:: HFDIFF_1 semantic presentation begin theorem :: HFDIFF_1:1 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "Z"))))) ; theorem :: HFDIFF_1:2 (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "f1")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))))) ; theorem :: HFDIFF_1:3 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k4_xboole_0 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ))) & (Bool (Set (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ")" )) ; theorem :: HFDIFF_1:4 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set (Var "p")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "p")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ))))) ; theorem :: HFDIFF_1:5 (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "n")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "m")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Set (Var "m")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f1")))))) ; theorem :: HFDIFF_1:6 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))))) ; theorem :: HFDIFF_1:7 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))) "holds" (Bool (Set (Var "f1")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z")))))) ; theorem :: HFDIFF_1:8 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) ; theorem :: HFDIFF_1:9 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set ($#k16_sin_cos :::"sin"::: ) ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))))) ; theorem :: HFDIFF_1:10 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set ($#k16_sin_cos :::"sin"::: ) ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 3) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: HFDIFF_1:11 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set ($#k16_sin_cos :::"sin"::: ) ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )))))) ; theorem :: HFDIFF_1:12 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))))) ; theorem :: HFDIFF_1:13 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 3) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")))))) ; theorem :: HFDIFF_1:14 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )))))) ; theorem :: HFDIFF_1:15 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" )))))) ; theorem :: HFDIFF_1:16 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" )))))) ; theorem :: HFDIFF_1:17 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "i"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "i")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "i")) ")" )))))) ; theorem :: HFDIFF_1:18 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "n")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "i"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "i")) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "i")) ")" )))))) ; theorem :: HFDIFF_1:19 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))))) ; theorem :: HFDIFF_1:20 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))))) ; theorem :: HFDIFF_1:21 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ))))))) ; theorem :: HFDIFF_1:22 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))))))) ; theorem :: HFDIFF_1:23 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 1)) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")))))) ; theorem :: HFDIFF_1:24 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 1)) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))))))) ; theorem :: HFDIFF_1:25 (Bool "for" (Set (Var "x")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" )))))) ; theorem :: HFDIFF_1:26 (Bool "for" (Set (Var "x")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" )))))) ; theorem :: HFDIFF_1:27 (Bool "for" (Set (Var "x")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" )))))) ; theorem :: HFDIFF_1:28 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")))))) ; theorem :: HFDIFF_1:29 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k5_prepower :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ))) ; theorem :: HFDIFF_1:30 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")))))) ; theorem :: HFDIFF_1:31 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 2))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 3)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 2) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 3) ")" ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")))))) ; theorem :: HFDIFF_1:32 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "m"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set (Var "m")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k3_newton :::"!"::: ) ")" ) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Set (Var "m")) ")" ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")))))) ; theorem :: HFDIFF_1:33 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n"))) ($#r2_relset_1 :::"="::: ) (Set ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ))) & (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set (Var "f"))) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))) ")" )))) ; theorem :: HFDIFF_1:34 (Bool "for" (Set (Var "x0")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_taylor_1 :::"Taylor"::: ) "(" (Set ($#k16_sin_cos :::"sin"::: ) ) "," (Set (Var "Z")) "," (Set (Var "x0")) "," (Set (Var "x")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k3_newton :::"!"::: ) ")" ))) & (Bool (Set (Set "(" ($#k6_taylor_1 :::"Taylor"::: ) "(" (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set (Var "Z")) "," (Set (Var "x0")) "," (Set (Var "x")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x0")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k3_newton :::"!"::: ) ")" ))) ")" )))) ; theorem :: HFDIFF_1:35 (Bool "for" (Set (Var "r")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_taylor_2 :::"Maclaurin"::: ) "(" (Set ($#k16_sin_cos :::"sin"::: ) ) "," (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "r")) ($#k2_rcomp_1 :::".["::: ) ) "," (Set (Var "x")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k3_newton :::"!"::: ) ")" ))) & (Bool (Set (Set "(" ($#k1_taylor_2 :::"Maclaurin"::: ) "(" (Set ($#k19_sin_cos :::"cos"::: ) ) "," (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "r")) ($#k2_rcomp_1 :::".["::: ) ) "," (Set (Var "x")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set ($#k32_sin_cos :::"PI"::: ) ) ")" ) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k2_newton :::"|^"::: ) (Set (Var "n")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "n")) ($#k3_newton :::"!"::: ) ")" ))) ")" ))) ; theorem :: HFDIFF_1:36 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "m"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "m")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set (Var "m")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k3_newton :::"!"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Set (Var "m")) ")" ) ")" )))))) ; theorem :: HFDIFF_1:37 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "m")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "m")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k3_newton :::"!"::: ) ))))) ; theorem :: HFDIFF_1:38 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n"))) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Set (Var "n")) "," (Set (Var "Z"))))) ; theorem :: HFDIFF_1:39 (Bool "for" (Set (Var "x")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "m")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set (Var "m")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "m")) ($#k3_newton :::"!"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Set (Var "m")) ")" ) ")" )))))) ; theorem :: HFDIFF_1:40 (Bool "for" (Set (Var "x")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Set (Var "n")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k3_newton :::"!"::: ) ")" )))))) ; theorem :: HFDIFF_1:41 (Bool "for" (Set (Var "x0")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "m")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_taylor_1 :::"Taylor"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) "," (Set (Var "Z")) "," (Set (Var "x0")) "," (Set (Var "x")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "n")) ($#k6_newton :::"choose"::: ) (Set (Var "m")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "x0")) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Set (Var "m")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k2_newton :::"|^"::: ) (Set (Var "m")) ")" ))) & (Bool (Set (Set "(" ($#k6_taylor_1 :::"Taylor"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) "," (Set (Var "Z")) "," (Set (Var "x0")) "," (Set (Var "x")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ($#k2_newton :::"|^"::: ) (Set (Var "n")))) ")" )))) ; theorem :: HFDIFF_1:42 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "m"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_taylor_2 :::"Maclaurin"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) "," (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "r")) ($#k2_rcomp_1 :::".["::: ) ) "," (Set (Var "x")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "m"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" ($#k1_taylor_2 :::"Maclaurin"::: ) "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) "," (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set (Var "r")) ")" ) "," (Set (Var "r")) ($#k2_rcomp_1 :::".["::: ) ) "," (Set (Var "x")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k2_newton :::"|^"::: ) (Set (Var "n")))) ")" ))) ; theorem :: HFDIFF_1:43 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))))) "holds" (Bool (Set (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))))) ; theorem :: HFDIFF_1:44 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set (Var "n")) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" ) ")" )))))) ; theorem :: HFDIFF_1:45 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k6_partfun1 :::"id"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" )) ; theorem :: HFDIFF_1:46 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z"))))) ; theorem :: HFDIFF_1:47 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x"))) & (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k5_prepower :::"#Z"::: ) (Num 2) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" )) ; theorem :: HFDIFF_1:48 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 2) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Num 3) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z"))))) ; theorem :: HFDIFF_1:49 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")))))) ; theorem :: HFDIFF_1:50 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Num 2) "," (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Num 2) "," (Set (Var "Z")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2) ")" ) ")" ))))) ; theorem :: HFDIFF_1:51 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" )))) "holds" (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k6_partfun1 :::"id"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z"))))) ; theorem :: HFDIFF_1:52 (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))) & (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))))) "holds" (Bool (Set (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k4_nat_1 :::"*"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 2) ")" ) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )))))) ; theorem :: HFDIFF_1:53 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" ) ")" )))) ; theorem :: HFDIFF_1:54 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 2) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" ) ")" )))) ; theorem :: HFDIFF_1:55 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Num 4) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" )))) ; theorem :: HFDIFF_1:56 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k29_sin_cos :::"tan"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k41_valued_1 :::"^2"::: ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")))) ")" )) ; theorem :: HFDIFF_1:57 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")))) ")" )) ; theorem :: HFDIFF_1:58 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set ($#k29_sin_cos :::"tan"::: ) ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" )))) ; theorem :: HFDIFF_1:59 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k30_sin_cos :::"cot"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")))) ")" )) ; theorem :: HFDIFF_1:60 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k32_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")))) ")" )) ; theorem :: HFDIFF_1:61 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set ($#k30_sin_cos :::"cot"::: ) ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" )))) ; theorem :: HFDIFF_1:62 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" )))) ; theorem :: HFDIFF_1:63 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" )))) ; theorem :: HFDIFF_1:64 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Num 3) "," (Set (Var "Z"))) & (Bool (Set (Var "f2")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Num 3) "," (Set (Var "Z")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 3)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 3) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set "(" (Num 3) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f1")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Num 3) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2) ")" ) ")" ) ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f2")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 3) ")" ) ")" ))))) ; theorem :: HFDIFF_1:65 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 3)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Num 8) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z")) ")" )))) ; theorem :: HFDIFF_1:66 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Num 2) "," (Set (Var "Z")))) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f")) ($#k41_valued_1 :::"^2"::: ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2) ")" ) ")" ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k41_valued_1 :::"^2"::: ) ")" ) ")" ))))) ; theorem :: HFDIFF_1:67 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_taylor_1 :::"is_differentiable_on"::: ) (Num 2) "," (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x0"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set (Var "f")) ($#k6_rfunct_1 :::"^"::: ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 2) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ))))) ; theorem :: HFDIFF_1:68 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_taylor_1 :::"diff"::: ) "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) "," (Set (Var "Z")) ")" ")" ) ($#k1_seqfunc :::"."::: ) (Num 3)) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k32_valued_1 :::"-"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k3_valued_1 :::"+"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ")" ) ($#k5_relset_1 :::"|"::: ) (Set (Var "Z"))))) ;