theorem :: INTEGR12:38

for a, b being Real
for A being non empty closed_interval Subset of REAL
for f, f1 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a / (sqrt (1 - (((a * x) + b) ^2))) & f1 . x = (a * x) + b & f1 . x > - 1 & f1 . x < 1 ) ) & Z c= dom (arccos * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- (arccos * f1)) . (upper_bound A)) - ((- (arccos * f1)) . (lower_bound A))