let a, b be Real; :: thesis: for A being non empty 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 = x + b & f1 . x > 0 ) ) & dom ((id Z) - ((a + b) (#) f)) = Z & Z = dom f2 & ( for x being Real st x in Z holds
f2 . x = (x - a) / (x + b) ) & f2 | A is continuous holds
integral (f2,A) = (((id Z) - ((a + b) (#) f)) . (upper_bound A)) - (((id Z) - ((a + b) (#) f)) . (lower_bound A))
let A be non empty 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 = x + b & f1 . x > 0 ) ) & dom ((id Z) - ((a + b) (#) f)) = Z & Z = dom f2 & ( for x being Real st x in Z holds
f2 . x = (x - a) / (x + b) ) & f2 | A is continuous holds
integral (f2,A) = (((id Z) - ((a + b) (#) f)) . (upper_bound A)) - (((id Z) - ((a + b) (#) f)) . (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 = x + b & f1 . x > 0 ) ) & dom ((id Z) - ((a + b) (#) f)) = Z & Z = dom f2 & ( for x being Real st x in Z holds
f2 . x = (x - a) / (x + b) ) & f2 | A is continuous holds
integral (f2,A) = (((id Z) - ((a + b) (#) f)) . (upper_bound A)) - (((id Z) - ((a + b) (#) f)) . (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 = x + b & f1 . x > 0 ) ) & dom ((id Z) - ((a + b) (#) f)) = Z & Z = dom f2 & ( for x being Real st x in Z holds
f2 . x = (x - a) / (x + b) ) & f2 | A is continuous implies integral (f2,A) = (((id Z) - ((a + b) (#) f)) . (upper_bound A)) - (((id Z) - ((a + b) (#) f)) . (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 = x + b & f1 . x > 0 ) ) & dom ((id Z) - ((a + b) (#) f)) = Z ) and
A3: Z = dom f2 and
A4: for x being Real st x in Z holds
f2 . x = (x - a) / (x + b) and
A5: f2 | A is continuous ; :: thesis: integral (f2,A) = (((id Z) - ((a + b) (#) f)) . (upper_bound A)) - (((id Z) - ((a + b) (#) f)) . (lower_bound A))
A6: f2 is_integrable_on A by A1, A3, A5, INTEGRA5:11;
A7: (id Z) - ((a + b) (#) f) is_differentiable_on Z by A2, FDIFF_4:10;
A8: for x being Element of REAL st x in dom (((id Z) - ((a + b) (#) f)) `| Z) holds
(((id Z) - ((a + b) (#) f)) `| Z) . x = f2 . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (((id Z) - ((a + b) (#) f)) `| Z) implies (((id Z) - ((a + b) (#) f)) `| Z) . x = f2 . x )
assume x in dom (((id Z) - ((a + b) (#) f)) `| Z) ; :: thesis: (((id Z) - ((a + b) (#) f)) `| Z) . x = f2 . x
then A9: x in Z by A7, FDIFF_1:def 7;
then (((id Z) - ((a + b) (#) f)) `| Z) . x = (x - a) / (x + b) by A2, FDIFF_4:10
.= f2 . x by A4, A9 ;
hence (((id Z) - ((a + b) (#) f)) `| Z) . x = f2 . x ; :: thesis: verum

end;

dom (((id Z) - ((a + b) (#) f)) `| Z) = dom f2 by A3, A7, FDIFF_1:def 7;
then ((id Z) - ((a + b) (#) f)) `| Z = f2 by A8, PARTFUN1:5;
hence integral (f2,A) = (((id Z) - ((a + b) (#) f)) . (upper_bound A)) - (((id Z) - ((a + b) (#) f)) . (lower_bound A)) by A1, A3, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; :: thesis: verum