let a be Real; 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; 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; 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; ( 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
; 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
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; verum