let a be Real; :: thesis: for A being non empty 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 & 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 (arccos * f3) = Z & Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & (arccos * f3) | A is continuous holds
integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A))
let A be non empty 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 & 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 (arccos * f3) = Z & Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & (arccos * f3) | A is continuous holds
integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: for f, f1, f2, f3 being PartFunc of REAL,REAL st A c= Z & 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 (arccos * f3) = Z & Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & (arccos * f3) | A is continuous holds
integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A))
let f, f1, f2, f3 be PartFunc of REAL,REAL; :: thesis: ( A c= Z & 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 (arccos * f3) = Z & Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) & (arccos * f3) | A is continuous implies integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A)) )
assume that
A1: A c= Z and
A2: ( f = f1 - f2 & f2 = #Z 2 ) and
A3: 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 ) and
A4: dom (arccos * f3) = Z and
A5: Z = dom (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) and
A6: (arccos * f3) | A is continuous ; :: thesis: integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A))
A7: arccos * f3 is_integrable_on A by A1, A4, A6, INTEGRA5:11;
A8: ((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f) is_differentiable_on Z by A2, A3, A5, FDIFF_7:29;
A9: for x being Element of REAL st x in dom ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) holds
((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = (arccos * f3) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) implies ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = (arccos * f3) . x )
assume x in dom ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) ; :: thesis: ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = (arccos * f3) . x
then A10: x in Z by A8, FDIFF_1:def 7;
then ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = arccos . (x / a) by A2, A3, A5, FDIFF_7:29
.= arccos . (f3 . x) by A3, A10
.= (arccos * f3) . x by A4, A10, FUNCT_1:12 ;
hence ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) . x = (arccos * f3) . x ; :: thesis: verum

end;

dom ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z) = dom (arccos * f3) by A4, A8, FDIFF_1:def 7;
then (((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) `| Z = arccos * f3 by A9, PARTFUN1:5;
hence integral ((arccos * f3),A) = ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (upper_bound A)) - ((((id Z) (#) (arccos * f3)) - ((#R (1 / 2)) * f)) . (lower_bound A)) by A1, A4, A6, A7, A8, INTEGRA5:10, INTEGRA5:13; :: thesis: verum