let n be Element of NAT ; :: thesis: for A being closed-interval Subset of REAL st n <> 0 holds
integral (cos * (AffineMap n,0 )),A = (((1 / n) (#) (sin * (AffineMap n,0 ))) . (sup A)) - (((1 / n) (#) (sin * (AffineMap n,0 ))) . (inf A))
let A be closed-interval Subset of REAL ; :: thesis: ( n <> 0 implies integral (cos * (AffineMap n,0 )),A = (((1 / n) (#) (sin * (AffineMap n,0 ))) . (sup A)) - (((1 / n) (#) (sin * (AffineMap n,0 ))) . (inf A)) )
assume B: n <> 0 ; :: thesis: integral (cos * (AffineMap n,0 )),A = (((1 / n) (#) (sin * (AffineMap n,0 ))) . (sup A)) - (((1 / n) (#) (sin * (AffineMap n,0 ))) . (inf A))
A1: [#] REAL = dom (cos * (AffineMap n,0 )) by FUNCT_2:def 1;
A2: [#] REAL = dom (AffineMap n,0 ) by FUNCT_2:def 1;
for x being Real st x in REAL holds
(AffineMap n,0 ) . x = (n * x) + 0 by JORDAN16:def 3;
then (AffineMap n,0 ) | REAL is continuous by A2, FDIFF_1:31, FDIFF_1:33;
then A3: (AffineMap n,0 ) | A is continuous by FCONT_1:17;
B1: cos | REAL is continuous by SIN_COS:72, FDIFF_1:33;
cos | ((AffineMap n,0 ) .: A) is continuous by B1, FCONT_1:17;
then (cos * (AffineMap n,0 )) | A is continuous by A3, FCONT_1:26;
then A4: ( cos * (AffineMap n,0 ) is_integrable_on A & (cos * (AffineMap n,0 )) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A5: 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 JORDAN16:def 3
.= n * x ;
hence (AffineMap n,0 ) . x = n * x ; :: thesis: verum

end;

A6: (1 / n) (#) (sin * (AffineMap n,0 )) is_differentiable_on REAL by B, Th12;
A7: 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 B, Th12
.= cos . ((AffineMap n,0 ) . x) by A5
.= (cos * (AffineMap n,0 )) . x by A1, FUNCT_1:22 ;
hence (((1 / n) (#) (sin * (AffineMap n,0 ))) `| REAL ) . x = (cos * (AffineMap n,0 )) . x ; :: thesis: verum

end;

dom (((1 / n) (#) (sin * (AffineMap n,0 ))) `| REAL ) = dom (cos * (AffineMap n,0 )) by A1, A6, FDIFF_1:def 8;
then ((1 / n) (#) (sin * (AffineMap n,0 ))) `| REAL = cos * (AffineMap n,0 ) by A7, PARTFUN1:34;
hence integral (cos * (AffineMap n,0 )),A = (((1 / n) (#) (sin * (AffineMap n,0 ))) . (sup A)) - (((1 / n) (#) (sin * (AffineMap n,0 ))) . (inf A)) by B, A4, Th12, INTEGRA5:13; :: thesis: verum