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

for f being PartFunc of REAL,REAL st Z c= dom (ln * f) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) holds

( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (ln * f) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) holds

( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (ln * f) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) implies ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x) ) ) )

assume that

A1: Z c= dom (ln * f) and

A2: for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ; :: thesis: ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x) ) )

for y being object st y in Z holds

y in dom f by A1, FUNCT_1:11;

then A3: Z c= dom f by TARSKI:def 3;

A4: for x being Real st x in Z holds

f . x = (1 * x) + a by A2;

then A5: f is_differentiable_on Z by A3, FDIFF_1:23;

A6: for x being Real st x in Z holds

ln * f is_differentiable_in x

for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x)

((ln * f) `| Z) . x = 1 / (a + x) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum

for f being PartFunc of REAL,REAL st Z c= dom (ln * f) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) holds

( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x) ) )

let Z be open Subset of REAL; :: thesis: for f being PartFunc of REAL,REAL st Z c= dom (ln * f) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) holds

( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x) ) )

let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (ln * f) & ( for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ) implies ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x) ) ) )

assume that

A1: Z c= dom (ln * f) and

A2: for x being Real st x in Z holds

( f . x = a + x & f . x > 0 ) ; :: thesis: ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x) ) )

for y being object st y in Z holds

y in dom f by A1, FUNCT_1:11;

then A3: Z c= dom f by TARSKI:def 3;

A4: for x being Real st x in Z holds

f . x = (1 * x) + a by A2;

then A5: f is_differentiable_on Z by A3, FDIFF_1:23;

A6: for x being Real st x in Z holds

ln * f is_differentiable_in x

proof

then A7:
ln * f is_differentiable_on Z
by A1, FDIFF_1:9;
let x be Real; :: thesis: ( x in Z implies ln * f is_differentiable_in x )

assume x in Z ; :: thesis: ln * f is_differentiable_in x

then ( f is_differentiable_in x & f . x > 0 ) by A2, A5, FDIFF_1:9;

hence ln * f is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

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

then ( f is_differentiable_in x & f . x > 0 ) by A2, A5, FDIFF_1:9;

hence ln * f is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

for x being Real st x in Z holds

((ln * f) `| Z) . x = 1 / (a + x)

proof

hence
( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
let x be Real; :: thesis: ( x in Z implies ((ln * f) `| Z) . x = 1 / (a + x) )

assume A8: x in Z ; :: thesis: ((ln * f) `| Z) . x = 1 / (a + x)

then A9: f . x = a + x by A2;

( f is_differentiable_in x & f . x > 0 ) by A2, A5, A8, FDIFF_1:9;

then diff ((ln * f),x) = (diff (f,x)) / (f . x) by TAYLOR_1:20

.= ((f `| Z) . x) / (f . x) by A5, A8, FDIFF_1:def 7

.= 1 / (a + x) by A3, A4, A8, A9, FDIFF_1:23 ;

hence ((ln * f) `| Z) . x = 1 / (a + x) by A7, A8, FDIFF_1:def 7; :: thesis: verum

end;assume A8: x in Z ; :: thesis: ((ln * f) `| Z) . x = 1 / (a + x)

then A9: f . x = a + x by A2;

( f is_differentiable_in x & f . x > 0 ) by A2, A5, A8, FDIFF_1:9;

then diff ((ln * f),x) = (diff (f,x)) / (f . x) by TAYLOR_1:20

.= ((f `| Z) . x) / (f . x) by A5, A8, FDIFF_1:def 7

.= 1 / (a + x) by A3, A4, A8, A9, FDIFF_1:23 ;

hence ((ln * f) `| Z) . x = 1 / (a + x) by A7, A8, FDIFF_1:def 7; :: thesis: verum

((ln * f) `| Z) . x = 1 / (a + x) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum