let A be non empty closed_interval Subset of REAL; :: thesis: for Z being open Subset of REAL st A c= Z holds
integral (((id Z) (#) sin),A) = ((((- (id Z)) (#) cos) + sin) . (upper_bound A)) - ((((- (id Z)) (#) cos) + sin) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z implies integral (((id Z) (#) sin),A) = ((((- (id Z)) (#) cos) + sin) . (upper_bound A)) - ((((- (id Z)) (#) cos) + sin) . (lower_bound A)) )
assume A1: A c= Z ; :: thesis: integral (((id Z) (#) sin),A) = ((((- (id Z)) (#) cos) + sin) . (upper_bound A)) - ((((- (id Z)) (#) cos) + sin) . (lower_bound A))
A2: dom (- (id Z)) = dom (id Z) by VALUED_1:8;
A3: dom (((- (id Z)) (#) cos) + sin) = (dom ((- (id Z)) (#) cos)) /\ REAL by SIN_COS:24, VALUED_1:def 1
.= dom ((- (id Z)) (#) cos) by XBOOLE_1:28
.= (dom (- (id Z))) /\ REAL by SIN_COS:24, VALUED_1:def 4
.= dom (- (id Z)) by XBOOLE_1:28
.= Z by A2, RELAT_1:45 ;
then A4: ((- (id Z)) (#) cos) + sin is_differentiable_on Z by FDIFF_4:46;
A5: for x being Real st x in Z holds
((id Z) (#) sin) . x = x * (sin . x)

proof

let x be Real; :: thesis: ( x in Z implies ((id Z) (#) sin) . x = x * (sin . x) )
assume A6: x in Z ; :: thesis: ((id Z) (#) sin) . x = x * (sin . x)
((id Z) (#) sin) . x = ((id Z) . x) * (sin . x) by VALUED_1:5
.= x * (sin . x) by A6, FUNCT_1:18 ;
hence ((id Z) (#) sin) . x = x * (sin . x) ; :: thesis: verum

end;

A7: for x being Element of REAL st x in dom ((((- (id Z)) (#) cos) + sin) `| Z) holds
((((- (id Z)) (#) cos) + sin) `| Z) . x = ((id Z) (#) sin) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((((- (id Z)) (#) cos) + sin) `| Z) implies ((((- (id Z)) (#) cos) + sin) `| Z) . x = ((id Z) (#) sin) . x )
assume x in dom ((((- (id Z)) (#) cos) + sin) `| Z) ; :: thesis: ((((- (id Z)) (#) cos) + sin) `| Z) . x = ((id Z) (#) sin) . x
then A8: x in Z by A4, FDIFF_1:def 7;
then ((((- (id Z)) (#) cos) + sin) `| Z) . x = x * (sin . x) by A3, FDIFF_4:46
.= ((id Z) (#) sin) . x by A5, A8 ;
hence ((((- (id Z)) (#) cos) + sin) `| Z) . x = ((id Z) (#) sin) . x ; :: thesis: verum

end;

A9: dom ((id Z) (#) sin) = (dom (id Z)) /\ REAL by SIN_COS:24, VALUED_1:def 4
.= dom (id Z) by XBOOLE_1:28
.= Z by RELAT_1:45 ;
then dom ((((- (id Z)) (#) cos) + sin) `| Z) = dom ((id Z) (#) sin) by A4, FDIFF_1:def 7;
then A10: (((- (id Z)) (#) cos) + sin) `| Z = (id Z) (#) sin by A7, PARTFUN1:5;
((id Z) (#) sin) | A is continuous ;
then A11: (id Z) (#) sin is_integrable_on A by A1, A9, INTEGRA5:11;
((id Z) (#) sin) | A is bounded by A1, A9, INTEGRA5:10;
hence integral (((id Z) (#) sin),A) = ((((- (id Z)) (#) cos) + sin) . (upper_bound A)) - ((((- (id Z)) (#) cos) + sin) . (lower_bound A)) by A1, A3, A11, A10, FDIFF_4:46, INTEGRA5:13; :: thesis: verum