let Z be open Subset of REAL; :: thesis: ( Z c= dom (sin (#) cos) implies ( sin (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) cos) `| Z) . x = cos (2 * x) ) ) )
A1: ( sin is_differentiable_on Z & cos is_differentiable_on Z ) by FDIFF_1:26, SIN_COS:67, SIN_COS:68;
assume A2: Z c= dom (sin (#) cos) ; :: thesis: ( sin (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) cos) `| Z) . x = cos (2 * x) ) )

now :: thesis: for x being Real st x in Z holds
((sin (#) cos) `| Z) . x = cos (2 * x)

let x be Real; :: thesis: ( x in Z implies ((sin (#) cos) `| Z) . x = cos (2 * x) )
assume x in Z ; :: thesis: ((sin (#) cos) `| Z) . x = cos (2 * x)
hence ((sin (#) cos) `| Z) . x = ((cos . x) * (diff (sin,x))) + ((sin . x) * (diff (cos,x))) by A2, A1, FDIFF_1:21
.= ((cos . x) * (cos . x)) + ((sin . x) * (diff (cos,x))) by SIN_COS:64
.= ((cos . x) * (cos . x)) + ((sin . x) * (- (sin . x))) by SIN_COS:63
.= ((cos . x) ^2) - ((sin . x) * (sin . x))
.= ((cos x) ^2) - ((sin . x) ^2) by SIN_COS:def 19
.= ((cos x) ^2) - ((sin x) ^2) by SIN_COS:def 17
.= cos (2 * x) by SIN_COS5:7 ;
:: thesis: verum

end;

hence ( sin (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((sin (#) cos) `| Z) . x = cos (2 * x) ) ) by A2, A1, FDIFF_1:21; :: thesis: verum