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 cD(sin(#)sin,h).x = (1/2)*(cos(2*x-h)-cos(2*x+h))
proof
  let x;
  cD(sin(#)sin,h).x = (sin(#)sin).(x+h/2) - (sin(#)sin).(x-h/2) by DIFF_1:5
    .= (sin.(x+h/2))*(sin.(x+h/2)) -(sin(#)sin).(x-h/2) by VALUED_1:5
    .= sin(x+h/2)*sin(x+h/2) -sin(x-h/2)*sin(x-h/2) by VALUED_1:5
    .= -(1/2)*(cos((x+h/2)+(x+h/2))-cos((x+h/2)-(x+h/2))) -sin(x-h/2)*sin(x-
  h/2) by SIN_COS4:29
    .= -(1/2)*(cos(2*(x+h/2))-cos(0)) -(-(1/2)*(cos((x-h/2)+(x-h/2)) -cos((x
  -h/2)-(x-h/2)))) by SIN_COS4:29;
  hence thesis;
end;
