let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> 1 / number_e holds
( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) )
let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> 1 / number_e holds
( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> 1 / number_e implies ( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) ) )
assume that
A1: Z c= dom ((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) and
A2: for x being Real st x in Z holds
f . x = x * (log (number_e,a)) and
A3: a > 0 and
A4: a <> 1 / number_e ; :: thesis: ( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) )
A5: Z c= dom ((exp_R * f) (#) exp_R) by A1, VALUED_1:def 5;
then Z c= (dom (exp_R * f)) /\ (dom exp_R) by VALUED_1:def 4;
then A6: Z c= dom (exp_R * f) by XBOOLE_1:18;
then A7: exp_R * f is_differentiable_on Z by A2, A3, Th11;
A8: exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
then A9: (exp_R * f) (#) exp_R is_differentiable_on Z by A5, A7, FDIFF_1:21;
A10: 1 + (log (number_e,a)) <> 0

proof

A11: number_e * a > 0 * a by A3, TAYLOR_1:11, XREAL_1:68;
assume A12: 1 + (log (number_e,a)) = 0 ; :: thesis: contradiction
A13: number_e <> 1 by TAYLOR_1:11;
log (number_e,1) = 0 by SIN_COS2:13, TAYLOR_1:13
.= (log (number_e,number_e)) + (log (number_e,a)) by A12, A13, POWER:52, TAYLOR_1:11
.= log (number_e,(number_e * a)) by A3, A13, POWER:53, TAYLOR_1:11 ;
then number_e * a = number_e to_power (log (number_e,1)) by A13, A11, POWER:def 3, TAYLOR_1:11
.= 1 by A13, POWER:def 3, TAYLOR_1:11 ;
hence contradiction by A4, XCMPLX_1:73; :: thesis: verum

end;

for x being Real st x in Z holds
(((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x)

proof

let x be Real; :: thesis: ( x in Z implies (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) )
assume A14: x in Z ; :: thesis: (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x)
then A15: (exp_R * f) . x = exp_R . (f . x) by A6, FUNCT_1:12
.= exp_R . (x * (log (number_e,a))) by A2, A14
.= a #R x by A3, Th1 ;
(((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (1 / (1 + (log (number_e,a)))) * (diff (((exp_R * f) (#) exp_R),x)) by A1, A9, A14, FDIFF_1:20
.= (1 / (1 + (log (number_e,a)))) * ((((exp_R * f) (#) exp_R) `| Z) . x) by A9, A14, FDIFF_1:def 7
.= (1 / (1 + (log (number_e,a)))) * (((exp_R . x) * (diff ((exp_R * f),x))) + (((exp_R * f) . x) * (diff (exp_R,x)))) by A5, A7, A8, A14, FDIFF_1:21
.= (1 / (1 + (log (number_e,a)))) * (((exp_R . x) * (((exp_R * f) `| Z) . x)) + (((exp_R * f) . x) * (diff (exp_R,x)))) by A7, A14, FDIFF_1:def 7
.= (1 / (1 + (log (number_e,a)))) * (((exp_R . x) * (((exp_R * f) `| Z) . x)) + (((exp_R * f) . x) * (exp_R . x))) by TAYLOR_1:16
.= (1 / (1 + (log (number_e,a)))) * (((((exp_R * f) `| Z) . x) + ((exp_R * f) . x)) * (exp_R . x))
.= (1 / (1 + (log (number_e,a)))) * ((((a #R x) * (log (number_e,a))) + ((exp_R * f) . x)) * (exp_R . x)) by A2, A3, A6, A14, Th11
.= (((1 / (1 + (log (number_e,a)))) * ((log (number_e,a)) + 1)) * (a #R x)) * (exp_R . x) by A15
.= (1 * (a #R x)) * (exp_R . x) by A10, XCMPLX_1:106
.= (a #R x) * (exp_R . x) ;
hence (((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ; :: thesis: verum

end;

hence ( (1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 + (log (number_e,a)))) (#) ((exp_R * f) (#) exp_R)) `| Z) . x = (a #R x) * (exp_R . x) ) ) by A1, A9, FDIFF_1:20; :: thesis: verum