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) ) & a > 0 ) and
A3: 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 4;
then A4: Z c= dom (exp_R * f) by XBOOLE_1:1;
A5: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
A6: ( 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, A4, 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 A4, FUNCT_1:22;
hence (exp_R * f) . x <> 0 by SIN_COS:59; :: thesis: verum

end;

then A8: exp_R / (exp_R * f) is_differentiable_on Z by A5, A6, FDIFF_2:21;
A9: 1 - (log number_e ,a) <> 0

proof

assume A10: 1 - (log number_e ,a) = 0 ; :: thesis: contradiction
A11: ( number_e > 0 & number_e <> 1 ) by TAYLOR_1:11;
then log number_e ,a = log number_e ,number_e by A10, POWER:60;
then a = number_e to_power (log number_e ,number_e ) by A2, A11, POWER:def 3
.= number_e by A11, POWER:def 3 ;
hence contradiction by A3; :: 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) )
assume A12: x in Z ; :: thesis: (((1 / (1 - (log number_e ,a))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x)
then A13: (exp_R * f) . x = exp_R . (f . x) by A4, FUNCT_1:22
.= exp_R . (x * (log number_e ,a)) by A2, A12
.= a #R x by A2, Th1 ;
A14: a #R x > 0 by A2, PREPOWER:95;
A15: exp_R is_differentiable_in x by SIN_COS:70;
A16: exp_R * f is_differentiable_in x by A6, A12, FDIFF_1:16;
A17: (exp_R * f) . x <> 0 by A7, A12;
(((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, A12, FDIFF_1:28
.= (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 A15, A16, A17, 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 A13, SIN_COS:70
.= (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, A12, FDIFF_1:def 8
.= (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, A4, A12, Th11
.= ((1 / (1 - (log number_e ,a))) * (((1 - (log number_e ,a)) * (exp_R . x)) * (a #R x))) / ((a #R x) ^2 ) by XCMPLX_1:75
.= ((((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_1:107
.= (exp_R . x) / (a #R x) by A14, XCMPLX_1:92 ;
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_1:28; :: thesis: verum