let A be closed-interval Subset of REAL ; :: thesis: for f1, f being PartFunc of REAL ,REAL
for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = arctan + ((id Z) / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral f,A = (((id Z) (#) arctan ) . (upper_bound A)) - (((id Z) (#) arctan ) . (lower_bound A))
let f1, f be PartFunc of REAL ,REAL ; :: thesis: for Z being open Subset of REAL st A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = arctan + ((id Z) / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous holds
integral f,A = (((id Z) (#) arctan ) . (upper_bound A)) - (((id Z) (#) arctan ) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = arctan + ((id Z) / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous implies integral f,A = (((id Z) (#) arctan ) . (upper_bound A)) - (((id Z) (#) arctan ) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = arctan + ((id Z) / (f1 + (#Z 2))) & Z c= ].(- 1),1.[ & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = (((id Z) (#) arctan ) . (upper_bound A)) - (((id Z) (#) arctan ) . (lower_bound A))
then A2: ( f is_integrable_on A & f | A is bounded ) by INTEGRA5:10, INTEGRA5:11;
Z = (dom arctan ) /\ (dom ((id Z) / (f1 + (#Z 2)))) by VALUED_1:def 1, A1;
then A4: ( Z c= dom arctan & Z c= dom ((id Z) / (f1 + (#Z 2))) ) by XBOOLE_1:18;
then Z c= (dom (id Z)) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0 })) by RFUNCT_1:def 4;
then A6: Z c= (dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0 }) by XBOOLE_1:18;
A8: (id Z) (#) arctan is_differentiable_on Z by A1, SIN_COS9:95;
A9: Z c= dom ((f1 + (#Z 2)) ^ ) by RFUNCT_1:def 8, A6;
dom ((f1 + (#Z 2)) ^ ) c= dom (f1 + (#Z 2)) by RFUNCT_1:11;
then A10: Z c= dom (f1 + (#Z 2)) by XBOOLE_1:1, A9;
B: for x being Real st x in Z holds
f . x = (arctan . x) + (x / (1 + (x ^2 )))

proof

let x be Real; :: thesis: ( x in Z implies f . x = (arctan . x) + (x / (1 + (x ^2 ))) )
assume A11: x in Z ; :: thesis: f . x = (arctan . x) + (x / (1 + (x ^2 )))
then (arctan + ((id Z) / (f1 + (#Z 2)))) . x = (arctan . x) + (((id Z) / (f1 + (#Z 2))) . x) by A1, VALUED_1:def 1
.= (arctan . x) + (((id Z) . x) / ((f1 + (#Z 2)) . x)) by RFUNCT_1:def 4, A4, A11
.= (arctan . x) + (x / ((f1 + (#Z 2)) . x)) by FUNCT_1:35, A11
.= (arctan . x) + (x / ((f1 . x) + ((#Z 2) . x))) by VALUED_1:def 1, A10, A11
.= (arctan . x) + (x / (1 + ((#Z 2) . x))) by A1, A11
.= (arctan . x) + (x / (1 + (x #Z 2))) by TAYLOR_1:def 1
.= (arctan . x) + (x / (1 + (x ^2 ))) by FDIFF_7:1 ;
hence f . x = (arctan . x) + (x / (1 + (x ^2 ))) by A1; :: thesis: verum

end;

A12: for x being Real st x in dom (((id Z) (#) arctan ) `| Z) holds
(((id Z) (#) arctan ) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom (((id Z) (#) arctan ) `| Z) implies (((id Z) (#) arctan ) `| Z) . x = f . x )
assume x in dom (((id Z) (#) arctan ) `| Z) ; :: thesis: (((id Z) (#) arctan ) `| Z) . x = f . x
then A13: x in Z by A8, FDIFF_1:def 8;
then (((id Z) (#) arctan ) `| Z) . x = (arctan . x) + (x / (1 + (x ^2 ))) by A1, SIN_COS9:95
.= f . x by B, A13 ;
hence (((id Z) (#) arctan ) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((id Z) (#) arctan ) `| Z) = dom f by A1, A8, FDIFF_1:def 8;
then ((id Z) (#) arctan ) `| Z = f by A12, PARTFUN1:34;
hence integral f,A = (((id Z) (#) arctan ) . (upper_bound A)) - (((id Z) (#) arctan ) . (lower_bound A)) by A1, A2, SIN_COS9:95, INTEGRA5:13; :: thesis: verum