let A be closed-interval Subset of REAL ; :: thesis: for Z being open Subset of REAL st A c= Z holds
integral ((id Z) (#) cos ),A = ((((id Z) (#) sin ) + cos ) . (upper_bound A)) - ((((id Z) (#) sin ) + cos ) . (lower_bound A))
let Z be open Subset of REAL ; :: thesis: ( A c= Z implies integral ((id Z) (#) cos ),A = ((((id Z) (#) sin ) + cos ) . (upper_bound A)) - ((((id Z) (#) sin ) + cos ) . (lower_bound A)) )
assume A1: A c= Z ; :: thesis: integral ((id Z) (#) cos ),A = ((((id Z) (#) sin ) + cos ) . (upper_bound A)) - ((((id Z) (#) sin ) + cos ) . (lower_bound A))
D1: dom (((id Z) (#) sin ) + cos ) = Z

proof

dom (((id Z) (#) sin ) + cos ) = (dom ((id Z) (#) sin )) /\ REAL by SIN_COS:27, VALUED_1:def 1
.= dom ((id Z) (#) sin ) by XBOOLE_1:28
.= (dom (id Z)) /\ REAL by SIN_COS:27, VALUED_1:def 4
.= dom (id Z) by XBOOLE_1:28 ;
hence dom (((id Z) (#) sin ) + cos ) = Z by RELAT_1:71; :: thesis: verum

end;

D2: Z = dom ((id Z) (#) cos )

proof

dom ((id Z) (#) cos ) = (dom (id Z)) /\ REAL by SIN_COS:27, VALUED_1:def 4
.= dom (id Z) by XBOOLE_1:28
.= Z by RELAT_1:71 ;
hence Z = dom ((id Z) (#) cos ) ; :: thesis: verum

end;

then Z = (dom (id Z)) /\ (dom cos ) by VALUED_1:def 4;
then AA: ( Z c= dom (id Z) & Z c= dom cos ) by XBOOLE_1:18;
A3: for x being Real st x in Z holds
(id Z) . x = (1 * x) + 0 by FUNCT_1:35;
B: for x being Real st x in Z holds
((id Z) (#) cos ) . x = x * (cos . x)

proof

let x be Real; :: thesis: ( x in Z implies ((id Z) (#) cos ) . x = x * (cos . x) )
assume C: x in Z ; :: thesis: ((id Z) (#) cos ) . x = x * (cos . x)
((id Z) (#) cos ) . x = ((id Z) . x) * (cos . x) by VALUED_1:5
.= x * (cos . x) by C, FUNCT_1:35 ;
hence ((id Z) (#) cos ) . x = x * (cos . x) ; :: thesis: verum

end;

C: id Z is_differentiable_on Z by AA, A3, FDIFF_1:31;
cos is_differentiable_on Z by FDIFF_1:34, SIN_COS:72;
then ((id Z) (#) cos ) | Z is continuous by D2, C, FDIFF_1:29, FDIFF_1:33;
then ((id Z) (#) cos ) | A is continuous by A1, FCONT_1:17;
then A4: ( (id Z) (#) cos is_integrable_on A & ((id Z) (#) cos ) | A is bounded ) by A1, D2, INTEGRA5:10, INTEGRA5:11;
A5: ((id Z) (#) sin ) + cos is_differentiable_on Z by D1, FDIFF_4:47;
A6: for x being Real st x in dom ((((id Z) (#) sin ) + cos ) `| Z) holds
((((id Z) (#) sin ) + cos ) `| Z) . x = ((id Z) (#) cos ) . x

proof

let x be Real; :: thesis: ( x in dom ((((id Z) (#) sin ) + cos ) `| Z) implies ((((id Z) (#) sin ) + cos ) `| Z) . x = ((id Z) (#) cos ) . x )
assume x in dom ((((id Z) (#) sin ) + cos ) `| Z) ; :: thesis: ((((id Z) (#) sin ) + cos ) `| Z) . x = ((id Z) (#) cos ) . x
then A7: x in Z by A5, FDIFF_1:def 8;
then ((((id Z) (#) sin ) + cos ) `| Z) . x = x * (cos . x) by D1, FDIFF_4:47
.= ((id Z) (#) cos ) . x by B, A7 ;
hence ((((id Z) (#) sin ) + cos ) `| Z) . x = ((id Z) (#) cos ) . x ; :: thesis: verum

end;

dom ((((id Z) (#) sin ) + cos ) `| Z) = dom ((id Z) (#) cos ) by D2, A5, FDIFF_1:def 8;
then (((id Z) (#) sin ) + cos ) `| Z = (id Z) (#) cos by A6, PARTFUN1:34;
hence integral ((id Z) (#) cos ),A = ((((id Z) (#) sin ) + cos ) . (upper_bound A)) - ((((id Z) (#) sin ) + cos ) . (lower_bound A)) by A1, D1, A4, FDIFF_4:47, INTEGRA5:13; :: thesis: verum