let a be Real; :: thesis: for A being closed-interval Subset of REAL
for f, f1, f2 being PartFunc of REAL ,REAL
for Z being open Subset of REAL st A c= Z & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) & dom (((2 * a) (#) f) - (id Z)) = Z & Z = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a - x) / (a + x) ) & f2 | A is continuous holds
integral f2,A = ((((2 * a) (#) f) - (id Z)) . (upper_bound A)) - ((((2 * a) (#) f) - (id Z)) . (lower_bound A))
let A be closed-interval Subset of REAL ; :: thesis: for f, f1, f2 being PartFunc of REAL ,REAL
for Z being open Subset of REAL st A c= Z & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) & dom (((2 * a) (#) f) - (id Z)) = Z & Z = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a - x) / (a + x) ) & f2 | A is continuous holds
integral f2,A = ((((2 * a) (#) f) - (id Z)) . (upper_bound A)) - ((((2 * a) (#) f) - (id Z)) . (lower_bound A))
let f, f1, f2 be PartFunc of REAL ,REAL ; :: thesis: for Z being open Subset of REAL st A c= Z & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) & dom (((2 * a) (#) f) - (id Z)) = Z & Z = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a - x) / (a + x) ) & f2 | A is continuous holds
integral f2,A = ((((2 * a) (#) f) - (id Z)) . (upper_bound A)) - ((((2 * a) (#) f) - (id Z)) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) & dom (((2 * a) (#) f) - (id Z)) = Z & Z = dom f2 & ( for x being Real st x in Z holds
f2 . x = (a - x) / (a + x) ) & f2 | A is continuous implies integral f2,A = ((((2 * a) (#) f) - (id Z)) . (upper_bound A)) - ((((2 * a) (#) f) - (id Z)) . (lower_bound A)) )
assume that
A1: A c= Z and
A2: ( f = ln * f1 & ( for x being Real st x in Z holds
( f1 . x = a + x & f1 . x > 0 ) ) & dom (((2 * a) (#) f) - (id Z)) = Z ) and
A3: Z = dom f2 and
A4: for x being Real st x in Z holds
f2 . x = (a - x) / (a + x) and
A5: f2 | A is continuous ; :: thesis: integral f2,A = ((((2 * a) (#) f) - (id Z)) . (upper_bound A)) - ((((2 * a) (#) f) - (id Z)) . (lower_bound A))
A6: f2 is_integrable_on A by A1, A3, A5, INTEGRA5:11;
A7: ((2 * a) (#) f) - (id Z) is_differentiable_on Z by A2, FDIFF_4:5;
A8: for x being Real st x in dom ((((2 * a) (#) f) - (id Z)) `| Z) holds
((((2 * a) (#) f) - (id Z)) `| Z) . x = f2 . x

proof

let x be Real; :: thesis: ( x in dom ((((2 * a) (#) f) - (id Z)) `| Z) implies ((((2 * a) (#) f) - (id Z)) `| Z) . x = f2 . x )
assume x in dom ((((2 * a) (#) f) - (id Z)) `| Z) ; :: thesis: ((((2 * a) (#) f) - (id Z)) `| Z) . x = f2 . x
then A9: x in Z by A7, FDIFF_1:def 8;
then ((((2 * a) (#) f) - (id Z)) `| Z) . x = (a - x) / (a + x) by A2, FDIFF_4:5
.= f2 . x by A4, A9 ;
hence ((((2 * a) (#) f) - (id Z)) `| Z) . x = f2 . x ; :: thesis: verum

end;

dom ((((2 * a) (#) f) - (id Z)) `| Z) = dom f2 by A3, A7, FDIFF_1:def 8;
then (((2 * a) (#) f) - (id Z)) `| Z = f2 by A8, PARTFUN1:34;
hence integral f2,A = ((((2 * a) (#) f) - (id Z)) . (upper_bound A)) - ((((2 * a) (#) f) - (id Z)) . (lower_bound A)) by A1, A3, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; :: thesis: verum