let A be non empty closed_interval Subset of REAL; :: thesis: integral (((#Z 0) / ((#Z 0) + (#Z 2))),A) = (arctan . (upper_bound A)) - (arctan . (lower_bound A))
set Z0 = #Z 0;
set Z2 = #Z 2;
set f = (#Z 0) / ((#Z 0) + (#Z 2));
A1: dom ((#Z 0) / ((#Z 0) + (#Z 2))) = REAL by Th4;
(#Z 0) / ((#Z 0) + (#Z 2)) is continuous by Th4;
then A2: ((#Z 0) / ((#Z 0) + (#Z 2))) | A is continuous ;
A3: for r being Real st r in REAL holds
((#Z 0) / ((#Z 0) + (#Z 2))) . r = 1 / (1 + (r ^2))

proof

let r be Real; :: thesis: ( r in REAL implies ((#Z 0) / ((#Z 0) + (#Z 2))) . r = 1 / (1 + (r ^2)) )
r |^ 0 = 1 by NEWTON:4;
hence ( r in REAL implies ((#Z 0) / ((#Z 0) + (#Z 2))) . r = 1 / (1 + (r ^2)) ) by Th4; :: thesis: verum

end;

A4: [#] REAL is open ;
A5: dom arctan = REAL by FUNCT_2:def 1;
A6: arctan is_differentiable_on REAL by A5, FDIFF_1:def 8;
A7: for x being Element of REAL st x in dom (arctan `| REAL) holds
(arctan `| REAL) . x = ((#Z 0) / ((#Z 0) + (#Z 2))) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (arctan `| REAL) implies (arctan `| REAL) . x = ((#Z 0) / ((#Z 0) + (#Z 2))) . x )
assume x in dom (arctan `| REAL) ; :: thesis: (arctan `| REAL) . x = ((#Z 0) / ((#Z 0) + (#Z 2))) . x
(arctan `| REAL) . x = 1 / (1 + (x ^2)) by Th3, A4
.= ((#Z 0) / ((#Z 0) + (#Z 2))) . x by A3 ;
hence (arctan `| REAL) . x = ((#Z 0) / ((#Z 0) + (#Z 2))) . x ; :: thesis: verum

end;

A8: arctan `| REAL = (#Z 0) / ((#Z 0) + (#Z 2)) by A6, A1, FDIFF_1:def 7, A7;
( (#Z 0) / ((#Z 0) + (#Z 2)) is_integrable_on A & ((#Z 0) / ((#Z 0) + (#Z 2))) | A is bounded ) by A1, A2, INTEGRA5:10, INTEGRA5:11;
hence integral (((#Z 0) / ((#Z 0) + (#Z 2))),A) = (arctan . (upper_bound A)) - (arctan . (lower_bound A)) by A4, A8, INTEGRA5:13, Th3; :: thesis: verum