:: FDIFF_2 semantic presentation begin registrationlet "h" be ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v1_fdiff_1 :::"-convergent"::: ) ($#m1_subset_1 :::"Real_Sequence":::); cluster (Set ($#k30_valued_1 :::"-"::: ) "h") -> ($#v2_relat_1 :::"non-zero"::: ) ($#v2_comseq_2 :::"convergent"::: ) ; end; theorem :: FDIFF_2:1 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "a")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set (Var "b")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "b")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "d")) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "d")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "b")) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Var "d")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "d"))) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "a")))) ")" )) ; theorem :: FDIFF_2:2 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "a")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")))) ")" )) "holds" (Bool (Set (Var "a")) "is" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ))) ; theorem :: FDIFF_2:3 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "a")) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k4_nat_1 :::"*"::: ) (Set (Var "n")) ")" ) ($#k2_nat_1 :::"+"::: ) (Num 1))) ")" )) "holds" (Bool (Set (Var "a")) "is" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ))) ; theorem :: FDIFF_2:4 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (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" bbbadV3_FUNCT_1() ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x0")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "c")) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c"))) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) ")" )))) ; theorem :: FDIFF_2:5 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "b"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ))) "holds" (Bool (Set (Var "a")) ($#r2_relset_1 :::"="::: ) (Set (Var "b"))))) ; theorem :: FDIFF_2:6 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "h")) "being" ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v1_fdiff_1 :::"-convergent"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set (Var "a")) "is" ($#m2_valued_0 :::"subsequence"::: ) "of" (Set (Var "h")))) "holds" (Bool (Set (Var "a")) "is" ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v1_fdiff_1 :::"-convergent"::: ) ($#m1_subset_1 :::"Real_Sequence":::)))) ; theorem :: FDIFF_2:7 (Bool "for" (Set (Var "g")) "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 "(" "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" bbbadV3_FUNCT_1() ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "g")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "g")) ($#k1_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#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"::: ) )) ")" )) "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" bbbadV3_FUNCT_1() ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "g")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "g")) ($#k1_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#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_2:8 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "r")) "st" (Bool (Set (Var "N")) ($#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"::: ) ($#m1_subset_1 :::"Real_Sequence":::)(Bool "ex" (Set (Var "c")) "being" bbbadV3_FUNCT_1() ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#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 (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ))))) ; theorem :: FDIFF_2:9 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" )))) "holds" (Bool "(" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "a"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f1")) ($#k8_funct_2 :::"/*"::: ) (Set (Var "a")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")))) ")" ))) ; scheme :: FDIFF_2:sch 1 ExIncSeqofNat{ F1() -> ($#m1_subset_1 :::"Real_Sequence":::), P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "q")) "being" bbbadV5_VALUED_0() ($#m1_subset_1 :::"sequence":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool P1[(Set (Set "(" (Set F1 "(" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "q")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n")))]) ")" ) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set F1 "(" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))))) "holds" (Bool P1[(Set (Var "r"))]) ")" )) "holds" (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_recdef_1 :::"."::: ) (Set (Var "m"))))) ")" ) ")" )) provided (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool P1[(Set (Set F1 "(" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "m")))]) ")" ))) proof end; theorem :: FDIFF_2:10 (Bool "for" (Set (Var "x0")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set (Var "r"))) & (Bool (Set (Var "f")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#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 (Var "N")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "g"))) ($#r1_hidden :::"<>"::: ) (Set (Var "r"))) ")" ) ")" )))) ; theorem :: FDIFF_2:11 (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"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#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" bbbadV3_FUNCT_1() ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#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_2:12 (Bool "for" (Set (Var "x0")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "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")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "g"))) "iff" (Bool "(" (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#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" bbbadV3_FUNCT_1() ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#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 ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "h")) ($#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_2:13 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set "(" (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ))) ")" ))) ; theorem :: FDIFF_2:14 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f2")) "," (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "f1")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set (Var "f2")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2"))) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "x0")) ")" ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "x0")) ")" ")" ) ($#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_2:15 (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 (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "f")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ))) ; theorem :: FDIFF_2:16 (Bool "for" (Set (Var "A")) "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_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "A")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")))) ")" ))) ; theorem :: FDIFF_2:17 (Bool "for" (Set (Var "A")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f2")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ))) ")" ))) ; theorem :: FDIFF_2:18 (Bool "for" (Set (Var "A")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f2")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ))) ")" ))) ; theorem :: FDIFF_2:19 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "A")) "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_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ))) ")" )))) ; theorem :: FDIFF_2:20 (Bool "for" (Set (Var "A")) "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_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f2")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ) ")" ))) ")" ))) ; theorem :: FDIFF_2:21 (Bool "for" (Set (Var "A")) "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_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Var "f2")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (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"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "f1")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "f2")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f2")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ))) ")" ))) ; theorem :: FDIFF_2:22 (Bool "for" (Set (Var "A")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (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 :::"^"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A"))) ($#r2_relset_1 :::"="::: ) (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ))) ")" ))) ; theorem :: FDIFF_2:23 (Bool "for" (Set (Var "A")) "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_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "f1")) ($#k7_relset_1 :::".:"::: ) (Set (Var "A"))) "is" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool (Set (Var "f2")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Set (Var "f1")) ($#k7_relset_1 :::".:"::: ) (Set (Var "A"))))) "holds" (Bool "(" (Bool (Set (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool (Set (Set "(" (Set (Var "f2")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f2")) ($#k2_fdiff_1 :::"`|"::: ) (Set "(" (Set (Var "f1")) ($#k7_relset_1 :::".:"::: ) (Set (Var "A")) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f1")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "A")) ")" ))) ")" ))) ; theorem :: FDIFF_2:24 (Bool "for" (Set (Var "A")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "p")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "r")) ($#k9_real_1 :::"-"::: ) (Set (Var "p")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "A"))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" ))) ; theorem :: FDIFF_2:25 (Bool "for" (Set (Var "p")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "r2")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "r1")) ($#k9_real_1 :::"-"::: ) (Set (Var "r2")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" ) & (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) )) "is" bbbadV3_FUNCT_1()) ")" ))) ; theorem :: FDIFF_2:26 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")))) & (Bool (Set (Var "r2")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "r1")) ($#k9_real_1 :::"-"::: ) (Set (Var "r2")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )) "is" bbbadV3_FUNCT_1()) ")" ))) ; theorem :: FDIFF_2:27 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "r1")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")))) & (Bool (Set (Var "r2")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "r1")) ($#k9_real_1 :::"-"::: ) (Set (Var "r2")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )) "is" bbbadV3_FUNCT_1()) ")" ))) ; theorem :: FDIFF_2:28 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool "(" "for" (Set (Var "r1")) "," (Set (Var "r2")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r1")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "r2")) ")" ) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "r1")) ($#k9_real_1 :::"-"::: ) (Set (Var "r2")) ")" ) ($#k5_square_1 :::"^2"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" )) "is" bbbadV3_FUNCT_1()) ")" )) ; theorem :: FDIFF_2:29 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v5_valued_0 :::"increasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ))) ; theorem :: FDIFF_2:30 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v6_valued_0 :::"decreasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ))) ; theorem :: FDIFF_2:31 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v7_valued_0 :::"non-decreasing"::: ) ))) ; theorem :: FDIFF_2:32 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v8_valued_0 :::"non-increasing"::: ) ))) ; theorem :: FDIFF_2:33 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v5_valued_0 :::"increasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ))) ; theorem :: FDIFF_2:34 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v6_valued_0 :::"decreasing"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ))) ; theorem :: FDIFF_2:35 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v7_valued_0 :::"non-decreasing"::: ) ))) ; theorem :: FDIFF_2:36 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "r")) ")" )) "is" ($#v8_valued_0 :::"non-increasing"::: ) ))) ; theorem :: FDIFF_2:37 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" )) "is" ($#v5_valued_0 :::"increasing"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ; theorem :: FDIFF_2:38 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" )) "is" ($#v6_valued_0 :::"decreasing"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" )) ; theorem :: FDIFF_2:39 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" )) "is" ($#v7_valued_0 :::"non-decreasing"::: ) )) ; theorem :: FDIFF_2:40 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" )) "is" ($#v8_valued_0 :::"non-increasing"::: ) )) ; theorem :: FDIFF_2:41 (Bool "for" (Set (Var "p")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ))) "or" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ) ")" )) "is" ($#v3_rcomp_1 :::"open"::: ) ))) ; theorem :: FDIFF_2:42 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p")))) & (Bool "(" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ))) "or" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p")) ")" ) ")" )) "is" ($#v3_rcomp_1 :::"open"::: ) ))) ; theorem :: FDIFF_2:43 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p")))) & (Bool "(" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ))) "or" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p")) ")" ) ")" )) "is" ($#v3_rcomp_1 :::"open"::: ) ))) ; theorem :: FDIFF_2:44 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) & (Bool "(" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ))) "or" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) "holds" (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) "is" ($#v3_rcomp_1 :::"open"::: ) )) ; theorem :: FDIFF_2:45 (Bool "for" (Set (Var "f")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_subset_1 :::"[#]"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) & (Bool "(" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ))) "or" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun2 :::"""::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun2 :::"""::: ) ")" ))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun2 :::"""::: ) ")" )))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f")) ($#k2_partfun2 :::"""::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun2 :::"""::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ")" ))) ")" ) ")" )) ; theorem :: FDIFF_2:46 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p")))) & (Bool "(" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ))) "or" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p")) ")" )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ")" )))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k10_prob_1 :::"left_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_2:47 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p")))) & (Bool "(" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ))) "or" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p")) ")" )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ")" )))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k3_limfunc1 :::"right_open_halfline"::: ) (Set (Var "p")) ")" ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_2:48 (Bool "for" (Set (Var "p")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#v2_funct_1 :::"one-to-one"::: ) ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ))) "or" (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) )) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) ")" )))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ) "," (Set (Var "x0")) ")" ) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set "(" (Set "(" (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "g")) ($#k2_rcomp_1 :::".["::: ) ) ")" ) ($#k2_partfun2 :::"""::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_2:49 (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")) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0")))) "holds" (Bool "for" (Set (Var "h")) "being" ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v1_fdiff_1 :::"-convergent"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "c")) "being" bbbadV3_FUNCT_1() ($#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 (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "h")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "h")) ")" ) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "c")) ($#k3_valued_1 :::"+"::: ) (Set (Var "h")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "c")) ($#k47_valued_1 :::"-"::: ) (Set (Var "h")) ")" ) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "h")) ")" ) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "c")) ($#k3_valued_1 :::"+"::: ) (Set (Var "h")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "c")) ($#k47_valued_1 :::"-"::: ) (Set (Var "h")) ")" ) ")" ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x0")) ")" )) ")" ))))) ;