let n, m be Element of NAT ; :: thesis: for A being non empty closed_interval Subset of REAL st m + n <> 0 & m - n <> 0 holds
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))
let A be non empty closed_interval Subset of REAL; :: thesis: ( m + n <> 0 & m - n <> 0 implies 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)) )
assume A1: ( m + n <> 0 & m - n <> 0 ) ; :: thesis: 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))
A2: 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;

A3: dom (cos * (AffineMap (n,0))) = [#] REAL by FUNCT_2:def 1;
A4: dom (sin * (AffineMap (m,0))) = [#] REAL by FUNCT_2:def 1;
A5: for x being Real st x in REAL holds
(AffineMap (m,0)) . x = m * x

proof

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

end;

A6: for x being Element of 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

proof

let x be Element of REAL ; :: thesis: ( x in dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) implies (((- ((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 )
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 * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x
(((- ((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 A5
.= (sin . ((AffineMap (m,0)) . x)) * (cos . ((AffineMap (n,0)) . x)) by A2
.= ((sin * (AffineMap (m,0))) . x) * (cos . ((AffineMap (n,0)) . x)) by A4, FUNCT_1:12
.= ((sin * (AffineMap (m,0))) . x) * ((cos * (AffineMap (n,0))) . x) by A3, 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: verum

end;

A7: [#] REAL = dom ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) by FUNCT_2:def 1;
((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is continuous ;
then A8: (sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))) is_integrable_on A by A7, 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 A7, FDIFF_1:def 7;
then A9: ((- ((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 A6, PARTFUN1:5;
((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is bounded by A7, 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, A8, A9, Th7, INTEGRA5:13; :: thesis: verum