reserve h,r,r1,r2,x0,x1,x2,x3,x4,x5,x,a,b,c,k for Real,
  f,f1,f2 for Function of REAL,REAL;

theorem
  for x holds bD(sin(#)sin(#)cos,h).x = (1/2)*(sin((6*x-3*h)/2)*sin(3*h/
  2))-(1/2)*(sin((2*x-h)/2)*sin(h/2))
proof
  let x;
  set y=3*x;
  set z=3*h;
  bD(sin(#)sin(#)cos,h).x = (sin(#)sin(#)cos).x -(sin(#)sin(#)cos).(x-h)
  by DIFF_1:4
    .= ((sin(#)sin).x)*(cos.x) -(sin(#)sin(#)cos).(x-h) by VALUED_1:5
    .= (sin.x)*(sin.x)*(cos.x) -(sin(#)sin(#)cos).(x-h) by VALUED_1:5
    .= (sin.x)*(sin.x)*(cos.x) -((sin(#)sin).(x-h))*(cos.(x-h)) by VALUED_1:5
    .= sin(x)*sin(x)*cos(x) -sin(x-h)*sin(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)) -sin(x-h)*sin(x-
  h)*cos(x-h) by SIN_COS4:34
    .= (1/4)*(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_COS4:34
    .= (1/4)*(cos(x)-cos(x-h))-(1/4)*(cos(3*x)-cos(3*(x-h)))
    .= (1/4)*(-2*(sin((x+(x-h))/2)*sin((x-(x-h))/2))) -(1/4)*(cos(3*x)-cos(3
  *(x-h))) by SIN_COS4:18
    .= (1/4)*(-2*(sin((2*x-h)/2)*sin(h/2))) -(1/4)*(-2*(sin((y+(y-z))/2)*
  sin((y-(y-z))/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 thesis;
end;
