let a be Real; :: thesis: for Z being open Subset of REAL

for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds

f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds

( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds

f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds

( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds

f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e implies ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) )

assume that

A1: Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (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 and

A4: a <> number_e ; :: thesis: ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) )

Z c= dom (exp_R / (exp_R * f)) by A1, VALUED_1:def 5;

then Z c= (dom exp_R) /\ ((dom (exp_R * f)) \ ((exp_R * f) " {0})) by RFUNCT_1:def 1;

then A5: Z c= dom (exp_R * f) by XBOOLE_1:1;

then A6: exp_R * f is_differentiable_on Z by A2, A3, Th11;

A7: for x being Real st x in Z holds

(exp_R * f) . x <> 0

then A8: exp_R / (exp_R * f) is_differentiable_on Z by A6, A7, FDIFF_2:21;

A9: 1 - (log (number_e,a)) <> 0

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x)

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) by A1, A8, FDIFF_1:20; :: thesis: verum

for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds

f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds

( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds

f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e holds

( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) & ( for x being Real st x in Z holds

f . x = x * (log (number_e,a)) ) & a > 0 & a <> number_e implies ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) )

assume that

A1: Z c= dom ((1 / (1 - (log (number_e,a)))) (#) (exp_R / (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 and

A4: a <> number_e ; :: thesis: ( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) )

Z c= dom (exp_R / (exp_R * f)) by A1, VALUED_1:def 5;

then Z c= (dom exp_R) /\ ((dom (exp_R * f)) \ ((exp_R * f) " {0})) by RFUNCT_1:def 1;

then A5: Z c= dom (exp_R * f) by XBOOLE_1:1;

then A6: exp_R * f is_differentiable_on Z by A2, A3, Th11;

A7: for x being Real st x in Z holds

(exp_R * f) . x <> 0

proof

exp_R is_differentiable_on Z
by FDIFF_1:26, TAYLOR_1:16;
let x be Real; :: thesis: ( x in Z implies (exp_R * f) . x <> 0 )

assume x in Z ; :: thesis: (exp_R * f) . x <> 0

then (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12;

hence (exp_R * f) . x <> 0 by SIN_COS:54; :: thesis: verum

end;assume x in Z ; :: thesis: (exp_R * f) . x <> 0

then (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12;

hence (exp_R * f) . x <> 0 by SIN_COS:54; :: thesis: verum

then A8: exp_R / (exp_R * f) is_differentiable_on Z by A6, A7, FDIFF_2:21;

A9: 1 - (log (number_e,a)) <> 0

proof

for x being Real st x in Z holds
A10:
number_e <> 1
by TAYLOR_1:11;

assume 1 - (log (number_e,a)) = 0 ; :: thesis: contradiction

then log (number_e,a) = log (number_e,number_e) by A10, POWER:52, TAYLOR_1:11;

then a = number_e to_power (log (number_e,number_e)) by A3, A10, POWER:def 3, TAYLOR_1:11

.= number_e by A10, POWER:def 3, TAYLOR_1:11 ;

hence contradiction by A4; :: thesis: verum

end;assume 1 - (log (number_e,a)) = 0 ; :: thesis: contradiction

then log (number_e,a) = log (number_e,number_e) by A10, POWER:52, TAYLOR_1:11;

then a = number_e to_power (log (number_e,number_e)) by A3, A10, POWER:def 3, TAYLOR_1:11

.= number_e by A10, POWER:def 3, TAYLOR_1:11 ;

hence contradiction by A4; :: thesis: verum

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x)

proof

hence
( (1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f)) is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) )

A11: exp_R is_differentiable_in x by SIN_COS:65;

A12: a #R x > 0 by A3, PREPOWER:81;

assume A13: x in Z ; :: thesis: (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x)

then A14: (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12

.= exp_R . (x * (log (number_e,a))) by A2, A13

.= a #R x by A3, Th1 ;

A15: ( exp_R * f is_differentiable_in x & (exp_R * f) . x <> 0 ) by A6, A7, A13, FDIFF_1:9;

