let h, x be Real; :: thesis: (fD ((() (#) cos),h)) . x = (1 / 2) * (((sin (((6 * x) + (3 * h)) / 2)) * (sin ((3 * h) / 2))) - ((sin (((2 * x) + h) / 2)) * (sin (h / 2))))
set y = 3 * x;
set z = 3 * h;
(fD ((() (#) cos),h)) . x = ((() (#) cos) . (x + h)) - ((() (#) cos) . x) by DIFF_1:3
.= ((() . (x + h)) * (cos . (x + h))) - ((() (#) cos) . x) by VALUED_1:5
.= (((sin . (x + h)) * (sin . (x + h))) * (cos . (x + h))) - ((() (#) cos) . x) by VALUED_1:5
.= (((sin . (x + h)) * (sin . (x + h))) * (cos . (x + h))) - ((() . x) * (cos . x)) by VALUED_1:5
.= (((sin (x + h)) * (sin (x + h))) * (cos (x + h))) - (((sin x) * (sin x)) * (cos x)) by VALUED_1:5
.= ((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))))) - (((sin x) * (sin x)) * (cos x)) by SIN_COS4:34
.= ((1 / 4) * ((cos (x + h)) - (cos (3 * (x + h))))) - ((1 / 4) * ((((- (cos ((x + x) - x))) + (cos ((x + x) - x))) + (cos ((x + x) - x))) - (cos ((x + x) + x)))) by SIN_COS4:34
.= ((1 / 4) * ((cos (x + h)) - (cos x))) - ((1 / 4) * ((cos (3 * (x + h))) - (cos (3 * x))))
.= ((1 / 4) * (- (2 * ((sin (((x + h) + x) / 2)) * (sin (((x + h) - x) / 2)))))) - ((1 / 4) * ((cos (3 * (x + h))) - (cos (3 * x)))) by SIN_COS4:18
.= ((1 / 4) * (- (2 * ((sin (((2 * x) + h) / 2)) * (sin (h / 2)))))) - ((1 / 4) * (- (2 * ((sin ((((3 * x) + (3 * h)) + (3 * x)) / 2)) * (sin ((((3 * x) + (3 * h)) - (3 * x)) / 2)))))) by SIN_COS4:18
.= ((1 / 2) * ((sin (((6 * x) + (3 * h)) / 2)) * (sin ((3 * h) / 2)))) - ((1 / 2) * ((sin (((2 * x) + h) / 2)) * (sin (h / 2)))) ;
hence (fD ((() (#) cos),h)) . x = (1 / 2) * (((sin (((6 * x) + (3 * h)) / 2)) * (sin ((3 * h) / 2))) - ((sin (((2 * x) + h) / 2)) * (sin (h / 2)))) ; :: thesis: verum