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

proof

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

end;

assume A2: Z c= dom (sin * exp_R) ; :: thesis: ( sin * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((sin * exp_R) `| Z) . x = (exp_R . x) * (cos . (exp_R . x)) ) )
then A3: sin * exp_R is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((sin * exp_R) `| Z) . x = (exp_R . x) * (cos . (exp_R . x))

proof

let x be Real; :: thesis: ( x in Z implies ((sin * exp_R) `| Z) . x = (exp_R . x) * (cos . (exp_R . x)) )
( exp_R is_differentiable_in x & sin is_differentiable_in exp_R . x ) by SIN_COS:69, SIN_COS:70;
then A4: diff ((sin * exp_R),x) = (diff (sin,(exp_R . x))) * (diff (exp_R,x)) by FDIFF_2:13
.= (cos . (exp_R . x)) * (diff (exp_R,x)) by SIN_COS:69
.= (exp_R . x) * (cos . (exp_R . x)) by SIN_COS:70 ;
assume x in Z ; :: thesis: ((sin * exp_R) `| Z) . x = (exp_R . x) * (cos . (exp_R . x))
hence ((sin * exp_R) `| Z) . x = (exp_R . x) * (cos . (exp_R . x)) by A3, A4, FDIFF_1:def 8; :: thesis: verum

end;

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