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 = x - a & f . x > 0 ) ) holds
( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) `| Z) . x = 1 / (x - a) ) )
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 = x - a & f . x > 0 ) ) holds
( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) `| Z) . x = 1 / (x - a) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( Z c= dom (ln * f) & ( for x being Real st x in Z holds
( f . x = x - a & f . x > 0 ) ) implies ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) `| Z) . x = 1 / (x - a) ) ) )
assume that
A1: Z c= dom (ln * f) and
A2: for x being Real st x in Z holds
( f . x = x - a & f . x > 0 ) ; :: thesis: ( ln * f is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * f) `| Z) . x = 1 / (x - a) ) )
A3: for x being Real st x in Z holds
f . x = (1 * x) + (- a)

proof

let x be Real; :: thesis: ( x in Z implies f . x = (1 * x) + (- a) )
A4: (1 * x) + (- a) = (1 * x) - a ;
assume x in Z ; :: thesis: f . x = (1 * x) + (- a)
hence f . x = (1 * x) + (- a) by A2, A4; :: thesis: verum

end;

for y being object st y in Z holds
y in dom f by A1, FUNCT_1:11;
then A5: Z c= dom f by TARSKI:def 3;
then A6: f is_differentiable_on Z by A3, FDIFF_1:23;
A7: 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, A6, FDIFF_1:9;
hence ln * f is_differentiable_in x by TAYLOR_1:20; :: thesis: verum

end;

then A8: 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 / (x - a)

proof

let x be Real; :: thesis: ( x in Z implies ((ln * f) `| Z) . x = 1 / (x - a) )
assume A9: x in Z ; :: thesis: ((ln * f) `| Z) . x = 1 / (x - a)
then A10: f . x = x - a by A2;
( f is_differentiable_in x & f . x > 0 ) by A2, A6, A9, FDIFF_1:9;
then diff ((ln * f),x) = (diff (f,x)) / (f . x) by TAYLOR_1:20
.= ((f `| Z) . x) / (f . x) by A6, A9, FDIFF_1:def 7
.= 1 / (x - a) by A5, A3, A9, A10, FDIFF_1:23 ;
hence ((ln * f) `| Z) . x = 1 / (x - a) by A8, A9, 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 / (x - a) ) ) by A1, A7, FDIFF_1:9; :: thesis: verum