let Z be open Subset of REAL; :: thesis: ( Z c= dom (exp_R (#) cos) implies ( exp_R (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) cos) `| Z) . x = (exp_R . x) * ((cos . x) - (sin . x)) ) ) )
A1: ( cos is_differentiable_on Z & exp_R is_differentiable_on Z ) by FDIFF_1:26, SIN_COS:67, TAYLOR_1:16;
assume A2: Z c= dom (exp_R (#) cos) ; :: thesis: ( exp_R (#) cos is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) cos) `| Z) . x = (exp_R . x) * ((cos . x) - (sin . x)) ) )

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

let x be Real; :: thesis: ( x in Z implies ((exp_R (#) cos) `| Z) . x = (exp_R . x) * ((cos . x) - (sin . x)) )
assume x in Z ; :: thesis: ((exp_R (#) cos) `| Z) . x = (exp_R . x) * ((cos . x) - (sin . x))
hence ((exp_R (#) cos) `| Z) . x = ((cos . x) * (diff (exp_R,x))) + ((exp_R . x) * (diff (cos,x))) by A2, A1, FDIFF_1:21
.= ((cos . x) * (exp_R . x)) + ((exp_R . x) * (diff (cos,x))) by TAYLOR_1:16
.= ((cos . x) * (exp_R . x)) + ((exp_R . x) * (- (sin . x))) by SIN_COS:63
.= (exp_R . x) * ((cos . x) - (sin . x)) ;
:: thesis: verum

end;

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