theorem :: FDIFF_7:29

for a being Real
for Z being open Subset of REAL
for f, f1, f2, f3 being PartFunc of REAL,REAL st Z c= dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) holds
( ((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . (x / a) ) )