theorem :: FDIFF_4:7

for a being Real
for Z being open Subset of REAL
for f, f1 being PartFunc of REAL,REAL st Z c= dom ((id Z) + ((2 * a) (#) f)) & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = x - a & f1 . x > 0 ) ) holds
( (id Z) + ((2 * a) (#) f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((id Z) + ((2 * a) (#) f)) `| Z) . x = (x + a) / (x - a) ) )