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)) ) ) )
assume A1: 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)) ) )
A2: 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
A3: exp_R is_differentiable_in x by SIN_COS:70;
sin is_differentiable_in exp_R . x by SIN_COS:69;
hence sin * exp_R is_differentiable_in x by A3, FDIFF_2:13; :: thesis: verum

end;

then A4: 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)) )
assume A5: x in Z ; :: thesis: ((sin * exp_R ) `| Z) . x = (exp_R . x) * (cos . (exp_R . x))
A6: exp_R is_differentiable_in x by SIN_COS:70;
sin is_differentiable_in exp_R . x by SIN_COS:69;
then diff (sin * exp_R ),x = (diff sin ,(exp_R . x)) * (diff exp_R ,x) by A6, 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 ;
hence ((sin * exp_R ) `| Z) . x = (exp_R . x) * (cos . (exp_R . x)) by A4, A5, 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 A1, A2, FDIFF_1:16; :: thesis: verum