theorem :: FDIFF_6:6

for a being Real
for Z being open Subset of REAL
for f, f1, f2 being PartFunc of REAL,REAL st Z c= dom ((1 / (4 * (a ^2))) (#) f) & f = ln * ((f1 + f2) / (f1 - f2)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 - f2) . x > 0 & a <> 0 ) ) holds
( (1 / (4 * (a ^2))) (#) f is_differentiable_on Z & ( for x being Real st x in Z holds
(((1 / (4 * (a ^2))) (#) f) `| Z) . x = x / ((a |^ 4) - (x |^ 4)) ) )