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))
A2: - cos is_differentiable_on Z by FDIFF_1:26, INTEGRA8:26;
A3: sin is_differentiable_on Z by FDIFF_1:26, SIN_COS:68;
A4: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:18;
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 A5: Z = dom ((- cos) + ((id Z) (#) sin)) by RELAT_1:45;
then Z = (dom (- cos)) /\ (dom ((id Z) (#) sin)) by VALUED_1:def 1;
then A6: Z c= dom ((id Z) (#) sin) by XBOOLE_1:18;
then Z c= (dom (id Z)) /\ (dom sin) by VALUED_1:def 4;
then Z c= dom (id Z) by XBOOLE_1:18;
then id Z is_differentiable_on Z by A4, FDIFF_1:23;
then (id Z) (#) sin is_differentiable_on Z by A6, A3, FDIFF_1:21;
then A7: ((- cos) + ((id Z) (#) sin)) | Z is continuous by A5, A2, FDIFF_1:18, FDIFF_1:25;
then ((- cos) + ((id Z) (#) sin)) | A is continuous by A1, FCONT_1:16;
then A8: (- cos) + ((id Z) (#) sin) is_integrable_on A by A1, A5, 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 A9: dom ((- (id Z)) (#) cos) = Z by RELAT_1:45;
then A10: (- (id Z)) (#) cos is_differentiable_on Z by FDIFF_4:44;
A11: 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 A12: 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 A5, VALUED_1:def 1
.= ((- cos) . x) + (((id Z) . x) * (sin . x)) by VALUED_1:5
.= ((- cos) . x) + (x * (sin . x)) by A12, 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;

A13: for x being Real st x in dom (((- (id Z)) (#) cos) `| Z) holds
(((- (id Z)) (#) cos) `| Z) . x = ((- cos) + ((id Z) (#) sin)) . x

proof

let x be 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 A14: x in Z by A10, FDIFF_1:def 7;
then (((- (id Z)) (#) cos) `| Z) . x = (- (cos . x)) + (x * (sin . x)) by A9, FDIFF_4:44
.= ((- cos) + ((id Z) (#) sin)) . x by A11, A14 ;
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 A5, A10, FDIFF_1:def 7;
then A15: ((- (id Z)) (#) cos) `| Z = (- cos) + ((id Z) (#) sin) by A13, PARTFUN1:5;
((- cos) + ((id Z) (#) sin)) | A is bounded by A1, A5, A7, FCONT_1:16, INTEGRA5:10;
hence integral (((- cos) + ((id Z) (#) sin)),A) = (((- (id Z)) (#) cos) . (upper_bound A)) - (((- (id Z)) (#) cos) . (lower_bound A)) by A1, A9, A8, A15, FDIFF_4:44, INTEGRA5:13; :: thesis: verum