let a be Real; :: thesis: for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) & ( for x being Real st x in Z holds
( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds
( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) )
let Z be open Subset of REAL; :: thesis: for f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) & ( for x being Real st x in Z holds
( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 holds
( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) )
let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) & ( for x being Real st x in Z holds
( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) ) & a > 0 & a <> 1 implies ( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) ) )
assume that
A1: Z c= dom ((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) and
A2: for x being Real st x in Z holds
( f1 . x = x * (log (number_e,a)) & f2 . x = x - (1 / (log (number_e,a))) ) and
A3: a > 0 and
A4: a <> 1 ; :: thesis: ( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) )
A5: Z c= dom ((exp_R * f1) (#) f2) by A1, VALUED_1:def 5;
then A6: Z c= (dom (exp_R * f1)) /\ (dom f2) by VALUED_1:def 4;
then A7: Z c= dom (exp_R * f1) by XBOOLE_1:18;
A8: for x being Real st x in Z holds
f2 . x = (1 * x) + (- (1 / (log (number_e,a))))

proof

let x be Real; :: thesis: ( x in Z implies f2 . x = (1 * x) + (- (1 / (log (number_e,a)))) )
A9: (1 * x) + (- (1 / (log (number_e,a)))) = (1 * x) - (1 / (log (number_e,a))) ;
assume x in Z ; :: thesis: f2 . x = (1 * x) + (- (1 / (log (number_e,a))))
hence f2 . x = (1 * x) + (- (1 / (log (number_e,a)))) by A2, A9; :: thesis: verum

end;

A10: for x being Real st x in Z holds
f1 . x = x * (log (number_e,a)) by A2;
then A11: exp_R * f1 is_differentiable_on Z by A3, A7, Th11;
A12: Z c= dom f2 by A6, XBOOLE_1:18;
then A13: f2 is_differentiable_on Z by A8, FDIFF_1:23;
then A14: (exp_R * f1) (#) f2 is_differentiable_on Z by A5, A11, FDIFF_1:21;
A15: log (number_e,a) <> 0

proof

A16: number_e <> 1 by TAYLOR_1:11;
assume log (number_e,a) = 0 ; :: thesis: contradiction
then log (number_e,a) = log (number_e,1) by SIN_COS2:13, TAYLOR_1:13;
then a = number_e to_power (log (number_e,1)) by A3, A16, POWER:def 3, TAYLOR_1:11
.= 1 by A16, 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))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x)

proof

let x be Real; :: thesis: ( x in Z implies (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) )
assume A17: x in Z ; :: thesis: (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x)
then A18: (exp_R * f1) . x = exp_R . (f1 . x) by A7, FUNCT_1:12
.= exp_R . (x * (log (number_e,a))) by A2, A17
.= a #R x by A3, Th1 ;
(((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = (1 / (log (number_e,a))) * (diff (((exp_R * f1) (#) f2),x)) by A1, A14, A17, FDIFF_1:20
.= (1 / (log (number_e,a))) * ((((exp_R * f1) (#) f2) `| Z) . x) by A14, A17, FDIFF_1:def 7
.= (1 / (log (number_e,a))) * (((f2 . x) * (diff ((exp_R * f1),x))) + (((exp_R * f1) . x) * (diff (f2,x)))) by A5, A11, A13, A17, FDIFF_1:21
.= (1 / (log (number_e,a))) * (((f2 . x) * (((exp_R * f1) `| Z) . x)) + (((exp_R * f1) . x) * (diff (f2,x)))) by A11, A17, FDIFF_1:def 7
.= (1 / (log (number_e,a))) * (((f2 . x) * (((exp_R * f1) `| Z) . x)) + (((exp_R * f1) . x) * ((f2 `| Z) . x))) by A13, A17, FDIFF_1:def 7
.= (1 / (log (number_e,a))) * (((f2 . x) * ((a #R x) * (log (number_e,a)))) + (((exp_R * f1) . x) * ((f2 `| Z) . x))) by A3, A10, A7, A17, Th11
.= (1 / (log (number_e,a))) * (((f2 . x) * ((a #R x) * (log (number_e,a)))) + (((exp_R * f1) . x) * 1)) by A12, A8, A17, FDIFF_1:23
.= (1 / (log (number_e,a))) * ((((f2 . x) * (log (number_e,a))) + 1) * (a #R x)) by A18
.= (1 / (log (number_e,a))) * ((((x - (1 / (log (number_e,a)))) * (log (number_e,a))) + 1) * (a #R x)) by A2, A17
.= ((1 / (log (number_e,a))) * (((x * (log (number_e,a))) - ((1 / (log (number_e,a))) * (log (number_e,a)))) + 1)) * (a #R x)
.= ((1 / (log (number_e,a))) * (((x * (log (number_e,a))) - 1) + 1)) * (a #R x) by A15, XCMPLX_1:106
.= (((1 / (log (number_e,a))) * (log (number_e,a))) * x) * (a #R x)
.= (1 * x) * (a #R x) by A15, XCMPLX_1:106
.= x * (a #R x) ;
hence (((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ; :: thesis: verum

end;

hence ( (1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2) is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (log (number_e,a))) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x) ) ) by A1, A14, FDIFF_1:20; :: thesis: verum