let Z be open Subset of REAL ; :: thesis: ( Z c= dom (exp_R * sin ) implies ( exp_R * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * sin ) `| Z) . x = (exp_R . (sin . x)) * (cos . x) ) ) )
assume A1: Z c= dom (exp_R * sin ) ; :: thesis: ( exp_R * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * sin ) `| Z) . x = (exp_R . (sin . x)) * (cos . x) ) )
A2: for x being Real st x in Z holds
exp_R * sin is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies exp_R * sin is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * sin is_differentiable_in x
A3: sin is_differentiable_in x by SIN_COS:69;
exp_R is_differentiable_in sin . x by SIN_COS:70;
hence exp_R * sin is_differentiable_in x by A3, FDIFF_2:13; :: thesis: verum

end;

then A4: exp_R * sin is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((exp_R * sin ) `| Z) . x = (exp_R . (sin . x)) * (cos . x)

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * sin ) `| Z) . x = (exp_R . (sin . x)) * (cos . x) )
assume A5: x in Z ; :: thesis: ((exp_R * sin ) `| Z) . x = (exp_R . (sin . x)) * (cos . x)
A6: sin is_differentiable_in x by SIN_COS:69;
exp_R is_differentiable_in sin . x by SIN_COS:70;
then diff (exp_R * sin ),x = (diff exp_R ,(sin . x)) * (diff sin ,x) by A6, FDIFF_2:13
.= (diff exp_R ,(sin . x)) * (cos . x) by SIN_COS:69
.= (exp_R . (sin . x)) * (cos . x) by SIN_COS:70 ;
hence ((exp_R * sin ) `| Z) . x = (exp_R . (sin . x)) * (cos . x) by A4, A5, FDIFF_1:def 8; :: thesis: verum

end;

hence ( exp_R * sin is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * sin ) `| Z) . x = (exp_R . (sin . x)) * (cos . x) ) ) by A1, A2, FDIFF_1:16; :: thesis: verum