let a be Real; :: thesis: for A being closed-interval Subset of REAL
for f, f2 being PartFunc of REAL ,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) & dom (2 (#) ((#R (1 / 2)) * f)) = Z & dom (2 (#) ((#R (1 / 2)) * f)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a + x) #R (- (1 / 2)) ) & f2 | A is continuous holds
integral f2,A = ((2 (#) ((#R (1 / 2)) * f)) . (upper_bound A)) - ((2 (#) ((#R (1 / 2)) * f)) . (lower_bound A))
let A be closed-interval Subset of REAL ; :: thesis: for f, f2 being PartFunc of REAL ,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) & dom (2 (#) ((#R (1 / 2)) * f)) = Z & dom (2 (#) ((#R (1 / 2)) * f)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a + x) #R (- (1 / 2)) ) & f2 | A is continuous holds
integral f2,A = ((2 (#) ((#R (1 / 2)) * f)) . (upper_bound A)) - ((2 (#) ((#R (1 / 2)) * f)) . (lower_bound A))
let f, f2 be PartFunc of REAL ,REAL ; :: thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) & dom (2 (#) ((#R (1 / 2)) * f)) = Z & dom (2 (#) ((#R (1 / 2)) * f)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a + x) #R (- (1 / 2)) ) & f2 | A is continuous holds
integral f2,A = ((2 (#) ((#R (1 / 2)) * f)) . (upper_bound A)) - ((2 (#) ((#R (1 / 2)) * f)) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) & dom (2 (#) ((#R (1 / 2)) * f)) = Z & dom (2 (#) ((#R (1 / 2)) * f)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a + x) #R (- (1 / 2)) ) & f2 | A is continuous implies integral f2,A = ((2 (#) ((#R (1 / 2)) * f)) . (upper_bound A)) - ((2 (#) ((#R (1 / 2)) * f)) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
( f . x = a + x & f . x > 0 ) ) & dom (2 (#) ((#R (1 / 2)) * f)) = Z & dom (2 (#) ((#R (1 / 2)) * f)) = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a + x) #R (- (1 / 2)) ) & f2 | A is continuous ) ; :: thesis: integral f2,A = ((2 (#) ((#R (1 / 2)) * f)) . (upper_bound A)) - ((2 (#) ((#R (1 / 2)) * f)) . (lower_bound A))
then A2: ( f2 is_integrable_on A & f2 | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
A3: 2 (#) ((#R (1 / 2)) * f) is_differentiable_on Z by A1, FDIFF_4:30;
A4: for x being Real st x in dom ((2 (#) ((#R (1 / 2)) * f)) `| Z) holds
((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = f2 . x

proof

let x be Real; :: thesis: ( x in dom ((2 (#) ((#R (1 / 2)) * f)) `| Z) implies ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = f2 . x )
assume x in dom ((2 (#) ((#R (1 / 2)) * f)) `| Z) ; :: thesis: ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = f2 . x
then A5: x in Z by A3, FDIFF_1:def 8;
then ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = (a + x) #R (- (1 / 2)) by A1, FDIFF_4:30
.= f2 . x by A1, A5 ;
hence ((2 (#) ((#R (1 / 2)) * f)) `| Z) . x = f2 . x ; :: thesis: verum

end;

dom ((2 (#) ((#R (1 / 2)) * f)) `| Z) = dom f2 by A1, A3, FDIFF_1:def 8;
then (2 (#) ((#R (1 / 2)) * f)) `| Z = f2 by A4, PARTFUN1:34;
hence integral f2,A = ((2 (#) ((#R (1 / 2)) * f)) . (upper_bound A)) - ((2 (#) ((#R (1 / 2)) * f)) . (lower_bound A)) by A1, A2, A3, INTEGRA5:13; :: thesis: verum