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)) ) ) )
assume A1: 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)) ) )
A2: cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;
A3: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;

now

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 A1, A2, A3, FDIFF_1:29
.= ((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:68
.= (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 A1, A2, A3, FDIFF_1:29; :: thesis: verum