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 / (f1 + (#Z 2)) & Z c= ].(- 1),1.[ & Z c= dom ((#Z 2) * arctan ) & Z = dom f & f | A is continuous holds
integral f,A = (((1 / 2) (#) ((#Z 2) * arctan )) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * 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 / (f1 + (#Z 2)) & Z c= ].(- 1),1.[ & Z c= dom ((#Z 2) * arctan ) & Z = dom f & f | A is continuous holds
integral f,A = (((1 / 2) (#) ((#Z 2) * arctan )) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * 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 / (f1 + (#Z 2)) & Z c= ].(- 1),1.[ & Z c= dom ((#Z 2) * arctan ) & Z = dom f & f | A is continuous implies integral f,A = (((1 / 2) (#) ((#Z 2) * arctan )) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arctan )) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & f = arctan / (f1 + (#Z 2)) & Z c= ].(- 1),1.[ & Z c= dom ((#Z 2) * arctan ) & Z = dom f & f | A is continuous ) ; :: thesis: integral f,A = (((1 / 2) (#) ((#Z 2) * arctan )) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arctan )) . (lower_bound A))
then A2: Z c= dom ((1 / 2) (#) ((#Z 2) * arctan )) by VALUED_1:def 5;
A3: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A4: (1 / 2) (#) ((#Z 2) * arctan ) is_differentiable_on Z by A1, A2, SIN_COS9:93;
Z c= (dom arctan ) /\ ((dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0 })) by RFUNCT_1:def 4, A1;
then Z c= (dom (f1 + (#Z 2))) \ ((f1 + (#Z 2)) " {0 }) by XBOOLE_1:18;
then A6: Z c= dom ((f1 + (#Z 2)) ^ ) by RFUNCT_1:def 8;
dom ((f1 + (#Z 2)) ^ ) c= dom (f1 + (#Z 2)) by RFUNCT_1:11;
then A7: Z c= dom (f1 + (#Z 2)) by XBOOLE_1:1, A6;
A8: for x being Real st x in Z holds
f . x = (arctan . x) / (1 + (x ^2 ))

proof

let x be Real; :: thesis: ( x in Z implies f . x = (arctan . x) / (1 + (x ^2 )) )
assume A9: x in Z ; :: thesis: f . x = (arctan . x) / (1 + (x ^2 ))
then (arctan / (f1 + (#Z 2))) . x = (arctan . x) / ((f1 + (#Z 2)) . x) by A1, RFUNCT_1:def 4
.= (arctan . x) / ((f1 . x) + ((#Z 2) . x)) by VALUED_1:def 1, A7, A9
.= (arctan . x) / ((f1 . x) + (x #Z 2)) by TAYLOR_1:def 1
.= (arctan . x) / (1 + (x #Z 2)) by A1, A9
.= (arctan . x) / (1 + (x ^2 )) by FDIFF_7:1 ;
hence f . x = (arctan . x) / (1 + (x ^2 )) by A1; :: thesis: verum

end;

A10: for x being Real st x in dom (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) holds
(((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) implies (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = f . x )
assume x in dom (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) ; :: thesis: (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = f . x
then A11: x in Z by A4, FDIFF_1:def 8;
then (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = (arctan . x) / (1 + (x ^2 )) by A1, A2, SIN_COS9:93
.= f . x by A8, A11 ;
hence (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) . x = f . x ; :: thesis: verum

end;

dom (((1 / 2) (#) ((#Z 2) * arctan )) `| Z) = dom f by A1, A4, FDIFF_1:def 8;
then ((1 / 2) (#) ((#Z 2) * arctan )) `| Z = f by A10, PARTFUN1:34;
hence integral f,A = (((1 / 2) (#) ((#Z 2) * arctan )) . (upper_bound A)) - (((1 / 2) (#) ((#Z 2) * arctan )) . (lower_bound A)) by A1, A2, A3, SIN_COS9:93, INTEGRA5:13; :: thesis: verum