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)) ) ) & a > 0 & 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) ) )
A3: ( ( for x being Real st x in Z holds
f1 . x = x * (log number_e ,a) ) & a > 0 ) by A2;
A4: Z c= dom ((exp_R * f1) (#) f2) by A1, VALUED_1:def 5;
then Z c= (dom (exp_R * f1)) /\ (dom f2) by VALUED_1:def 4;
then A5: ( Z c= dom (exp_R * f1) & Z c= dom f2 ) by XBOOLE_1:18;
then A6: ( exp_R * f1 is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R * f1) `| Z) . x = (a #R x) * (log number_e ,a) ) ) by A3, Th11;
A7: 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))) )
assume A8: x in Z ; :: thesis: f2 . x = (1 * x) + (- (1 / (log number_e ,a)))
(1 * x) + (- (1 / (log number_e ,a))) = (1 * x) - (1 / (log number_e ,a)) ;
hence f2 . x = (1 * x) + (- (1 / (log number_e ,a))) by A2, A8; :: thesis: verum

end;

then A9: ( f2 is_differentiable_on Z & ( for x being Real st x in Z holds
(f2 `| Z) . x = 1 ) ) by A5, FDIFF_1:31;
then A10: (exp_R * f1) (#) f2 is_differentiable_on Z by A4, A6, FDIFF_1:29;
A11: log number_e ,a <> 0

proof

assume log number_e ,a = 0 ; :: thesis: contradiction
then A12: log number_e ,a = log number_e ,1 by SIN_COS2:13, TAYLOR_1:13;
A13: ( number_e > 0 & number_e <> 1 ) by TAYLOR_1:11;
then a = number_e to_power (log number_e ,1) by A2, A12, POWER:def 3
.= 1 by A13, POWER:def 3 ;
hence contradiction by A2; :: 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 A14: x in Z ; :: thesis: (((1 / (log number_e ,a)) (#) ((exp_R * f1) (#) f2)) `| Z) . x = x * (a #R x)
then A15: (exp_R * f1) . x = exp_R . (f1 . x) by A5, FUNCT_1:22
.= exp_R . (x * (log number_e ,a)) by A2, A14
.= a #R x by A2, 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, A10, A14, FDIFF_1:28
.= (1 / (log number_e ,a)) * ((((exp_R * f1) (#) f2) `| Z) . x) by A10, A14, FDIFF_1:def 8
.= (1 / (log number_e ,a)) * (((f2 . x) * (diff (exp_R * f1),x)) + (((exp_R * f1) . x) * (diff f2,x))) by A4, A6, A9, A14, FDIFF_1:29
.= (1 / (log number_e ,a)) * (((f2 . x) * (((exp_R * f1) `| Z) . x)) + (((exp_R * f1) . x) * (diff f2,x))) by A6, A14, FDIFF_1:def 8
.= (1 / (log number_e ,a)) * (((f2 . x) * (((exp_R * f1) `| Z) . x)) + (((exp_R * f1) . x) * ((f2 `| Z) . x))) by A9, A14, FDIFF_1:def 8
.= (1 / (log number_e ,a)) * (((f2 . x) * ((a #R x) * (log number_e ,a))) + (((exp_R * f1) . x) * ((f2 `| Z) . x))) by A3, A5, A14, Th11
.= (1 / (log number_e ,a)) * (((f2 . x) * ((a #R x) * (log number_e ,a))) + (((exp_R * f1) . x) * 1)) by A5, A7, A14, FDIFF_1:31
.= (1 / (log number_e ,a)) * ((((f2 . x) * (log number_e ,a)) + 1) * (a #R x)) by A15
.= (1 / (log number_e ,a)) * ((((x - (1 / (log number_e ,a))) * (log number_e ,a)) + 1) * (a #R x)) by A2, A14
.= ((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 A11, XCMPLX_1:107
.= (((1 / (log number_e ,a)) * (log number_e ,a)) * x) * (a #R x)
.= (1 * x) * (a #R x) by A11, XCMPLX_1:107
.= 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, A10, FDIFF_1:28; :: thesis: verum