theorem :: FDIFF_4:25

for a, b being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (a - b)) (#) f) & f = ln * (f1 / f2) & ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 & f2 . x = x - b & f2 . x > 0 & a - b <> 0 ) ) holds
( (1 / (a - b)) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (a - b)) (#) f) `| Z) . x = 1 / ((x - a) * (x - b)) ) )