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 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

for x being Real st x in Z holds

exp_R * f is_differentiable_in x

A9: for x being Real st x in Z holds

((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a))

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_1:20; :: thesis: verum

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 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

then A7:
f is_differentiable_on Z
by A5, FDIFF_1:23;
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;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

for x being Real st x in Z holds

exp_R * f is_differentiable_in x

proof

then A8:
exp_R * f is_differentiable_on Z
by A4, FDIFF_1:9;
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_1:9;

hence exp_R * f is_differentiable_in x by TAYLOR_1:19; :: thesis: verum

end;assume x in Z ; :: thesis: exp_R * f is_differentiable_in x

then f is_differentiable_in x by A7, FDIFF_1:9;

hence exp_R * f is_differentiable_in x by TAYLOR_1:19; :: thesis: verum

A9: for x being Real st x in Z holds

((- (exp_R * f)) `| Z) . x = (a #R (- x)) * (log (number_e,a))

proof

Z c= dom ((- 1) (#) (exp_R * f))
by A1;
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_1:9;

((- (exp_R * f)) `| Z) . x = (- 1) * (diff ((exp_R * f),x)) by A1, A8, A10, FDIFF_1:20

.= (- 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 7

.= (- 1) * ((exp_R . (f . x)) * (- (log (number_e,a)))) by A5, A6, A10, FDIFF_1:23

.= (- 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;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_1:9;

((- (exp_R * f)) `| Z) . x = (- 1) * (diff ((exp_R * f),x)) by A1, A8, A10, FDIFF_1:20

.= (- 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 7

.= (- 1) * ((exp_R . (f . x)) * (- (log (number_e,a)))) by A5, A6, A10, FDIFF_1:23

.= (- 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

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_1:20; :: thesis: verum