let A be non empty closed_interval Subset of REAL; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (arcsin . x) / (sqrt (1 - (x ^2))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A))
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (arcsin . x) / (sqrt (1 - (x ^2))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) & Z = dom f & f | A is continuous holds
integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A))
let f be PartFunc of REAL,REAL; ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = (arcsin . x) / (sqrt (1 - (x ^2))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin)) & Z = dom f & f | A is continuous implies integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A)) )
assume that
A1:
A c= Z
and
A2:
Z c= ].(- 1),1.[
and
A3:
for x being Real st x in Z holds
f . x = (arcsin . x) / (sqrt (1 - (x ^2)))
and
A4:
Z c= dom ((1 / 2) (#) ((#Z 2) * arcsin))
and
A5:
Z = dom f
and
A6:
f | A is continuous
; integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A))
A7:
(1 / 2) (#) ((#Z 2) * arcsin) is_differentiable_on Z
by A2, A4, FDIFF_7:12;
A8:
for x being Element of REAL st x in dom (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) holds
(((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) . x = f . x
dom (((1 / 2) (#) ((#Z 2) * arcsin)) `| Z) = dom f
by A5, A7, FDIFF_1:def 7;
then A10:
((1 / 2) (#) ((#Z 2) * arcsin)) `| Z = f
by A8, PARTFUN1:5;
( f is_integrable_on A & f | A is bounded )
by A1, A5, A6, INTEGRA5:10, INTEGRA5:11;
hence
integral (f,A) = (((1 / 2) (#) ((#Z 2) * arcsin)) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arcsin)) . (lower_bound A))
by A1, A2, A4, A10, FDIFF_7:12, INTEGRA5:13; verum