theorem Th25: :: FDIFF_6:25

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 (2 * x)) ) )