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (1 / (1 - (log (number_e,a)))) * (diff ((exp_R / (exp_R * f)),x)) by A1, A8, A13, FDIFF_1:20

.= (1 / (1 - (log (number_e,a)))) * ((((diff (exp_R,x)) * ((exp_R * f) . x)) - ((diff ((exp_R * f),x)) * (exp_R . x))) / (((exp_R * f) . x) ^2)) by A11, A15, FDIFF_2:14

.= (1 / (1 - (log (number_e,a)))) * ((((exp_R . x) * (a #R x)) - ((diff ((exp_R * f),x)) * (exp_R . x))) / ((a #R x) ^2)) by A14, SIN_COS:65

.= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - (diff ((exp_R * f),x)))) / ((a #R x) ^2))

.= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - (((exp_R * f) `| Z) . x))) / ((a #R x) ^2)) by A6, A13, FDIFF_1:def 7

.= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - ((a #R x) * (log (number_e,a))))) / ((a #R x) ^2)) by A2, A3, A5, A13, Th11

.= ((1 / (1 - (log (number_e,a)))) * (((1 - (log (number_e,a))) * (exp_R . x)) * (a #R x))) / ((a #R x) ^2) by XCMPLX_1:74

.= ((((1 / (1 - (log (number_e,a)))) * (1 - (log (number_e,a)))) * (exp_R . x)) * (a #R x)) / ((a #R x) ^2)

.= ((1 * (exp_R . x)) * (a #R x)) / ((a #R x) ^2) by A9, XCMPLX_1:106

.= (exp_R . x) / (a #R x) by A12, XCMPLX_1:91 ;

hence (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ; :: thesis: verum

end;A11: exp_R is_differentiable_in x by SIN_COS:65;

A12: a #R x > 0 by A3, PREPOWER:81;

assume A13: x in Z ; :: thesis: (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x)

then A14: (exp_R * f) . x = exp_R . (f . x) by A5, FUNCT_1:12

.= exp_R . (x * (log (number_e,a))) by A2, A13

.= a #R x by A3, Th1 ;

A15: ( exp_R * f is_differentiable_in x & (exp_R * f) . x <> 0 ) by A6, A7, A13, FDIFF_1:9;

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (1 / (1 - (log (number_e,a)))) * (diff ((exp_R / (exp_R * f)),x)) by A1, A8, A13, FDIFF_1:20

.= (1 / (1 - (log (number_e,a)))) * ((((diff (exp_R,x)) * ((exp_R * f) . x)) - ((diff ((exp_R * f),x)) * (exp_R . x))) / (((exp_R * f) . x) ^2)) by A11, A15, FDIFF_2:14

.= (1 / (1 - (log (number_e,a)))) * ((((exp_R . x) * (a #R x)) - ((diff ((exp_R * f),x)) * (exp_R . x))) / ((a #R x) ^2)) by A14, SIN_COS:65

.= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - (diff ((exp_R * f),x)))) / ((a #R x) ^2))

.= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - (((exp_R * f) `| Z) . x))) / ((a #R x) ^2)) by A6, A13, FDIFF_1:def 7

.= (1 / (1 - (log (number_e,a)))) * (((exp_R . x) * ((a #R x) - ((a #R x) * (log (number_e,a))))) / ((a #R x) ^2)) by A2, A3, A5, A13, Th11

.= ((1 / (1 - (log (number_e,a)))) * (((1 - (log (number_e,a))) * (exp_R . x)) * (a #R x))) / ((a #R x) ^2) by XCMPLX_1:74

.= ((((1 / (1 - (log (number_e,a)))) * (1 - (log (number_e,a)))) * (exp_R . x)) * (a #R x)) / ((a #R x) ^2)

.= ((1 * (exp_R . x)) * (a #R x)) / ((a #R x) ^2) by A9, XCMPLX_1:106

.= (exp_R . x) / (a #R x) by A12, XCMPLX_1:91 ;

hence (((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ; :: thesis: verum

(((1 / (1 - (log (number_e,a)))) (#) (exp_R / (exp_R * f))) `| Z) . x = (exp_R . x) / (a #R x) ) ) by A1, A8, FDIFF_1:20; :: thesis: verum