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) ) & a > 0 ) and
A3: 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) ) )
A4: 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 A5: ( Z c= dom (exp_R * f) & Z c= dom exp_R ) by XBOOLE_1:18;
then 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, Th11;
A7: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
then A8: (exp_R * f) (#) exp_R is_differentiable_on Z by A4, A6, FDIFF_1:29;
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;
A12: number_e * a > 0 * a by A2, TAYLOR_1:11, XREAL_1:70;
log number_e ,1 = 0 by SIN_COS2:13, TAYLOR_1:13
.= (log number_e ,number_e ) + (log number_e ,a) by A10, A11, POWER:60
.= log number_e ,(number_e * a) by A2, A11, POWER:61 ;
then number_e * a = number_e to_power (log number_e ,1) by A11, A12, POWER:def 3
.= 1 by A11, POWER:def 3 ;
hence contradiction by A3, XCMPLX_1:74; :: 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 A13: x in Z ; :: thesis: (((1 / (1 + (log number_e ,a))) (#) ((exp_R * f) (#) exp_R )) `| Z) . x = (a #R x) * (exp_R . x)
then A14: (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:22
.= exp_R . (x * (log number_e ,a)) by A2, A13
.= a #R x by A2, 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, A8, A13, FDIFF_1:28
.= (1 / (1 + (log number_e ,a))) * ((((exp_R * f) (#) exp_R ) `| Z) . x) by A8, A13, FDIFF_1:def 8
.= (1 / (1 + (log number_e ,a))) * (((exp_R . x) * (diff (exp_R * f),x)) + (((exp_R * f) . x) * (diff exp_R ,x))) by A4, A6, A7, A13, FDIFF_1:29
.= (1 / (1 + (log number_e ,a))) * (((exp_R . x) * (((exp_R * f) `| Z) . x)) + (((exp_R * f) . x) * (diff exp_R ,x))) by A6, A13, FDIFF_1:def 8
.= (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, A5, A13, Th11
.= (((1 / (1 + (log number_e ,a))) * ((log number_e ,a) + 1)) * (a #R x)) * (exp_R . x) by A14
.= (1 * (a #R x)) * (exp_R . x) by A9, XCMPLX_1:107
.= (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, A8, FDIFF_1:28; :: thesis: verum