let h, x be Real; :: thesis:
(bD (((cos (#) cos) (#) cos),h)) . x = (((cos (#) cos) (#) cos) . x) - (((cos (#) cos) (#) cos) . (x - h)) by DIFF_1:4
.= (((cos (#) cos) . x) * (cos . x)) - (((cos (#) cos) (#) cos) . (x - h)) by VALUED_1:5
.= (((cos . x) * (cos . x)) * (cos . x)) - (((cos (#) cos) (#) cos) . (x - h)) by VALUED_1:5
.= (((cos . x) * (cos . x)) * (cos . x)) - (((cos (#) cos) . (x - h)) * (cos . (x - h))) by VALUED_1:5
.= (((cos x) * (cos x)) * (cos x)) - (((cos (x - h)) * (cos (x - h))) * (cos (x - h))) by VALUED_1:5
.= ((1 / 4) * ((((cos ((x + x) - x)) + (cos ((x + x) - x))) + (cos ((x + x) - x))) + (cos ((x + x) + x)))) - (((cos (x - h)) * (cos (x - h))) * (cos (x - h))) by SIN_COS4:36
.= ((1 / 4) * ((((cos x) + (cos x)) + (cos x)) + (cos (3 * x)))) - ((1 / 4) * ((((cos (((x - h) + (x - h)) - (x - h))) + (cos (((x - h) + (x - h)) - (x - h)))) + (cos (((x - h) + (x - h)) - (x - h)))) + (cos (((x - h) + (x - h)) + (x - h))))) by SIN_COS4:36
.= (1 / 4) * ((3 * ((cos x) - (cos (x - h)))) + ((cos (3 * x)) - (cos (3 * (x - h)))))
.= (1 / 4) * ((3 * (- (2 * ((sin ((x + (x - h)) / 2)) * (sin ((x - (x - h)) / 2)))))) + ((cos (3 * x)) - (cos (3 * (x - h))))) by SIN_COS4:18
.= (1 / 4) * ((3 * (- (2 * ((sin (((2 * x) - h) / 2)) * (sin (h / 2)))))) + (- (2 * ((sin (((3 * x) + (3 * (x - h))) / 2)) * (sin (((3 * x) - (3 * (x - h))) / 2)))))) by SIN_COS4:18
.= (- (1 / 2)) * ((3 * ((sin (((2 * x) - h) / 2)) * (sin (h / 2)))) + ((sin ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2)))) ;
hence (bD (((cos (#) cos) (#) cos),h)) . x = - ((1 / 2) * (((3 * (sin (((2 * x) - h) / 2))) * (sin (h / 2))) + ((sin ((3 * ((2 * x) - h)) / 2)) * (sin ((3 * h) / 2))))) ; :: thesis: