let a be Real; :: thesis: for A being closed-interval Subset of REAL
for Z being open Subset of REAL
for f, f1, f2, f3 being PartFunc of REAL ,REAL st A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous holds
integral (arcsin * f3),A = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (sup A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (inf A))
let A be closed-interval Subset of REAL ; :: thesis: for Z being open Subset of REAL
for f, f1, f2, f3 being PartFunc of REAL ,REAL st A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous holds
integral (arcsin * f3),A = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (sup A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (inf A))
let Z be open Subset of REAL ; :: thesis: for f, f1, f2, f3 being PartFunc of REAL ,REAL st A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous holds
integral (arcsin * f3),A = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (sup A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (inf A))
let f, f1, f2, f3 be PartFunc of REAL ,REAL ; :: thesis: ( A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous implies integral (arcsin * f3),A = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (sup A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (inf A)) )
assume A1: ( A c= Z & Z c= ].(- 1),1.[ & f = f1 - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f1 . x = a ^2 & f . x > 0 & f3 . x = x / a & f3 . x > - 1 & f3 . x < 1 & x <> 0 & a > 0 ) ) & dom (arcsin * f3) = Z & Z c= dom (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) & (arcsin * f3) | A is continuous ) ; :: thesis: integral (arcsin * f3),A = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (sup A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (inf A))
then A2: ( arcsin * f3 is_integrable_on A & (arcsin * f3) | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: ((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f) is_differentiable_on Z by A1, FDIFF_7:28;
A4: for x being Real st x in dom ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) holds
((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = (arcsin * f3) . x

proof

let x be Real; :: thesis: ( x in dom ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) implies ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = (arcsin * f3) . x )
assume x in dom ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) ; :: thesis: ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = (arcsin * f3) . x
then A5: x in Z by A3, FDIFF_1:def 8;
((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = arcsin . (x / a) by A1, A5, FDIFF_7:28
.= arcsin . (f3 . x) by A1, A5
.= (arcsin * f3) . x by A5, A1, FUNCT_1:22 ;
hence ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) . x = (arcsin * f3) . x ; :: thesis: verum

end;

dom ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z) = dom (arcsin * f3) by A1, A3, FDIFF_1:def 8;
then (((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) `| Z = arcsin * f3 by A4, PARTFUN1:34;
hence integral (arcsin * f3),A = ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (sup A)) - ((((id Z) (#) (arcsin * f3)) + ((#R (1 / 2)) * f)) . (inf A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum