let h, x be Real; :: thesis: (bD (((sin (#) cos) (#) cos),h)) . x = (1 / 2) * (((cos (((2 * x) - h) / 2)) * (sin (h / 2))) + ((cos (((6 * x) - (3 * h)) / 2)) * (sin ((3 * h) / 2))))
set y = 3 * x;
set z = 3 * h;
(bD (((sin (#) cos) (#) cos),h)) . x = (((sin (#) cos) (#) cos) . x) - (((sin (#) cos) (#) cos) . (x - h)) by DIFF_1:4
.= (((sin (#) cos) . x) * (cos . x)) - (((sin (#) cos) (#) cos) . (x - h)) by VALUED_1:5
.= (((sin . x) * (cos . x)) * (cos . x)) - (((sin (#) cos) (#) cos) . (x - h)) by VALUED_1:5
.= (((sin . x) * (cos . x)) * (cos . x)) - (((sin (#) cos) . (x - h)) * (cos . (x - h))) by VALUED_1:5
.= (((sin x) * (cos x)) * (cos x)) - (((sin (x - h)) * (cos (x - h))) * (cos (x - h))) by VALUED_1:5
.= ((1 / 4) * ((((sin ((x + x) - x)) - (sin ((x + x) - x))) + (sin ((x + x) - x))) + (sin ((x + x) + x)))) - (((sin (x - h)) * (cos (x - h))) * (cos (x - h))) by SIN_COS4:35
.= ((1 / 4) * ((sin x) + (sin (3 * x)))) - ((1 / 4) * ((((sin (((x - h) + (x - h)) - (x - h))) - (sin (((x - h) + (x - h)) - (x - h)))) + (sin (((x - h) + (x - h)) - (x - h)))) + (sin (((x - h) + (x - h)) + (x - h))))) by SIN_COS4:35
.= ((1 / 4) * ((sin x) - (sin (x - h)))) + ((1 / 4) * ((sin (3 * x)) - (sin (3 * (x - h)))))
.= ((1 / 4) * (2 * ((cos ((x + (x - h)) / 2)) * (sin ((x - (x - h)) / 2))))) + ((1 / 4) * ((sin (3 * x)) - (sin (3 * (x - h))))) by SIN_COS4:16
.= ((1 / 4) * (2 * ((cos (((2 * x) - h) / 2)) * (sin (h / 2))))) + ((1 / 4) * (2 * ((cos (((3 * x) + ((3 * x) - (3 * h))) / 2)) * (sin (((3 * x) - ((3 * x) - (3 * h))) / 2))))) by SIN_COS4:16
.= ((1 / 2) * ((cos (((2 * x) - h) / 2)) * (sin (h / 2)))) + ((1 / 2) * ((cos (((6 * x) - (3 * h)) / 2)) * (sin ((3 * h) / 2)))) ;
hence (bD (((sin (#) cos) (#) cos),h)) . x = (1 / 2) * (((cos (((2 * x) - h) / 2)) * (sin (h / 2))) + ((cos (((6 * x) - (3 * h)) / 2)) * (sin ((3 * h) / 2)))) ; :: thesis: verum