let n be Element of NAT ; :: thesis: for A being non empty closed_interval Subset of REAL st n <> 0 holds
integral ((cos * (AffineMap (n,0))),A) = (((1 / n) (#) (sin * (AffineMap (n,0)))) . (upper_bound A)) - (((1 / n) (#) (sin * (AffineMap (n,0)))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; :: thesis: ( n <> 0 implies integral ((cos * (AffineMap (n,0))),A) = (((1 / n) (#) (sin * (AffineMap (n,0)))) . (upper_bound A)) - (((1 / n) (#) (sin * (AffineMap (n,0)))) . (lower_bound A)) )
assume A1: n <> 0 ; :: thesis: integral ((cos * (AffineMap (n,0))),A) = (((1 / n) (#) (sin * (AffineMap (n,0)))) . (upper_bound A)) - (((1 / n) (#) (sin * (AffineMap (n,0)))) . (lower_bound A))
A2: [#] REAL = dom (cos * (AffineMap (n,0))) by FUNCT_2:def 1;
A3: for x being Real st x in REAL holds
(AffineMap (n,0)) . x = n * x

proof

let x be Real; :: thesis: ( x in REAL implies (AffineMap (n,0)) . x = n * x )
assume x in REAL ; :: thesis: (AffineMap (n,0)) . x = n * x
(AffineMap (n,0)) . x = (n * x) + 0 by FCONT_1:def 4
.= n * x ;
hence (AffineMap (n,0)) . x = n * x ; :: thesis: verum

end;

A4: for x being Real st x in dom (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) holds
(((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (cos * (AffineMap (n,0))) . x

proof

let x be Real; :: thesis: ( x in dom (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) implies (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (cos * (AffineMap (n,0))) . x )
assume x in dom (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) ; :: thesis: (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (cos * (AffineMap (n,0))) . x
(((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = cos (n * x) by A1, Th12
.= cos . ((AffineMap (n,0)) . x) by A3
.= (cos * (AffineMap (n,0))) . x by A2, FUNCT_1:12 ;
hence (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) . x = (cos * (AffineMap (n,0))) . x ; :: thesis: verum

end;

(1 / n) (#) (sin * (AffineMap (n,0))) is_differentiable_on REAL by A1, Th12;
then dom (((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL) = dom (cos * (AffineMap (n,0))) by A2, FDIFF_1:def 7;
then A5: ((1 / n) (#) (sin * (AffineMap (n,0)))) `| REAL = cos * (AffineMap (n,0)) by A4, PARTFUN1:5;
( [#] REAL = dom (AffineMap (n,0)) & ( for x being Real st x in REAL holds
(AffineMap (n,0)) . x = (n * x) + 0 ) ) by FCONT_1:def 4, FUNCT_2:def 1;
then (AffineMap (n,0)) | REAL is continuous by FDIFF_1:23, FDIFF_1:25;
then A6: (AffineMap (n,0)) | A is continuous by FCONT_1:16;
cos | REAL is continuous by FDIFF_1:25, SIN_COS:67;
then A7: cos | ((AffineMap (n,0)) .: A) is continuous by FCONT_1:16;
then (cos * (AffineMap (n,0))) | A is continuous by A6, FCONT_1:25;
then A8: cos * (AffineMap (n,0)) is_integrable_on A by A2, INTEGRA5:11;
(cos * (AffineMap (n,0))) | A is bounded by A2, A6, A7, FCONT_1:25, INTEGRA5:10;
hence integral ((cos * (AffineMap (n,0))),A) = (((1 / n) (#) (sin * (AffineMap (n,0)))) . (upper_bound A)) - (((1 / n) (#) (sin * (AffineMap (n,0)))) . (lower_bound A)) by A1, A8, A5, Th12, INTEGRA5:13; :: thesis: verum