theorem :: FDIFF_4:3

for a being Real
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) ) )