let r be Real; :: thesis: for A being non empty closed_interval Subset of REAL
for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arctan * g holds
integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; :: thesis: for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arctan * g holds
integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let f, g, 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)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arctan * g holds
integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arctan * g implies integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) )
assume A1: ( A c= Z & Z c= dom (ln * (f1 + f2)) & r <> 0 & ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 & f1 . x = 1 ) ) & f2 = (#Z 2) * g & Z = dom f & f = arctan * g ) ; :: thesis: integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A))
Z c= (dom (id Z)) /\ (dom f) by A1;
then A2: Z c= dom ((id Z) (#) (arctan * g)) by A1, VALUED_1:def 4;
Z c= dom ((r / 2) (#) (ln * (f1 + f2))) by A1, VALUED_1:def 5;
then Z c= (dom ((id Z) (#) (arctan * g))) /\ (dom ((r / 2) (#) (ln * (f1 + f2)))) by A2, XBOOLE_1:19;
then A3: Z c= dom (((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) by VALUED_1:12;
for x being Real st x in Z holds
g . x = ((1 / r) * x) + 0

proof

let x be Real; :: thesis: ( x in Z implies g . x = ((1 / r) * x) + 0 )
assume x in Z ; :: thesis: g . x = ((1 / r) * x) + 0
then g . x = x / r by A1;
hence g . x = ((1 / r) * x) + 0 ; :: thesis: verum

end;

then for x being Real st x in Z holds
( g . x = ((1 / r) * x) + 0 & g . x > - 1 & g . x < 1 ) by A1;
then f is_differentiable_on Z by A1, SIN_COS9:87;
then f | Z is continuous by FDIFF_1:25;
then f | A is continuous by A1, FCONT_1:16;
then A4: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A5: ( ( for x being Real st x in Z holds
( g . x = x / r & g . x > - 1 & g . x < 1 ) ) & ( for x being Real st x in Z holds
f1 . x = 1 ) & ( for x being Real st x in Z holds
g . x = x / r ) ) by A1;
then A6: ((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2))) is_differentiable_on Z by A1, A3, SIN_COS9:109;
A7: for x being Real st x in Z holds
f . x = arctan . (x / r)

proof

let x be Real; :: thesis: ( x in Z implies f . x = arctan . (x / r) )
assume A8: x in Z ; :: thesis: f . x = arctan . (x / r)
then (arctan * g) . x = arctan . (g . x) by A1, FUNCT_1:12
.= arctan . (x / r) by A1, A8 ;
hence f . x = arctan . (x / r) by A1; :: thesis: verum

end;

A9: for x being Element of REAL st x in dom ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) holds
((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) implies ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x )
assume x in dom ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) ; :: thesis: ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x
then A10: x in Z by A6, FDIFF_1:def 7;
then ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = arctan . (x / r) by A1, A3, A5, SIN_COS9:109
.= f . x by A7, A10 ;
hence ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z) = dom f by A1, A6, FDIFF_1:def 7;
then (((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) `| Z = f by A9, PARTFUN1:5;
hence integral (f,A) = ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (upper_bound A)) - ((((id Z) (#) (arctan * g)) - ((r / 2) (#) (ln * (f1 + f2)))) . (lower_bound A)) by A1, A4, A6, INTEGRA5:13; :: thesis: verum