let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom (exp_R * f) & ( for x being Real st x in Z holds
f . x = x * (log number_e ,a) ) & a > 0 holds
( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f) `| Z) . x = (a #R x) * (log number_e ,a) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (exp_R * f) & ( for x being Real st x in Z holds
f . x = x * (log number_e ,a) ) & a > 0 holds
( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f) `| Z) . x = (a #R x) * (log number_e ,a) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (exp_R * f) & ( for x being Real st x in Z holds
f . x = x * (log number_e ,a) ) & a > 0 implies ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f) `| Z) . x = (a #R x) * (log number_e ,a) ) ) )
assume that
A1: Z c= dom (exp_R * f) and
A2: for x being Real st x in Z holds
f . x = x * (log number_e ,a) and
A3: a > 0 ; :: thesis: ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f) `| Z) . x = (a #R x) * (log number_e ,a) ) )
for y being set st y in Z holds
y in dom f by A1, FUNCT_1:21;
then A4: Z c= dom f by TARSKI:def 3;
A5: for x being Real st x in Z holds
f . x = ((log number_e ,a) * x) + 0 by A2;
then A6: f is_differentiable_on Z by A4, FDIFF_1:31;
A7: for x being Real st x in Z holds
exp_R * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies exp_R * f is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * f is_differentiable_in x
then f is_differentiable_in x by A6, FDIFF_1:16;
hence exp_R * f is_differentiable_in x by TAYLOR_1:19; :: thesis: verum

end;

then A8: exp_R * f is_differentiable_on Z by A1, FDIFF_1:16;
for x being Real st x in Z holds
((exp_R * f) `| Z) . x = (a #R x) * (log number_e ,a)

proof

let x be Real; :: thesis: ( x in Z implies ((exp_R * f) `| Z) . x = (a #R x) * (log number_e ,a) )
assume A9: x in Z ; :: thesis: ((exp_R * f) `| Z) . x = (a #R x) * (log number_e ,a)
then f is_differentiable_in x by A6, FDIFF_1:16;
then diff (exp_R * f),x = (exp_R . (f . x)) * (diff f,x) by TAYLOR_1:19
.= (exp_R . (f . x)) * ((f `| Z) . x) by A6, A9, FDIFF_1:def 8
.= (exp_R . (f . x)) * (log number_e ,a) by A4, A5, A9, FDIFF_1:31
.= (exp_R . (x * (log number_e ,a))) * (log number_e ,a) by A2, A9
.= (a #R x) * (log number_e ,a) by A3, Th1 ;
hence ((exp_R * f) `| Z) . x = (a #R x) * (log number_e ,a) by A8, A9, FDIFF_1:def 8; :: thesis: verum

end;

hence ( exp_R * f is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f) `| Z) . x = (a #R x) * (log number_e ,a) ) ) by A1, A7, FDIFF_1:16; :: thesis: verum