let a be Real; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: 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)))

proof

let x be Real; :: thesis: ( x in Z implies f . x = x / (a * (1 + ((x / a) ^2))) )
assume A8: x in Z ; :: thesis: f . x = x / (a * (1 + ((x / a) ^2)))
then A9: x in dom (f1 + f2) by A1, FUNCT_1:11;
dom (f1 + f2) = (dom f1) /\ (dom f2) by VALUED_1:def 1;
then dom (f1 + f2) c= dom f2 by XBOOLE_1:18;
then A10: x in dom f2 by A9;
((id Z) / (a (#) (f1 + f2))) . x = ((id Z) . x) / ((a (#) (f1 + f2)) . x) by A1, A8, RFUNCT_1:def 1
.= x / ((a (#) (f1 + f2)) . x) by A8, FUNCT_1:18
.= x / (a * ((f1 + f2) . x)) by VALUED_1:6
.= x / (a * ((f1 . x) + (f2 . x))) by A9, VALUED_1:def 1
.= x / (a * (1 + (f2 . x))) by A4, A8
.= x / (a * (1 + ((#Z 2) . (h . x)))) by A1, A10, FUNCT_1:12
.= x / (a * (1 + ((h . x) #Z 2))) by TAYLOR_1:def 1
.= x / (a * (1 + ((h . x) ^2))) by FDIFF_7:1
.= x / (a * (1 + ((x / a) ^2))) by A5, A8 ;
hence f . x = x / (a * (1 + ((x / a) ^2))) by A1; :: thesis: verum

end;

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

proof

let x be Element of REAL ; :: thesis: ( x in dom (((a / 2) (#) (ln * (f1 + f2))) `| Z) implies (((a / 2) (#) (ln * (f1 + f2))) `| Z) . x = f . x )
assume x in dom (((a / 2) (#) (ln * (f1 + f2))) `| Z) ; :: thesis: (((a / 2) (#) (ln * (f1 + f2))) `| Z) . x = f . x
then A12: x in Z by A6, FDIFF_1:def 7;
then (((a / 2) (#) (ln * (f1 + f2))) `| Z) . x = x / (a * (1 + ((x / a) ^2))) by A1, A3, A4, A5, SIN_COS9:108
.= f . x by A7, A12 ;
hence (((a / 2) (#) (ln * (f1 + f2))) `| Z) . x = f . x ; :: thesis: verum

end;

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; :: thesis: verum