let a be Real; for A being non empty closed_interval Subset of REAL
for f, h, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & f = (id Z) / (a (#) (f1 + f2)) & ( for x being Real st x in Z holds
( h . x = x / a & f1 . x = 1 ) ) & a <> 0 & f2 = (#Z 2) * h & Z = dom f & f | A is continuous holds
integral (f,A) = (((a / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((a / 2) (#) (ln * (f1 + f2))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; for f, h, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & f = (id Z) / (a (#) (f1 + f2)) & ( for x being Real st x in Z holds
( h . x = x / a & f1 . x = 1 ) ) & a <> 0 & f2 = (#Z 2) * h & Z = dom f & f | A is continuous holds
integral (f,A) = (((a / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((a / 2) (#) (ln * (f1 + f2))) . (lower_bound A))
let f, h, f1, f2 be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & f = (id Z) / (a (#) (f1 + f2)) & ( for x being Real st x in Z holds
( h . x = x / a & f1 . x = 1 ) ) & a <> 0 & f2 = (#Z 2) * h & Z = dom f & f | A is continuous holds
integral (f,A) = (((a / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((a / 2) (#) (ln * (f1 + f2))) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & Z c= dom (ln * (f1 + f2)) & f = (id Z) / (a (#) (f1 + f2)) & ( for x being Real st x in Z holds
( h . x = x / a & f1 . x = 1 ) ) & a <> 0 & f2 = (#Z 2) * h & Z = dom f & f | A is continuous implies integral (f,A) = (((a / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((a / 2) (#) (ln * (f1 + f2))) . (lower_bound A)) )
assume A1:
( A c= Z & Z c= dom (ln * (f1 + f2)) & f = (id Z) / (a (#) (f1 + f2)) & ( for x being Real st x in Z holds
( h . x = x / a & f1 . x = 1 ) ) & a <> 0 & f2 = (#Z 2) * h & Z = dom f & f | A is continuous )
; integral (f,A) = (((a / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((a / 2) (#) (ln * (f1 + f2))) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
A3:
Z c= dom ((a / 2) (#) (ln * (f1 + f2)))
by A1, VALUED_1:def 5;
A4:
for x being Real st x in Z holds
f1 . x = 1
by A1;
A5:
for x being Real st x in Z holds
h . x = x / a
by A1;
then A6:
(a / 2) (#) (ln * (f1 + f2)) is_differentiable_on Z
by A1, A3, A4, SIN_COS9:108;
A7:
for x being Real st x in Z holds
f . x = x / (a * (1 + ((x / a) ^2)))
A11:
for x being Element of REAL st x in dom (((a / 2) (#) (ln * (f1 + f2))) `| Z) holds
(((a / 2) (#) (ln * (f1 + f2))) `| Z) . x = f . x
dom (((a / 2) (#) (ln * (f1 + f2))) `| Z) = dom f
by A1, A6, FDIFF_1:def 7;
then
((a / 2) (#) (ln * (f1 + f2))) `| Z = f
by A11, PARTFUN1:5;
hence
integral (f,A) = (((a / 2) (#) (ln * (f1 + f2))) . (upper_bound A)) - (((a / 2) (#) (ln * (f1 + f2))) . (lower_bound A))
by A1, A2, A6, INTEGRA5:13; verum