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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) holds
integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) 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 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) implies integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) arctan) . (lower_bound A)) )
assume A1: ( A c= Z & ( for x being Real st x in Z holds
f1 . x = 1 ) & Z c= ].(- 1),1.[ & Z = dom f & f = (exp_R (#) arctan) + (exp_R / (f1 + (#Z 2))) ) ; :: thesis: integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) arctan) . (lower_bound A))
then A3: Z = (dom (exp_R (#) arctan)) /\ (dom (exp_R / (f1 + (#Z 2)))) by VALUED_1:def 1;
A5: exp_R is_differentiable_on Z by FDIFF_1:34, TAYLOR_1:16;
A6: exp_R (#) arctan is_differentiable_on Z by A1, SIN_COS9:123;
A7: Z c= dom (exp_R / (f1 + (#Z 2))) by XBOOLE_1:18, A3;
then A8: Z c= dom (exp_R (#) ((f1 + (#Z 2)) ^)) by RFUNCT_1:47;
then Z c= (dom exp_R) /\ (dom ((f1 + (#Z 2)) ^)) by VALUED_1:def 4;
then A9: Z c= dom ((f1 + (#Z 2)) ^) by XBOOLE_1:18;
dom ((f1 + (#Z 2)) ^) c= dom (f1 + (#Z 2)) by RFUNCT_1:11;
then AA: Z c= dom (f1 + (#Z 2)) by XBOOLE_1:1, A9;
(f1 + (#Z 2)) ^ is_differentiable_on Z by A1, A9, Th1;
then exp_R (#) ((f1 + (#Z 2)) ^) is_differentiable_on Z by A5, A8, FDIFF_1:29;
then exp_R / (f1 + (#Z 2)) is_differentiable_on Z by RFUNCT_1:47;
then f | Z is continuous by FDIFF_1:33, A1, A6, FDIFF_1:26;
then f | A is continuous by A1, FCONT_1:17;
then A15: ( f is_integrable_on A & f | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
B1: for x being Real st x in Z holds
f . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))

proof

let x be Real; :: thesis: ( x in Z implies f . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) )
assume B2: x in Z ; :: thesis: f . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2)))
then ((exp_R (#) arctan) + (exp_R / (f1 + (#Z 2)))) . x = ((exp_R (#) arctan) . x) + ((exp_R / (f1 + (#Z 2))) . x) by VALUED_1:def 1, A1
.= ((exp_R . x) * (arctan . x)) + ((exp_R / (f1 + (#Z 2))) . x) by VALUED_1:5
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / ((f1 + (#Z 2)) . x)) by RFUNCT_1:def 4, A7, B2
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / ((f1 . x) + ((#Z 2) . x))) by VALUED_1:def 1, AA, B2
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / ((f1 . x) + (x #Z 2))) by TAYLOR_1:def 1
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / ((f1 . x) + (x ^2))) by FDIFF_7:1
.= ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) by A1, B2 ;
hence f . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) by A1; :: thesis: verum

end;

A16: for x being Real st x in dom ((exp_R (#) arctan) `| Z) holds
((exp_R (#) arctan) `| Z) . x = f . x

proof

let x be Real; :: thesis: ( x in dom ((exp_R (#) arctan) `| Z) implies ((exp_R (#) arctan) `| Z) . x = f . x )
assume x in dom ((exp_R (#) arctan) `| Z) ; :: thesis: ((exp_R (#) arctan) `| Z) . x = f . x
then A17: x in Z by A6, FDIFF_1:def 8;
then ((exp_R (#) arctan) `| Z) . x = ((exp_R . x) * (arctan . x)) + ((exp_R . x) / (1 + (x ^2))) by A1, SIN_COS9:123
.= f . x by A17, B1 ;
hence ((exp_R (#) arctan) `| Z) . x = f . x ; :: thesis: verum

end;

dom ((exp_R (#) arctan) `| Z) = dom f by A1, A6, FDIFF_1:def 8;
then (exp_R (#) arctan) `| Z = f by A16, PARTFUN1:34;
hence integral (f,A) = ((exp_R (#) arctan) . (upper_bound A)) - ((exp_R (#) arctan) . (lower_bound A)) by A1, A15, SIN_COS9:123, INTEGRA5:13; :: thesis: verum