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) ) ) )
assume A1: 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) ) )
A2: sin is_differentiable_on Z by FDIFF_1:34, SIN_COS:73;
A3: cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;

now

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 A1, A2, A3, FDIFF_1:29
.= ((cos . x) * (cos . x)) + ((sin . x) * (diff cos ,x)) by SIN_COS:69
.= ((cos . x) * (cos . x)) + ((sin . x) * (- (sin . x))) by SIN_COS:68
.= ((cos . x) ^2 ) - ((sin . x) * (sin . x))
.= ((cos x) ^2 ) - ((sin . x) ^2 ) by SIN_COS:def 23
.= ((cos x) ^2 ) - ((sin x) ^2 ) by SIN_COS:def 21
.= 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 A1, A2, A3, FDIFF_1:29; :: thesis: verum