let a be Real; :: thesis: for Z being open Subset of REAL
for f being PartFunc of REAL ,REAL st Z c= dom (- (exp_R * f)) & ( for x being Real st x in Z holds
f . x = - (x * (log number_e ,a)) ) & a > 0 holds
( - (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) ) )
let Z be open Subset of REAL ; :: thesis: for f being PartFunc of REAL ,REAL st Z c= dom (- (exp_R * f)) & ( for x being Real st x in Z holds
f . x = - (x * (log number_e ,a)) ) & a > 0 holds
( - (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) ) )
let f be PartFunc of REAL ,REAL ; :: thesis: ( Z c= dom (- (exp_R * f)) & ( for x being Real st x in Z holds
f . x = - (x * (log number_e ,a)) ) & a > 0 implies ( - (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) ) ) )
assume that
A1: Z c= dom (- (exp_R * f)) and
A2: for x being Real st x in Z holds
f . x = - (x * (log number_e ,a)) and
A3: a > 0 ; :: thesis: ( - (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) ) )
A4: Z c= dom (exp_R * f) by A1, VALUED_1:8;
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
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 A5, FDIFF_1:31;
for x being Real st x in Z holds
exp_R * f is_differentiable_in x

proof

let x be Real; :: thesis: ( x in Z implies exp_R * f is_differentiable_in x )
assume x in Z ; :: thesis: exp_R * f is_differentiable_in x
then f is_differentiable_in x by A7, FDIFF_1:16;
hence exp_R * f is_differentiable_in x by TAYLOR_1:19; :: thesis: verum

end;

then A8: exp_R * f is_differentiable_on Z by A4, FDIFF_1:16;
A9: for x being Real st x in Z holds
((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log number_e ,a)

proof

let x be Real; :: thesis: ( x in Z implies ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log number_e ,a) )
assume A10: x in Z ; :: thesis: ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log number_e ,a)
then A11: f is_differentiable_in x by A7, FDIFF_1:16;
((- (exp_R * f)) `| Z) . x = (- 1) * (diff (exp_R * f),x) by A1, A8, A10, FDIFF_1:28
.= (- 1) * ((exp_R . (f . x)) * (diff f,x)) by A11, TAYLOR_1:19
.= (- 1) * ((exp_R . (f . x)) * ((f `| Z) . x)) by A7, A10, FDIFF_1:def 8
.= (- 1) * ((exp_R . (f . x)) * (- (log number_e ,a))) by A5, A6, A10, FDIFF_1:31
.= (- 1) * ((exp_R . (- (x * (log number_e ,a)))) * (- (log number_e ,a))) by A2, A10
.= (- 1) * ((a #R (- x)) * (- (log number_e ,a))) by A3, Th2
.= (a #R (- x)) * (log number_e ,a) ;
hence ((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log number_e ,a) ; :: thesis: verum

end;

Z c= dom ((- 1) (#) (exp_R * f)) by A1;
hence ( - (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 A8, A9, FDIFF_1:28; :: thesis: verum