theorem :: FDIFF_5:23

for a being Real
for Z being open Subset of REAL
for f1, f2 being PartFunc of REAL,REAL st Z c= dom (ln * ((f1 + f2) / (f1 - f2))) & ( for x being Real st x in Z holds
f1 . x = a ) & f2 = #Z 2 & ( for x being Real st x in Z holds
(f1 - f2) . x > 0 ) & ( for x being Real st x in Z holds
(f1 + f2) . x > 0 ) holds
( ln * ((f1 + f2) / (f1 - f2)) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * ((f1 + f2) / (f1 - f2))) `| Z) . x = ((4 * a) * x) / ((a |^ 2) - (x |^ 4)) ) )