:: FDIFF_11 semantic presentation begin theorem :: FDIFF_11:1 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ))) & (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 ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:2 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k16_sin_cos :::"sin"::: ) ) ")" ))) & (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 ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:3 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:4 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k19_sin_cos :::"cos"::: ) ) ")" ))) & (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 ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:5 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ))) & (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 ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" )) ; theorem :: FDIFF_11:6 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) ")" ) ")" )) ; theorem :: FDIFF_11:7 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) ")" ) ")" )) ; theorem :: FDIFF_11:8 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k30_sin_cos :::"cot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Num 1)) ")" ) ")" )) ; theorem :: FDIFF_11:9 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_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 "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:10 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#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 "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:11 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#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 "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:12 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:13 (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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:14 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:15 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:16 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:17 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:18 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:19 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:20 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:21 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:22 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:23 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:24 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:25 (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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#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")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:26 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:27 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:28 (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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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 "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:29 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#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"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:30 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:31 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:32 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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"::: ) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:33 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#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")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:34 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#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")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:35 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) )) ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#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"::: ) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:36 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:37 (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 ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) ; theorem :: FDIFF_11:38 (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 ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:39 (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 ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:40 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#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")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:41 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#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"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:42 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:43 (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 ($#k1_relset_1 :::"dom"::: ) (Set ($#k29_sin_cos :::"tan"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:44 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k29_sin_cos :::"tan"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k29_sin_cos :::"tan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:45 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k30_sin_cos :::"cot"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:46 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k30_sin_cos :::"cot"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k30_sin_cos :::"cot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:47 (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 ($#k1_relset_1 :::"dom"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#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_11:48 (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 ($#k1_relset_1 :::"dom"::: ) (Set ($#k1_fdiff_9 :::"sec"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k1_fdiff_9 :::"sec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#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"::: ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k1_fdiff_9 :::"sec"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:49 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#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_11:50 (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 ($#k1_relset_1 :::"dom"::: ) (Set ($#k2_fdiff_9 :::"cosec"::: ) ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k2_fdiff_9 :::"cosec"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:51 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:52 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#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")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:53 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k24_sin_cos :::"exp_R"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:54 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#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 "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:55 (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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) ; theorem :: FDIFF_11:56 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k24_sin_cos :::"exp_R"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:57 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) ; theorem :: FDIFF_11:58 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:59 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k3_valued_1 :::"+"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" ) ")" )) ; theorem :: FDIFF_11:60 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k19_sin_cos :::"cos"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#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_sin_cos9 :::"arctan"::: ) ) ($#k47_valued_1 :::"-"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Num 2) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k16_sin_cos :::"sin"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:61 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) )) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:62 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#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 "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:63 (Bool "for" (Set (Var "Z")) "being" ($#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")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#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 "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#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"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "x")) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:64 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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")) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:65 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool (Set (Var "g")) ($#r2_relset_1 :::"="::: ) (Set ($#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 "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#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")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_11:66 (Bool "for" (Set (Var "Z")) "being" ($#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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool (Set (Var "g")) ($#r2_relset_1 :::"="::: ) (Set ($#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 "(" (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k1_real_1 :::"-"::: ) (Num 1))) & (Bool (Set (Set "(" (Set "(" ($#k1_partfun2 :::"id"::: ) (Set (Var "Z")) ")" ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<"::: ) (Num 1)) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Set (Var "g")) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#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")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "x")) ")" ) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_11:67 (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_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 ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#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_sin_cos9 :::"arctan"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#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 "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:68 (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 ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ) ($#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 ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_square_1 :::"^2"::: ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:69 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#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 "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ) ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_11:70 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#v3_rcomp_1 :::"open"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" )) ($#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 "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Set (Var "n")) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k1_taylor_1 :::"#Z"::: ) (Set (Var "n")) ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Num 1) ($#k10_real_1 :::"/"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_prepower :::"#Z"::: ) (Set "(" (Set (Var "n")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" ))) ; theorem :: FDIFF_11:71 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_prepower :::"#R"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:72 (Bool "for" (Set (Var "Z")) "being" ($#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 "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#r2_fdiff_1 :::"is_differentiable_on"::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set "(" (Set "(" (Num 2) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_prepower :::"#R"::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:73 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) )) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )) "holds" (Bool "(" (Bool (Set (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" )) ($#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 "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set ($#k1_sin_cos9 :::"arctan"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_prepower :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ))) ")" ) ")" )) ; theorem :: FDIFF_11:74 (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 ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ))) & (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set "(" ($#k1_real_1 :::"-"::: ) (Num 1) ")" ) "," (Num 1) ($#k2_rcomp_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Set "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" )) ($#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 "(" (Num 2) ($#k10_real_1 :::"/"::: ) (Num 3) ")" ) ($#k26_valued_1 :::"(#)"::: ) (Set "(" (Set "(" ($#k4_taylor_1 :::"#R"::: ) (Set "(" (Num 3) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ")" ) ")" ) ($#k2_fdiff_1 :::"`|"::: ) (Set (Var "Z")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_real_1 :::"-"::: ) (Set "(" (Set "(" (Set "(" (Set ($#k2_sin_cos9 :::"arccot"::: ) ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_prepower :::"#R"::: ) (Set "(" (Num 1) ($#k10_real_1 :::"/"::: ) (Num 2) ")" ) ")" ) ($#k10_real_1 :::"/"::: ) (Set "(" (Num 1) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "x")) ($#k5_square_1 :::"^2"::: ) ")" ) ")" ) ")" ))) ")" ) ")" )) ;