let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds
( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / ((log (number_e,a)) - 1)) (#) ((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 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds
( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / ((log (number_e,a)) - 1)) (#) ((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 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e implies ( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) ) )
assume that
A1: Z c= dom ((1 / ((log (number_e,a)) - 1)) (#) ((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 <> number_e ; :: thesis: ( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) )
Z c= dom ((exp_R * f) / exp_R) by A1, VALUED_1:def 5;
then Z c= (dom (exp_R * f)) /\ ((dom exp_R) \ (exp_R " {0})) by RFUNCT_1:def 1;
then A5: Z c= dom (exp_R * f) by XBOOLE_1:18;
then A6: exp_R * f is_differentiable_on Z by A2, A3, Th11;
( exp_R is_differentiable_on Z & ( for x being Real st x in Z holds
exp_R . x <> 0 ) ) by FDIFF_1:26, SIN_COS:54, TAYLOR_1:16;
then A7: (exp_R * f) / exp_R is_differentiable_on Z by A6, FDIFF_2:21;
A8: (log (number_e,a)) - 1 <> 0

proof

A9: number_e <> 1 by TAYLOR_1:11;
assume (log (number_e,a)) - 1 = 0 ; :: thesis: contradiction
then log (number_e,a) = log (number_e,number_e) by A9, POWER:52, TAYLOR_1:11;
then a = number_e to_power (log (number_e,number_e)) by A3, A9, POWER:def 3, TAYLOR_1:11
.= number_e by A9, POWER:def 3, TAYLOR_1:11 ;
hence contradiction by A4; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) )
A10: exp_R . x <> 0 by SIN_COS:54;
assume A11: x in Z ; :: thesis: (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x)
then A12: (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12
.= exp_R . (x * (log (number_e,a))) by A2, A11
.= a #R x by A3, Th1 ;
A13: ( exp_R is_differentiable_in x & exp_R * f is_differentiable_in x ) by A6, A11, FDIFF_1:9, SIN_COS:65;
(((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (1 / ((log (number_e,a)) - 1)) * (diff (((exp_R * f) / exp_R),x)) by A1, A7, A11, FDIFF_1:20
.= (1 / ((log (number_e,a)) - 1)) * ((((diff ((exp_R * f),x)) * (exp_R . x)) - ((diff (exp_R,x)) * ((exp_R * f) . x))) / ((exp_R . x) ^2)) by A13, A10, FDIFF_2:14
.= (1 / ((log (number_e,a)) - 1)) * ((((diff ((exp_R * f),x)) * (exp_R . x)) - ((exp_R . x) * (a #R x))) / ((exp_R . x) ^2)) by A12, SIN_COS:65
.= (1 / ((log (number_e,a)) - 1)) * ((((diff ((exp_R * f),x)) - (a #R x)) * (exp_R . x)) / ((exp_R . x) * (exp_R . x)))
.= (1 / ((log (number_e,a)) - 1)) * (((diff ((exp_R * f),x)) - (a #R x)) / (exp_R . x)) by A10, XCMPLX_1:91
.= ((1 / ((log (number_e,a)) - 1)) * ((diff ((exp_R * f),x)) - (a #R x))) / (exp_R . x) by XCMPLX_1:74
.= ((1 / ((log (number_e,a)) - 1)) * ((((exp_R * f) `| Z) . x) - (a #R x))) / (exp_R . x) by A6, A11, FDIFF_1:def 7
.= ((1 / ((log (number_e,a)) - 1)) * (((a #R x) * (log (number_e,a))) - (a #R x))) / (exp_R . x) by A2, A3, A5, A11, Th11
.= (((1 / ((log (number_e,a)) - 1)) * ((log (number_e,a)) - 1)) * (a #R x)) / (exp_R . x)
.= (1 * (a #R x)) / (exp_R . x) by A8, XCMPLX_1:106
.= (a #R x) / (exp_R . x) ;
hence (((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ; :: thesis: verum

end;

hence ( (1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / ((log (number_e,a)) - 1)) (#) ((exp_R * f) / exp_R)) `| Z) . x = (a #R x) / (exp_R . x) ) ) by A1, A7, FDIFF_1:20; :: thesis: verum