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

proof

let x be Real; :: thesis: ( x in Z implies exp_R * exp_R is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * exp_R is_differentiable_in x
A2: exp_R is_differentiable_in exp_R . x by SIN_COS:65;
exp_R is_differentiable_in x by SIN_COS:65;
hence exp_R * exp_R is_differentiable_in x by A2, FDIFF_2:13; :: thesis: verum

end;

assume A3: Z c= dom (exp_R * exp_R) ; :: thesis: ( exp_R * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * exp_R) `| Z) . x = (exp_R . (exp_R . x)) * (exp_R . x) ) )
then A4: exp_R * exp_R is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((exp_R * exp_R) `| Z) . x = (exp_R . (exp_R . x)) * (exp_R . x)

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * exp_R) `| Z) . x = (exp_R . (exp_R . x)) * (exp_R . x) )
assume A5: x in Z ; :: thesis: ((exp_R * exp_R) `| Z) . x = (exp_R . (exp_R . x)) * (exp_R . x)
A6: exp_R is_differentiable_in exp_R . x by SIN_COS:65;
exp_R is_differentiable_in x by SIN_COS:65;
then diff ((exp_R * exp_R),x) = (diff (exp_R,(exp_R . x))) * (diff (exp_R,x)) by A6, FDIFF_2:13
.= (exp_R . (exp_R . x)) * (diff (exp_R,x)) by SIN_COS:65
.= (exp_R . (exp_R . x)) * (exp_R . x) by SIN_COS:65 ;
hence ((exp_R * exp_R) `| Z) . x = (exp_R . (exp_R . x)) * (exp_R . x) by A4, A5, FDIFF_1:def 7; :: thesis: verum

end;

hence ( exp_R * exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * exp_R) `| Z) . x = (exp_R . (exp_R . x)) * (exp_R . x) ) ) by A3, A1, FDIFF_1:9; :: thesis: verum