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))) ) ) )
assume A1: 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))) ) )
A2: 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
A3: exp_R is_differentiable_in x by SIN_COS:70;
cos is_differentiable_in exp_R . x by SIN_COS:68;
hence cos * exp_R is_differentiable_in x by A3, FDIFF_2:13; :: thesis: verum

end;

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