let m, n be Element of NAT ; :: thesis:
assume that
A1: m + n <> 0 and
A2: m - n <> 0 ; :: thesis:
A3: ( dom (cos * (AffineMap ((m + n),0))) = [#] REAL & ( for x being Real st x in REAL holds
(AffineMap ((m + n),0)) . x = ((m + n) * x) + 0 ) ) by FCONT_1:def 4, FUNCT_2:def 1;
then A4: cos * (AffineMap ((m + n),0)) is_differentiable_on REAL by FDIFF_4:38;
A5: for x being Real st x in REAL holds
((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x = (1 / 2) * (sin ((m + n) * x))

let x be Real; :: thesis:
assume x in REAL ; :: thesis:
A6: dom (((- 1) / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) = [#] REAL by FUNCT_2:def 1;
((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x = ((((- 1) * (1 / (2 * (m + n)))) (#) (cos * (AffineMap ((m + n),0)))) `| REAL) . x by RFUNCT_1:17
.= (((- (1 / (2 * (m + n)))) (#) (cos * (AffineMap ((m + n),0)))) `| REAL) . x
.= ((((- 1) / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) `| REAL) . x by XCMPLX_1:187
.= ((- 1) / (2 * (m + n))) * (diff ((cos * (AffineMap ((m + n),0))),x)) by A4, A6, FDIFF_1:20
.= ((- 1) / (2 * (m + n))) * (((cos * (AffineMap ((m + n),0))) `| REAL) . x) by A4, FDIFF_1:def 7
.= ((- 1) / (2 * (m + n))) * (- ((m + n) * (sin . (((m + n) * x) + 0)))) by A3, FDIFF_4:38
.= ((- ((- 1) / (2 * (m + n)))) * (m + n)) * (sin . (((m + n) * x) + 0))
.= ((1 / (2 * (m + n))) * (m + n)) * (sin . (((m + n) * x) + 0)) by XCMPLX_1:190
.= ((1 * (m + n)) / (2 * (m + n))) * (sin . (((m + n) * x) + 0)) by XCMPLX_1:74
.= (1 / 2) * (sin ((m + n) * x)) by A1, XCMPLX_1:91 ;
hence ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x = (1 / 2) * (sin ((m + n) * x)) ; :: thesis:

end;

A7: dom (- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) = [#] REAL by FUNCT_2:def 1;
dom ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) = [#] REAL by FUNCT_2:def 1;
then ( - ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) = (- 1) (#) ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) & (1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))) is_differentiable_on REAL ) by A4, FDIFF_1:20;
then A8: - ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0)))) is_differentiable_on REAL by A7, FDIFF_1:20;
A9: REAL = dom ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) by FUNCT_2:def 1;
A10: ( dom (cos * (AffineMap ((m - n),0))) = [#] REAL & ( for x being Real st x in REAL holds
(AffineMap ((m - n),0)) . x = ((m - n) * x) + 0 ) ) by FCONT_1:def 4, FUNCT_2:def 1;
then A11: cos * (AffineMap ((m - n),0)) is_differentiable_on REAL by FDIFF_4:38;
A12: dom ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) = [#] REAL by FUNCT_2:def 1;
then A13: (1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))) is_differentiable_on REAL by A11, FDIFF_1:20;
A14: for x being Real st x in REAL holds
(((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x = - ((1 / 2) * (sin ((m - n) * x)))

let x be Real; :: thesis:
assume x in REAL ; :: thesis:
(((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x = (1 / (2 * (m - n))) * (diff ((cos * (AffineMap ((m - n),0))),x)) by A12, A11, FDIFF_1:20
.= (1 / (2 * (m - n))) * (((cos * (AffineMap ((m - n),0))) `| REAL) . x) by A11, FDIFF_1:def 7
.= (1 / (2 * (m - n))) * (- ((m - n) * (sin . (((m - n) * x) + 0)))) by A10, FDIFF_4:38
.= ((- (1 / (2 * (m - n)))) * (m - n)) * (sin . (((m - n) * x) + 0))
.= (((- 1) / (2 * (m - n))) * (m - n)) * (sin . (((m - n) * x) + 0)) by XCMPLX_1:187
.= (((- 1) * (m - n)) / (2 * (m - n))) * (sin . (((m - n) * x) + 0)) by XCMPLX_1:74
.= ((- 1) / 2) * (sin ((m - n) * x)) by A2, XCMPLX_1:91 ;
hence (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x = - ((1 / 2) * (sin ((m - n) * x))) ; :: thesis:

end;

for x being Real st x in REAL holds
(((- ((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))

let x be Real; :: thesis:
assume x in REAL ; :: thesis:
(((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = (diff ((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))),x)) - (diff (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))),x)) by A7, A9, A8, A13, FDIFF_1:19
.= (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x) - (diff (((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))),x)) by A8, FDIFF_1:def 7
.= (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) `| REAL) . x) - ((((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x) by A13, FDIFF_1:def 7
.= ((1 / 2) * (sin ((m + n) * x))) - ((((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) `| REAL) . x) by A5
.= ((1 / 2) * (sin ((m + n) * x))) - (- ((1 / 2) * (sin ((m - n) * x)))) by A14
.= (1 / 2) * ((sin ((m + n) * x)) + (sin ((m - n) * x)))
.= (1 / 2) * (2 * ((cos ((((m + n) * x) - ((m - n) * x)) / 2)) * (sin ((((m + n) * x) + ((m - n) * x)) / 2)))) by SIN_COS4:15
.= (sin . (m * x)) * (cos . (n * x)) ;
hence (((- ((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)) ; :: thesis:

end;

hence ( (- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) is_differentiable_on REAL & ( for x being Real holds (((- ((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 A7, A9, A8, A13, FDIFF_1:19; :: thesis: