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 A1: ( Z c= dom (ln * f) & ( 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) ) )
then for y being set st y in Z holds
y in dom f by FUNCT_1:21;
then A2: Z c= dom f by TARSKI:def 3;
A3: for x being Real st x in Z holds
f . x = (1 * x) + a by A1;
then A4: ( f is_differentiable_on Z & ( for x being Real st x in Z holds
(f `| Z) . x = 1 ) ) by A2, FDIFF_1:31;
A5: 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 A6: x in Z ; :: thesis: ln * f is_differentiable_in x
then A7: f is_differentiable_in x by A4, FDIFF_1:16;
f . x > 0 by A1, A6;
hence ln * f is_differentiable_in x by A7, TAYLOR_1:20; :: thesis: verum

end;

then A8: ln * f is_differentiable_on Z by A1, FDIFF_1:16;
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 A9: x in Z ; :: thesis: ((ln * f) `| Z) . x = 1 / (a + x)
then A10: f is_differentiable_in x by A4, FDIFF_1:16;
A11: ( f . x = a + x & f . x > 0 ) by A1, A9;
then diff (ln * f),x = (diff f,x) / (f . x) by A10, TAYLOR_1:20
.= ((f `| Z) . x) / (f . x) by A4, A9, FDIFF_1:def 8
.= 1 / (a + x) by A2, A3, A9, A11, FDIFF_1:31 ;
hence ((ln * f) `| Z) . x = 1 / (a + x) by A8, A9, FDIFF_1:def 8; :: 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, A5, FDIFF_1:16; :: thesis: verum