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

proof

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;

then A7: ln * f is_differentiable_on Z by A1, FDIFF_1:9;
for x being Real st x in Z holds
((ln * f) `| Z) . x = 1 / (a + x)

proof

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;

hence ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) `| Z) . x = 1 / (a + x) ) ) by A1, A6, FDIFF_1:9; :: thesis: verum