let m, n be Element of NAT ; :: thesis:
let A be non empty closed_interval Subset of REAL; :: thesis:
assume A1: ( m + n <> 0 & m - n <> 0 ) ; :: thesis:
( dom (AffineMap (n,0)) = [#] REAL & ( 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 A2: (AffineMap (n,0)) | A is continuous by FCONT_1:16;
( dom (AffineMap (m,0)) = [#] REAL & ( for x being Real st x in REAL holds
(AffineMap (m,0)) . x = (m * x) + 0 ) ) by FCONT_1:def 4, FUNCT_2:def 1;
then (AffineMap (m,0)) | REAL is continuous by FDIFF_1:23, FDIFF_1:25;
then A3: (AffineMap (m,0)) | A is continuous by FCONT_1:16;
sin | ((AffineMap (m,0)) .: A) is continuous ;
then A4: (sin * (AffineMap (m,0))) | A is continuous by A3, FCONT_1:25;
A5: for x being Real st x in REAL holds
(AffineMap (n,0)) . x = n * x

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

end;

A6: dom (cos * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def 1;
A7: dom (sin * (AffineMap (m,0))) = [#] REAL by FUNCT_2:def 1;
then A8: (dom (sin * (AffineMap (m,0)))) /\ (dom (cos * (AffineMap (n,0)))) = [#] REAL by A6;
A9: for x being Real st x in REAL holds
(AffineMap (m,0)) . x = m * x

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

end;

A10: for x being Real st x in dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) holds
(((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x

let x be Real; :: thesis:
assume x in dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) ; :: thesis:
(((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (sin . (m * x)) * (cos . (n * x)) by A1, Th7
.= (sin . ((AffineMap (m,0)) . x)) * (cos . (n * x)) by A9
.= (sin . ((AffineMap (m,0)) . x)) * (cos . ((AffineMap (n,0)) . x)) by A5
.= ((sin * (AffineMap (m,0))) . x) * (cos . ((AffineMap (n,0)) . x)) by A7, FUNCT_1:12
.= ((sin * (AffineMap (m,0))) . x) * ((cos * (AffineMap (n,0))) . x) by A6, FUNCT_1:12
.= ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x by VALUED_1:5 ;
hence (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x ; :: thesis:

end;

A11: [#] REAL = dom ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) by FUNCT_2:def 1;
cos | ((AffineMap (n,0)) .: A) is continuous ;
then A12: (cos * (AffineMap (n,0))) | A is continuous by A2, FCONT_1:25;
then ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is continuous by A8, A4, FCONT_1:18;
then A13: (sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))) is_integrable_on A by A11, INTEGRA5:11;
(- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) is_differentiable_on REAL by A1, Th7;
then dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) = dom ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) by A11, FDIFF_1:def 7;
then A14: ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL = (sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))) by A10, PARTFUN1:5;
((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is bounded by A11, A8, A12, A4, FCONT_1:18, INTEGRA5:10;
hence integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A)) by A1, A13, A14, Th7, INTEGRA5:13; :: thesis: