theorem :: FDIFF_4:18

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)) & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & (f1 + f2) . x > 0 ) ) holds
( ln * (f1 + f2) is_differentiable_on Z & ( for x being Real st x in Z holds
((ln * (f1 + f2)) `| Z) . x = (2 * x) / ((a ^2) + (x |^ 2)) ) )