theorem Th25: :: FDIFF_7:25

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