:: FDIFF_10 semantic presentation begin theorem :: FDIFF_10:1 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:2 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:3 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:4 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )) ($#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 ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:5 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:6 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:7 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:8 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:9 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:10 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:11 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )) ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:12 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )) ($#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 ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#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 (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:13 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:14 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )) ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:15 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:16 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#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 ($#k3_taylor_1 :::"ln"::: ) ) ($#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 "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:17 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:18 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) )) ($#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 ($#k3_taylor_1 :::"ln"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:19 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (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 ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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 "(" (Set "(" (Set ($#k3_taylor_1 :::"ln"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:20 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:21 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k21_sin_cos :::"cos"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:22 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k21_sin_cos :::"cos"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:23 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k18_sin_cos :::"sin"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:24 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_sin_cos :::"sin"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:25 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#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 ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:26 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:27 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:28 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:29 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#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 ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:30 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:31 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_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 ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:32 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k47_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 ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:33 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k47_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 ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:34 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_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 ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:35 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#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 "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:36 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:37 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:38 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:39 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#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 ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:40 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:41 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:42 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:43 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) )) ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:44 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:45 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_10:46 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" )))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ;