:: FDIFF_5 semantic presentation begin theorem :: FDIFF_5:1 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k7_real_1 :::"+"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )))) ; theorem :: FDIFF_5:2 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (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 "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set (Var "a")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )))) ; theorem :: FDIFF_5:3 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "a")))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")))) & (Bool (Set (Set (Var "f2")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2"))) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k3_rfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "b")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )))) ; theorem :: FDIFF_5:4 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Bool (Set (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:5 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:6 (Bool "for" (Set (Var "Z")) "being" ($#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 "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:7 (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 ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:8 (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 ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:9 (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 ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:10 (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 "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")))) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )))) ; theorem :: FDIFF_5:11 (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 "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")))) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )))) ; theorem :: FDIFF_5:12 (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 "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")))) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ")" ))) ")" ) ")" )))) ; theorem :: FDIFF_5:13 (Bool "for" (Set (Var "n")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Num 1)) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set (Var "x")))) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set (Var "f")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "n")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k9_real_1 :::"-"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "n")) ($#k7_real_1 :::"+"::: ) (Num 1) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )))) ; theorem :: FDIFF_5:14 (Bool "for" (Set (Var "Z")) "being" ($#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 "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:15 (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"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:16 (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 ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k10_prepower :::"#R"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:17 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2)))) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_5:18 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2)))) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_5:19 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) )))) "holds" (Bool "(" (Bool (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")))) ")" ) ")" )) ; theorem :: FDIFF_5:20 (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 ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:21 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "x")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_5:22 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" ) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 4) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k2_newton :::"|^"::: ) (Num 2) ")" ) ")" ) ($#k2_newton :::"|^"::: ) (Num 2) ")" ))) ")" ) ")" )))) ; theorem :: FDIFF_5:23 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "a"))) ")" ) & (Bool (Set (Var "f2")) ($#r1_hidden :::"="::: ) (Set ($#k1_taylor_1 :::"#Z"::: ) (Num 2))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k3_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k47_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Num 4) ($#k8_real_1 :::"*"::: ) (Set (Var "a")) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k2_newton :::"|^"::: ) (Num 2) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k2_newton :::"|^"::: ) (Num 4) ")" ) ")" ))) ")" ) ")" )))) ; theorem :: FDIFF_5:24 (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 ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "x")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_taylor_1 :::"ln"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_5:25 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set (Var "x")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ;