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) ) & a > 0 ) and
A3: 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 4;
then A4: Z c= dom (exp_R * f) by XBOOLE_1:18;
then A5: ( 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 A2, Th11;
A6: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
for x being Real st x in Z holds
exp_R . x <> 0 by SIN_COS:59;
then A7: (exp_R * f) / exp_R is_differentiable_on Z by A5, A6, FDIFF_2:21;
A8: (log number_e ,a) - 1 <> 0

proof

assume A9: (log number_e ,a) - 1 = 0 ; :: thesis: contradiction
A10: ( number_e > 0 & number_e <> 1 ) by TAYLOR_1:11;
then log number_e ,a = log number_e ,number_e by A9, POWER:60;
then a = number_e to_power (log number_e ,number_e ) by A2, A10, POWER:def 3
.= number_e by A10, POWER:def 3 ;
hence contradiction by A3; :: 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) )
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 A4, FUNCT_1:22
.= exp_R . (x * (log number_e ,a)) by A2, A11
.= a #R x by A2, Th1 ;
A13: exp_R is_differentiable_in x by SIN_COS:70;
A14: exp_R * f is_differentiable_in x by A5, A11, FDIFF_1:16;
A15: exp_R . x <> 0 by SIN_COS:59;
(((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_1:28
.= (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, A14, A15, 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:70
.= (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 A15, XCMPLX_1:92
.= ((1 / ((log number_e ,a) - 1)) * ((diff (exp_R * f),x) - (a #R x))) / (exp_R . x) by XCMPLX_1:75
.= ((1 / ((log number_e ,a) - 1)) * ((((exp_R * f) `| Z) . x) - (a #R x))) / (exp_R . x) by A5, A11, FDIFF_1:def 8
.= ((1 / ((log number_e ,a) - 1)) * (((a #R x) * (log number_e ,a)) - (a #R x))) / (exp_R . x) by A2, A4, 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_1:107
.= (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_1:28; :: thesis: verum