let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom (- ()) & ( for x being Real st x in Z holds
f . x = - (x * (log (number_e,a))) ) & a > 0 holds
( - () is_differentiable_on Z & ( for x being Real st x in Z holds
((- ()) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (- ()) & ( for x being Real st x in Z holds
f . x = - (x * (log (number_e,a))) ) & a > 0 holds
( - () is_differentiable_on Z & ( for x being Real st x in Z holds
((- ()) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (- ()) & ( for x being Real st x in Z holds
f . x = - (x * (log (number_e,a))) ) & a > 0 implies ( - () is_differentiable_on Z & ( for x being Real st x in Z holds
((- ()) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) ) )

assume that
A1: Z c= dom (- ()) and
A2: for x being Real st x in Z holds
f . x = - (x * (log (number_e,a))) and
A3: a > 0 ; :: thesis: ( - () is_differentiable_on Z & ( for x being Real st x in Z holds
((- ()) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) )

A4: Z c= dom () by ;
then for y being object st y in Z holds
y in dom f by FUNCT_1:11;
then A5: Z c= dom f by TARSKI:def 3;
A6: for x being Real st x in Z holds
f . x = ((- (log (number_e,a))) * x) + 0
proof
let x be Real; :: thesis: ( x in Z implies f . x = ((- (log (number_e,a))) * x) + 0 )
assume x in Z ; :: thesis: f . x = ((- (log (number_e,a))) * x) + 0
then f . x = - ((log (number_e,a)) * x) by A2
.= ((- (log (number_e,a))) * x) + 0 ;
hence f . x = ((- (log (number_e,a))) * x) + 0 ; :: thesis: verum
end;
then A7: f is_differentiable_on Z by ;
for x being Real st x in Z holds
exp_R * f is_differentiable_in x
proof end;
then A8: exp_R * f is_differentiable_on Z by ;
A9: for x being Real st x in Z holds
((- ()) `| Z) . x = (a #R (- x)) * (log (number_e,a))
proof
let x be Real; :: thesis: ( x in Z implies ((- ()) `| Z) . x = (a #R (- x)) * (log (number_e,a)) )
assume A10: x in Z ; :: thesis: ((- ()) `| Z) . x = (a #R (- x)) * (log (number_e,a))
then A11: f is_differentiable_in x by ;
((- ()) `| Z) . x = (- 1) * (diff ((),x)) by
.= (- 1) * ((exp_R . (f . x)) * (diff (f,x))) by
.= (- 1) * ((exp_R . (f . x)) * ((f `| Z) . x)) by
.= (- 1) * ((exp_R . (f . x)) * (- (log (number_e,a)))) by
.= (- 1) * ((exp_R . (- (x * (log (number_e,a))))) * (- (log (number_e,a)))) by
.= (- 1) * ((a #R (- x)) * (- (log (number_e,a)))) by
.= (a #R (- x)) * (log (number_e,a)) ;
hence ((- ()) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ; :: thesis: verum
end;
Z c= dom ((- 1) (#) ()) by A1;
hence ( - () is_differentiable_on Z & ( for x being Real st x in Z holds
((- ()) `| Z) . x = (a #R (- x)) * (log (number_e,a)) ) ) by ; :: thesis: verum