theorem :: INTEGR12:22

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
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = arccos - ((id Z) / ((#R (1 / 2)) * (f1 - (#Z 2)))) holds
integral (f,A) = (((id Z) (#) arccos) . (upper_bound A)) - (((id Z) (#) arccos) . (lower_bound A))