let a, b be Real; 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) + ((b - a) (#) 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) + ((b - a) (#) f)) . (upper_bound A)) - (((id Z) + ((b - a) (#) f)) . (lower_bound A))
let A be 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) + ((b - a) (#) 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) + ((b - a) (#) f)) . (upper_bound A)) - (((id Z) + ((b - a) (#) f)) . (lower_bound A))
let f, f1, f2 be 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) + ((b - a) (#) 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) + ((b - a) (#) f)) . (upper_bound A)) - (((id Z) + ((b - a) (#) f)) . (lower_bound A))
let Z be open Subset of REAL; ( 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) + ((b - a) (#) 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) + ((b - a) (#) f)) . (upper_bound A)) - (((id Z) + ((b - a) (#) 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) + ((b - a) (#) 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
; integral (f2,A) = (((id Z) + ((b - a) (#) f)) . (upper_bound A)) - (((id Z) + ((b - a) (#) f)) . (lower_bound A))
A6:
f2 is_integrable_on A
by A1, A3, A5, INTEGRA5:11;
A7:
(id Z) + ((b - a) (#) f) is_differentiable_on Z
by A2, FDIFF_4:11;
A8:
for x being Element of REAL st x in dom (((id Z) + ((b - a) (#) f)) `| Z) holds
(((id Z) + ((b - a) (#) f)) `| Z) . x = f2 . x
dom (((id Z) + ((b - a) (#) f)) `| Z) = dom f2
by A3, A7, FDIFF_1:def 7;
then
((id Z) + ((b - a) (#) f)) `| Z = f2
by A8, PARTFUN1:5;
hence
integral (f2,A) = (((id Z) + ((b - a) (#) f)) . (upper_bound A)) - (((id Z) + ((b - a) (#) f)) . (lower_bound A))
by A1, A3, A5, A6, A7, INTEGRA5:10, INTEGRA5:13; verum