theorem Th27: :: FDIFF_6:27

for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((#Z 2) * (exp_R - f)) & ( for x being Real st x in Z holds
f . x = 1 ) holds
( (#Z 2) * (exp_R - f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((#Z 2) * (exp_R - f)) `| Z) . x = (2 * (exp_R . x)) * ((exp_R . x) - 1) ) )