theorem :: FDIFF_6:9

for n being Element of NAT
for Z being open Subset of REAL st Z c= dom (ln * (#Z n)) & ( for x being Real st x in Z holds
x > 0 ) holds
( ln * (#Z n) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (#Z n)) `| Z) . x = n / x ) )