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 . (cos . x)) * (sin . x)) ) ) )
A1: for x being Real st x in Z holds
exp_R * cos is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies exp_R * cos is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * cos is_differentiable_in x
( cos is_differentiable_in x & exp_R is_differentiable_in cos . x ) by SIN_COS:63, SIN_COS:65;
hence exp_R * cos is_differentiable_in x by FDIFF_2:13; :: thesis: verum

end;

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 . (cos . x)) * (sin . x)) ) )
then A3: exp_R * cos is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((exp_R * cos) `| Z) . x = - ((exp_R . (cos . x)) * (sin . x))

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * cos) `| Z) . x = - ((exp_R . (cos . x)) * (sin . x)) )
( cos is_differentiable_in x & exp_R is_differentiable_in cos . x ) by SIN_COS:63, SIN_COS:65;
then A4: diff ((exp_R * cos),x) = (diff (exp_R,(cos . x))) * (diff (cos,x)) by FDIFF_2:13
.= (diff (exp_R,(cos . x))) * (- (sin . x)) by SIN_COS:63
.= (exp_R . (cos . x)) * (- (sin . x)) by SIN_COS:65
.= - ((exp_R . (cos . x)) * (sin . x)) ;
assume x in Z ; :: thesis: ((exp_R * cos) `| Z) . x = - ((exp_R . (cos . x)) * (sin . x))
hence ((exp_R * cos) `| Z) . x = - ((exp_R . (cos . x)) * (sin . x)) by A3, A4, FDIFF_1:def 7; :: 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 . (cos . x)) * (sin . x)) ) ) by A2, A1, FDIFF_1:9; :: thesis: verum