theorem Th102: :: SIN_COS9:102

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