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

proof

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

end;

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

proof

let x be Element of REAL ; :: thesis: ( x in dom (((id Z) (#) sin) `| Z) implies (((id Z) (#) sin) `| Z) . x = (sin + ((id Z) (#) cos)) . x )
assume x in dom (((id Z) (#) sin) `| Z) ; :: thesis: (((id Z) (#) sin) `| Z) . x = (sin + ((id Z) (#) cos)) . x
then A9: x in Z by A5, FDIFF_1:def 7;
then (((id Z) (#) sin) `| Z) . x = (sin . x) + (x * (cos . x)) by A4, FDIFF_4:45
.= (sin + ((id Z) (#) cos)) . x by A6, A9 ;
hence (((id Z) (#) sin) `| Z) . x = (sin + ((id Z) (#) cos)) . x ; :: thesis: verum

end;

dom (((id Z) (#) sin) `| Z) = dom (sin + ((id Z) (#) cos)) by A2, A5, FDIFF_1:def 7;
then A10: ((id Z) (#) sin) `| Z = sin + ((id Z) (#) cos) by A8, PARTFUN1:5;
(sin + ((id Z) (#) cos)) | A is bounded by A1, A2, INTEGRA5:10;
hence integral ((sin + ((id Z) (#) cos)),A) = (((id Z) (#) sin) . (upper_bound A)) - (((id Z) (#) sin) . (lower_bound A)) by A1, A4, A3, A10, FDIFF_4:45, INTEGRA5:13; :: thesis: verum