let A be non empty closed_interval Subset of REAL; :: thesis: for Z being open Subset of REAL st A c= Z holds
integral (((- cos) + ((id Z) (#) sin)),A) = (((- (id Z)) (#) cos) . (upper_bound A)) - (((- (id Z)) (#) cos) . (lower_bound A))
let Z be open Subset of REAL; :: thesis: ( A c= Z implies integral (((- cos) + ((id Z) (#) sin)),A) = (((- (id Z)) (#) cos) . (upper_bound A)) - (((- (id Z)) (#) cos) . (lower_bound A)) )
assume A1: A c= Z ; :: thesis: integral (((- cos) + ((id Z) (#) sin)),A) = (((- (id Z)) (#) cos) . (upper_bound A)) - (((- (id Z)) (#) cos) . (lower_bound A))
dom ((- cos) + ((id Z) (#) sin)) = (dom (- cos)) /\ (dom ((id Z) (#) sin)) by VALUED_1:def 1
.= REAL /\ (dom ((id Z) (#) sin)) by SIN_COS:24, VALUED_1:8
.= dom ((id Z) (#) sin) by XBOOLE_1:28
.= (dom (id Z)) /\ REAL by SIN_COS:24, VALUED_1:def 4
.= dom (id Z) by XBOOLE_1:28 ;
then A2: Z = dom ((- cos) + ((id Z) (#) sin)) by RELAT_1:45;
((- cos) + ((id Z) (#) sin)) | A is continuous ;
then A3: (- cos) + ((id Z) (#) sin) is_integrable_on A by A1, A2, INTEGRA5:11;
dom ((- (id Z)) (#) cos) = (dom (- (id Z))) /\ REAL by SIN_COS:24, VALUED_1:def 4
.= dom (- (id Z)) by XBOOLE_1:28
.= dom (id Z) by VALUED_1:8 ;
then A4: dom ((- (id Z)) (#) cos) = Z by RELAT_1:45;
then A5: (- (id Z)) (#) cos is_differentiable_on Z by FDIFF_4:44;
A6: for x being Real st x in Z holds
((- cos) + ((id Z) (#) sin)) . x = (- (cos . x)) + (x * (sin . x))

proof

let x be Real; :: thesis: ( x in Z implies ((- cos) + ((id Z) (#) sin)) . x = (- (cos . x)) + (x * (sin . x)) )
assume A7: x in Z ; :: thesis: ((- cos) + ((id Z) (#) sin)) . x = (- (cos . x)) + (x * (sin . x))
then ((- cos) + ((id Z) (#) sin)) . x = ((- cos) . x) + (((id Z) (#) sin) . x) by A2, VALUED_1:def 1
.= ((- cos) . x) + (((id Z) . x) * (sin . x)) by VALUED_1:5
.= ((- cos) . x) + (x * (sin . x)) by A7, FUNCT_1:18
.= (- (cos . x)) + (x * (sin . x)) by VALUED_1:8 ;
hence ((- cos) + ((id Z) (#) sin)) . x = (- (cos . x)) + (x * (sin . x)) ; :: thesis: verum

end;

A8: for x being Element of REAL st x in dom (((- (id Z)) (#) cos) `| Z) holds
(((- (id Z)) (#) cos) `| Z) . x = ((- cos) + ((id Z) (#) sin)) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom (((- (id Z)) (#) cos) `| Z) implies (((- (id Z)) (#) cos) `| Z) . x = ((- cos) + ((id Z) (#) sin)) . x )
assume x in dom (((- (id Z)) (#) cos) `| Z) ; :: thesis: (((- (id Z)) (#) cos) `| Z) . x = ((- cos) + ((id Z) (#) sin)) . x
then A9: x in Z by A5, FDIFF_1:def 7;
then (((- (id Z)) (#) cos) `| Z) . x = (- (cos . x)) + (x * (sin . x)) by A4, FDIFF_4:44
.= ((- cos) + ((id Z) (#) sin)) . x by A6, A9 ;
hence (((- (id Z)) (#) cos) `| Z) . x = ((- cos) + ((id Z) (#) sin)) . x ; :: thesis: verum

end;

dom (((- (id Z)) (#) cos) `| Z) = dom ((- cos) + ((id Z) (#) sin)) by A2, A5, FDIFF_1:def 7;
then A10: ((- (id Z)) (#) cos) `| Z = (- cos) + ((id Z) (#) sin) by A8, PARTFUN1:5;
((- cos) + ((id Z) (#) sin)) | A is bounded by A1, A2, INTEGRA5:10;
hence integral (((- cos) + ((id Z) (#) sin)),A) = (((- (id Z)) (#) cos) . (upper_bound A)) - (((- (id Z)) (#) cos) . (lower_bound A)) by A1, A4, A3, A10, FDIFF_4:44, INTEGRA5:13; :: thesis: verum