:: FDIFF_3 semantic presentation begin theorem :: FDIFF_3:1 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v1_fdiff_1 :::"-convergent"::: ) ($#m1_subset_1 :::"Real_Sequence":::)(Bool "ex" (Set (Var "c")) "being" ($#v3_funct_1 :::"constant"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ))))) ; theorem :: FDIFF_3:2 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ))) "holds" (Bool "ex" (Set (Var "h")) "being" ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v1_fdiff_1 :::"-convergent"::: ) ($#m1_subset_1 :::"Real_Sequence":::)(Bool "ex" (Set (Var "c")) "being" ($#v3_funct_1 :::"constant"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ))))) ; theorem :: FDIFF_3:3 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ")" ) & (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "h1")) "," (Set (Var "h2")) "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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h2")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h1")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h2")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h2")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))))))) ; theorem :: FDIFF_3:4 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool "(" "for" (Set (Var "h")) "being" ($#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"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ")" ) & (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "h1")) "," (Set (Var "h2")) "being" ($#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"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h2")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h1")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h2")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h1")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h2")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h2")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))))))) ; definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x0" be ($#m1_subset_1 :::"Real":::); pred "f" :::"is_Lcontinuous_in"::: "x0" means :: FDIFF_3:def 1 (Bool "(" (Bool "x0" ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) "x0" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ))) & (Bool (Set (Var "a")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) "x0")) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "a"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set "f" ($#k1_seq_1 :::"."::: ) "x0") ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "a")) ")" ))) ")" ) ")" ) ")" ); pred "f" :::"is_Rcontinuous_in"::: "x0" means :: FDIFF_3:def 2 (Bool "(" (Bool "x0" ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) "x0" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ))) & (Bool (Set (Var "a")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) "x0")) "holds" (Bool "(" (Bool (Set "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "a"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set "f" ($#k1_seq_1 :::"."::: ) "x0") ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "a")) ")" ))) ")" ) ")" ) ")" ); pred "f" :::"is_right_differentiable_in"::: "x0" means :: FDIFF_3:def 3 (Bool "(" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) "x0" "," (Set "(" "x0" ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) ")" )) & (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) "x0" ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ")" ) ")" ); pred "f" :::"is_left_differentiable_in"::: "x0" means :: FDIFF_3:def 4 (Bool "(" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" "x0" ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," "x0" ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) ")" )) & (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) "x0" ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"is_Lcontinuous_in"::: FDIFF_3:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_fdiff_3 :::"is_Lcontinuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "x0")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "a")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "a"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "a")) ")" ))) ")" ) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_Rcontinuous_in"::: FDIFF_3:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_fdiff_3 :::"is_Rcontinuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "x0")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ))) & (Bool (Set (Var "a")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "a"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "a")) ")" ))) ")" ) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_right_differentiable_in"::: FDIFF_3:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) & (Bool "(" "for" (Set (Var "h")) "being" ($#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"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_left_differentiable_in"::: FDIFF_3:def 4 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) & (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) )) ")" ) ")" ) ")" ))); theorem :: FDIFF_3:5 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Var "f")) ($#r1_fdiff_3 :::"is_Lcontinuous_in"::: ) (Set (Var "x0"))))) ; theorem :: FDIFF_3:6 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_fdiff_3 :::"is_Lcontinuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set (Var "g2"))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ))) "holds" (Bool "ex" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r1")) ")" ) "," (Set (Var "x0")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r1")) ")" ) "," (Set (Var "x0")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set (Var "g2"))) ")" ) ")" )))) ; theorem :: FDIFF_3:7 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool (Set (Var "f")) ($#r2_fdiff_3 :::"is_Rcontinuous_in"::: ) (Set (Var "x0"))))) ; theorem :: FDIFF_3:8 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "," (Set (Var "g2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_fdiff_3 :::"is_Rcontinuous_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set (Var "g2"))) & (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ))) "holds" (Bool "ex" (Set (Var "r1")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r1")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r1")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set (Var "g2"))) ")" ) ")" )))) ; definitionlet "x0" be ($#m1_subset_1 :::"Real":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); assume (Bool (Set (Const "f")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Const "x0"))) ; func :::"Ldiff"::: "(" "f" "," "x0" ")" -> ($#m1_subset_1 :::"Real":::) means :: FDIFF_3:def 5 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) "x0" ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))))); end; :: deftheorem defines :::"Ldiff"::: FDIFF_3:def 5 : (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) "iff" (Bool "for" (Set (Var "h")) "being" ($#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"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))))) ")" )))); definitionlet "x0" be ($#m1_subset_1 :::"Real":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); assume (Bool (Set (Const "f")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Const "x0"))) ; func :::"Rdiff"::: "(" "f" "," "x0" ")" -> ($#m1_subset_1 :::"Real":::) means :: FDIFF_3:def 6 (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) "x0" ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" "f" ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))))); end; :: deftheorem defines :::"Rdiff"::: FDIFF_3:def 6 : (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) "iff" (Bool "for" (Set (Var "h")) "being" ($#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"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))))) ")" )))); theorem :: FDIFF_3:9 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "g"))) "iff" (Bool "(" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set "(" (Set (Var "x0")) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "x0")) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) & (Bool "(" "for" (Set (Var "h")) "being" ($#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"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "g"))) ")" )) ")" ) ")" ) ")" ))) ; theorem :: FDIFF_3:10 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: FDIFF_3:11 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: FDIFF_3:12 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ))) ")" ))) ; theorem :: FDIFF_3:13 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2"))) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ))) ; theorem :: FDIFF_3:14 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ))) ; theorem :: FDIFF_3:15 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "," (Set (Var "g1")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "g1"))) "iff" (Bool "(" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "x0")) "," (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set (Var "r")) ")" ) ($#k1_rcomp_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" )) & (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 ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "h")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "g1"))) ")" )) ")" ) ")" ) ")" ))) ; theorem :: FDIFF_3:16 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: FDIFF_3:17 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: FDIFF_3:18 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ))) ")" ))) ; theorem :: FDIFF_3:19 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f1")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2"))) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ))) ; theorem :: FDIFF_3:20 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ))) ; theorem :: FDIFF_3:21 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) ")" ))) ; theorem :: FDIFF_3:22 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_fdiff_3 :::"is_right_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f")) ($#r4_fdiff_3 :::"is_left_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_fdiff_3 :::"Rdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_3 :::"Ldiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) ")" ))) ;