let Z be open Subset of REAL; :: thesis: ( Z c= dom (arctan * sin) & ( for x being Real st x in Z holds
( sin . x > - 1 & sin . x < 1 ) ) implies ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) ) )
assume that
A1: Z c= dom (arctan * sin) and
A2: for x being Real st x in Z holds
( sin . x > - 1 & sin . x < 1 ) ; :: thesis: ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) )
A3: for x being Real st x in Z holds
arctan * sin is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies arctan * sin is_differentiable_in x )
assume x in Z ; :: thesis: arctan * sin is_differentiable_in x
then A4: ( sin . x > - 1 & sin . x < 1 ) by A2;
sin is_differentiable_in x by SIN_COS:64;
hence arctan * sin is_differentiable_in x by A4, SIN_COS9:85; :: thesis: verum

end;

then A5: arctan * sin is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2))

proof

let x be Real; :: thesis: ( x in Z implies ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) )
A6: sin is_differentiable_in x by SIN_COS:64;
assume A7: x in Z ; :: thesis: ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2))
then A8: ( sin . x > - 1 & sin . x < 1 ) by A2;
((arctan * sin) `| Z) . x = diff ((arctan * sin),x) by A5, A7, FDIFF_1:def 7
.= (diff (sin,x)) / (1 + ((sin . x) ^2)) by A6, A8, SIN_COS9:85
.= (cos . x) / (1 + ((sin . x) ^2)) by SIN_COS:64 ;
hence ((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ; :: thesis: verum

end;

hence ( arctan * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((arctan * sin) `| Z) . x = (cos . x) / (1 + ((sin . x) ^2)) ) ) by A1, A3, FDIFF_1:9; :: thesis: verum