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 / (exp_R * f))) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds
( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #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 / (exp_R * f))) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds
( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds
f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e implies ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) )
assume that
A1: Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (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 and
A4: a <> number_e ; :: thesis: ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) )
Z c= dom (exp_R / (exp_R * f)) by A1, VALUED_1:def 5;
then Z c= (dom exp_R) /\ ((dom (exp_R * f)) \ ((exp_R * f) " {0})) by RFUNCT_1:def 1;
then A5: Z c= dom (exp_R * f) by XBOOLE_1:1;
then A6: exp_R * f is_differentiable_on Z by A2, A3, Th11;
A7: for x being Real st x in Z holds
(exp_R * f) . x <> 0

proof

let x be Real; :: thesis: ( x in Z implies (exp_R * f) . x <> 0 )
assume x in Z ; :: thesis: (exp_R * f) . x <> 0
then (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12;
hence (exp_R * f) . x <> 0 by SIN_COS:54; :: thesis: verum

end;

exp_R is_differentiable_on Z by FDIFF_1:26, TAYLOR_1:16;
then A8: exp_R / (exp_R * f) is_differentiable_on Z by A6, A7, FDIFF_2:21;
A9: 1 - (log (number_e,a)) <> 0

proof

A10: number_e <> 1 by TAYLOR_1:11;
assume 1 - (log (number_e,a)) = 0 ; :: thesis: contradiction
then log (number_e,a) = log (number_e,number_e) by A10, POWER:52, TAYLOR_1:11;
then a = number_e to_power (log (number_e,number_e)) by A3, A10, POWER:def 3, TAYLOR_1:11
.= number_e by A10, POWER:def 3, TAYLOR_1:11 ;
hence contradiction by A4; :: thesis: verum

end;

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

proof

let x be Real; :: thesis: ( x in Z implies (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) )
A11: exp_R is_differentiable_in x by SIN_COS:65;
A12: a #R x > 0 by A3, PREPOWER:81;
assume A13: x in Z ; :: thesis: (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x)
then A14: (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12
.= exp_R . (x * (log (number_e,a))) by A2, A13
.= a #R x by A3, Th1 ;
A15: ( exp_R * f is_differentiable_in x & (exp_R * f) . x <> 0 ) by A6, A7, A13, FDIFF_1:9;
(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (1 / (1 - (log (number_e,a)))) * (diff ((exp_R / (exp_R * f)),x)) by A1, A8, A13, FDIFF_1:20
.= (1 / (1 - (log (number_e,a)))) * ((((diff (exp_R,x)) * ((exp_R * f) . x)) - ((diff ((exp_R * f),x)) * (exp_R . x))) / (((exp_R * f) . x) ^2)) by A11, A15, FDIFF_2:14
.= (1 / (1 - (log (number_e,a)))) * ((((exp_R . x) * (a #R x)) - ((diff ((exp_R * f),x)) * (exp_R . x))) / ((a #R x) ^2)) by A14, SIN_COS:65
.= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - (diff ((exp_R * f),x)))) / ((a #R x) ^2))
.= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - (((exp_R * f) `| Z) . x))) / ((a #R x) ^2)) by A6, A13, FDIFF_1:def 7
.= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - ((a #R x) * (log (number_e,a))))) / ((a #R x) ^2)) by A2, A3, A5, A13, Th11
.= ((1 / (1 - (log (number_e,a)))) * (((1 - (log (number_e,a))) * (exp_R . x)) * (a #R x))) / ((a #R x) ^2) by XCMPLX_1:74
.= ((((1 / (1 - (log (number_e,a)))) * (1 - (log (number_e,a)))) * (exp_R . x)) * (a #R x)) / ((a #R x) ^2)
.= ((1 * (exp_R . x)) * (a #R x)) / ((a #R x) ^2) by A9, XCMPLX_1:106
.= (exp_R . x) / (a #R x) by A12, XCMPLX_1:91 ;
hence (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ; :: thesis: verum

end;

hence ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) by A1, A8, FDIFF_1:20; :: thesis: verum