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

proof

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

end;

assume A2: Z c= dom (cos * exp_R) ; :: thesis: ( cos * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * exp_R) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x))) ) )
then A3: cos * exp_R is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((cos * exp_R) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x)))

proof

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

end;

hence ( cos * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((cos * exp_R) `| Z) . x = - ((exp_R . x) * (sin . (exp_R . x))) ) ) by A2, A1, FDIFF_1:9; :: thesis: verum