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)) ) ) )
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 . (cos . x)) * (sin . x)) ) )
A2: 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
A3: cos is_differentiable_in x by SIN_COS:68;
exp_R is_differentiable_in cos . x by SIN_COS:70;
hence exp_R * cos is_differentiable_in x by A3, FDIFF_2:13; :: thesis: verum

end;

then A4: exp_R * cos is_differentiable_on Z by A1, FDIFF_1:16;
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)) )
assume A5: x in Z ; :: thesis: ((exp_R * cos ) `| Z) . x = - ((exp_R . (cos . x)) * (sin . x))
A6: cos is_differentiable_in x by SIN_COS:68;
exp_R is_differentiable_in cos . x by SIN_COS:70;
then diff (exp_R * cos ),x = (diff exp_R ,(cos . x)) * (diff cos ,x) by A6, FDIFF_2:13
.= (diff exp_R ,(cos . x)) * (- (sin . x)) by SIN_COS:68
.= (exp_R . (cos . x)) * (- (sin . x)) by SIN_COS:70
.= - ((exp_R . (cos . x)) * (sin . x)) ;
hence ((exp_R * cos ) `| Z) . x = - ((exp_R . (cos . x)) * (sin . x)) by A4, A5, FDIFF_1:def 8; :: 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 A1, A2, FDIFF_1:16; :: thesis: verum