ))))); theorem :: NDIFF_4:38 (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 1) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 1)) "st" (Bool (Bool (Set (Var "J")) ($#r2_funct_2 :::"="::: ) (Set ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ))) "holds" (Bool "(" (Bool (Set (Var "J")) ($#r1_pdiff_7 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_pdiff_7 :::"diff"::: ) "(" (Set (Var "J")) "," (Set (Var "x0")) ")" ) ($#r2_funct_2 :::"="::: ) (Set (Var "J"))) ")" ))) ; theorem :: NDIFF_4:39 (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "st" (Bool (Bool (Set (Var "J")) ($#r1_funct_2 :::"="::: ) (Set ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ))) "holds" (Bool "(" (Bool (Set (Var "J")) ($#r3_ndiff_4 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k3_ndiff_4 :::"diff"::: ) "(" (Set (Var "J")) "," (Set (Var "x0")) ")" ) ($#r2_funct_2 :::"="::: ) (Set (Var "J"))) ")" ))) ; theorem :: NDIFF_4:40 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k2_funct_1 :::"""::: ) ))) "holds" (Bool "(" (Bool "(" "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set (Var "R")) ($#k1_partfun1 :::"*"::: ) (Set (Var "I"))) "is" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearOperator":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set (Var "L")) ($#k1_partfun1 :::"*"::: ) (Set (Var "I"))) "is" ($#m1_subset_1 :::"LinearFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" )) ")" ) ")" ))) ; theorem :: NDIFF_4:41 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "J")) ($#r1_funct_2 :::"="::: ) (Set ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ))) "holds" (Bool "(" (Bool "(" "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set (Var "R")) ($#k1_partfun1 :::"*"::: ) (Set (Var "J"))) "is" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" )) ")" ) & (Bool "(" "for" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set (Var "L")) ($#k1_partfun1 :::"*"::: ) (Set (Var "J"))) "is" ($#v2_lopban_1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" )) ")" ) ")" ))) ; theorem :: NDIFF_4:42 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (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 "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k2_funct_1 :::"""::: ) )) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "x0")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "y0")) ($#k12_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "I"))) ($#r2_relset_1 :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "f")) ($#r1_ndiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "y0"))) & (Bool (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "y0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_ndiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Num 1) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_ndiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set "(" ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "y0")) ")" ")" ))) ")" ) ")" ))))))) ; theorem :: NDIFF_4:43 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (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 "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k2_funct_1 :::"""::: ) )) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "x0")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "y0")) ($#k12_finseq_1 :::"*>"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "I"))) ($#r2_relset_1 :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_ndiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) "iff" (Bool (Set (Var "g")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "y0"))) ")" ))))))) ; theorem :: NDIFF_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 "J")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "J")) ($#r1_funct_2 :::"="::: ) (Set ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" )) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "x0")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "y0")) ($#k12_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "J"))))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_ndiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) "iff" (Bool (Set (Var "g")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "y0"))) ")" ))))))) ; theorem :: NDIFF_4:45 (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 "J")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "J")) ($#r1_funct_2 :::"="::: ) (Set ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" )) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "y0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Var "x0")) ($#r1_hidden :::"="::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "y0")) ($#k12_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "J")))) & (Bool (Set (Var "g")) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "y0")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_ndiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "y0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_ndiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Num 1) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k3_ndiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "r")) ($#k12_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set "(" ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "y0")) ")" ")" ))) ")" ) ")" ))))))) ; theorem :: NDIFF_4:46 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "R")) ($#k7_partfun1 :::"/."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" )))) "holds" (Bool "for" (Set (Var "e")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "e")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "d")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "d")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k17_complex1 :::"|."::: ) (Set (Var "h")) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "R")) ($#k7_partfun1 :::"/."::: ) (Set (Var "h")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "e")) ($#k11_binop_2 :::"*"::: ) (Set ($#k17_complex1 :::"|."::: ) (Set (Var "h")) ($#k17_complex1 :::".|"::: ) ))) ")" ) ")" ))))) ; theorem :: NDIFF_4:47 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "L")) "being" ($#v2_lopban_1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "holds" (Bool (Set (Set (Var "L")) ($#k1_partfun1 :::"*"::: ) (Set (Var "R"))) "is" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ))))) ; theorem :: NDIFF_4:48 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R1")) "being" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "R1")) ($#k7_partfun1 :::"/."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" )))) "holds" (Bool "for" (Set (Var "R2")) "being" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Set (Var "R2")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" )))) "holds" (Bool "for" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set (Set (Var "R2")) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "L")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "R1")) ")" )) "is" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" )))))) ; theorem :: NDIFF_4:49 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "R1")) "being" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Set (Var "R1")) ($#k7_partfun1 :::"/."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" )))) "holds" (Bool "for" (Set (Var "R2")) "being" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Set (Var "R2")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" )))) "holds" (Bool "for" (Set (Var "L1")) "being" ($#m1_subset_1 :::"LinearFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "L2")) "being" ($#v2_lopban_1 :::"Lipschitzian"::: ) ($#m1_subset_1 :::"LinearOperator":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "holds" (Bool (Set (Set "(" (Set (Var "L2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "R1")) ")" ) ($#k6_vfunct_1 :::"+"::: ) (Set "(" (Set (Var "R2")) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set (Var "L1")) ($#k6_vfunct_1 :::"+"::: ) (Set (Var "R1")) ")" ) ")" )) "is" ($#m1_subset_1 :::"RestFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ))))))) ; theorem :: NDIFF_4:50 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (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_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_ndiff_1 :::"is_differentiable_in"::: ) (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))) ($#r1_ndiff_3 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_ndiff_3 :::"diff"::: ) "(" (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "g")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_ndiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ")" ) ($#k17_lopban_1 :::"."::: ) (Set "(" ($#k1_ndiff_3 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "x0")) ")" ")" ))) ")" ))))) ;