let A be closed-interval Subset of REAL ; :: thesis: 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 = - ((arccos . x) / (sqrt (1 - (x ^2 )))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arccos )) & Z = dom f & f | A is continuous holds
integral f,A = (((1 / 2) (#) ((#Z 2) * arccos )) . (sup A)) - (((1 / 2) (#) ((#Z 2) * arccos )) . (inf A))
let Z be open Subset of REAL ; :: thesis: 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 = - ((arccos . x) / (sqrt (1 - (x ^2 )))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arccos )) & Z = dom f & f | A is continuous holds
integral f,A = (((1 / 2) (#) ((#Z 2) * arccos )) . (sup A)) - (((1 / 2) (#) ((#Z 2) * arccos )) . (inf A))
let f be PartFunc of REAL ,REAL ; :: thesis: ( A c= Z & Z c= ].(- 1),1.[ & ( for x being Real st x in Z holds
f . x = - ((arccos . x) / (sqrt (1 - (x ^2 )))) ) & Z c= dom ((1 / 2) (#) ((#Z 2) * arccos )) & Z = dom f & f | A is continuous implies integral f,A = (((1 / 2) (#) ((#Z 2) * arccos )) . (sup A)) - (((1 / 2) (#) ((#Z 2) * arccos )) . (inf 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 = - ((arccos . x) / (sqrt (1 - (x ^2 )))) and
A4: Z c= dom ((1 / 2) (#) ((#Z 2) * arccos )) and
A5: Z = dom f and
A6: f | A is continuous ; :: thesis: integral f,A = (((1 / 2) (#) ((#Z 2) * arccos )) . (sup A)) - (((1 / 2) (#) ((#Z 2) * arccos )) . (inf A))
A7: (1 / 2) (#) ((#Z 2) * arccos ) is_differentiable_on Z by A2, A4, FDIFF_7:13;
A8: for x being Real st x in dom (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) holds
(((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) implies (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = f . x )
assume x in dom (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) ; :: thesis: (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = f . x
then A9: x in Z by A7, FDIFF_1:def 8;
then (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = - ((arccos . x) / (sqrt (1 - (x ^2 )))) by A2, A4, FDIFF_7:13
.= f . x by A3, A9 ;
hence (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((1 / 2) (#) ((#Z 2) * arccos )) `| Z) = dom f by A5, A7, FDIFF_1:def 8;
then A10: ((1 / 2) (#) ((#Z 2) * arccos )) `| Z = f by A8, PARTFUN1:34;
( 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) * arccos )) . (sup A)) - (((1 / 2) (#) ((#Z 2) * arccos )) . (inf A)) by A1, A2, A4, A10, FDIFF_7:13, INTEGRA5:13; :: thesis: verum