reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;
reserve S for Seq_Sequence;

theorem
  cD(sin(#)sin(#)sin,h).x = (1/2)*(3*cos(x)*sin(h/2)-cos(3*x)*sin(3*h/2))
proof
  cD(sin(#)sin(#)sin,h).x = (sin(#)sin(#)sin).(x+h/2)
       - (sin(#)sin(#)sin).(x-h/2) by DIFF_1:5
    .= ((sin(#)sin).(x+h/2))*(sin.(x+h/2))
       -(sin(#)sin(#)sin).(x-h/2) by VALUED_1:5
    .= (sin.(x+h/2))*(sin.(x+h/2))*(sin.(x+h/2))
       -(sin(#)sin(#)sin).(x-h/2) by VALUED_1:5
    .= (sin.(x+h/2))*(sin.(x+h/2))*(sin.(x+h/2))
       -((sin(#)sin).(x-h/2))*(sin.(x-h/2)) by VALUED_1:5
    .= sin(x+h/2)*sin(x+h/2)*sin(x+h/2)
       -sin(x-h/2)*sin(x-h/2)*sin(x-h/2) by VALUED_1:5
    .= (1/4)*(sin((x+h/2)+(x+h/2)-(x+h/2))+sin((x+h/2)+(x+h/2)-(x+h/2))
       +sin((x+h/2)+(x+h/2)-(x+h/2))-sin((x+h/2)+(x+h/2)+(x+h/2)))
       -sin(x-h/2)*sin(x-h/2)*sin(x-h/2) by SIN_COS4:33
    .= (1/4)*(sin(x+h/2)+sin(x+h/2)+sin(x+h/2)-sin(3*(x+h/2)))
       -(1/4)*(sin((x-h/2)+(x-h/2)-(x-h/2))+sin((x-h/2)+(x-h/2)-(x-h/2))
       +sin((x-h/2)+(x-h/2)-(x-h/2))-sin((x-h/2)+(x-h/2)+(x-h/2)))
                                                         by SIN_COS4:33
    .= (1/4)*((3*(sin(x+h/2)-sin(x-h/2))-(sin(3*(x+h/2))-sin(3*(x-h/2)))))
    .= (1/4)*((3*(2*(cos(((x+h/2)+(x-h/2))/2)*sin(((x+h/2)-(x-h/2))/2)))
       -(sin(3*(x+h/2))-sin(3*(x-h/2))))) by SIN_COS4:16
    .= (1/4)*((3*(2*(cos((2*x)/2)*sin(h/2)))
       -(2*(cos(((3*(x+h/2))+(3*(x-h/2)))/2)*sin(((3*(x+h/2))
       -(3*(x-h/2)))/2))))) by SIN_COS4:16
    .= (1/2)*(3*cos(x)*sin(h/2)-cos(3*x)*sin(3*h/2));
  hence thesis;
end;
