theorem Th7: :: INTEGR11:7

for n, m being Element of NAT st m + n <> 0 & m - n <> 0 holds
( (- ((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)) ) )