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(cos(#)cos,h).x = (1/2)*(cos(2*x+h)-cos(2*x-h))
proof
  let x;
  cD(cos(#)cos,h).x = (cos(#)cos).(x+h/2) - (cos(#)cos).(x-h/2) by DIFF_1:5
    .= (cos.(x+h/2))*(cos.(x+h/2)) -(cos(#)cos).(x-h/2) by VALUED_1:5
    .= cos(x+h/2)*cos(x+h/2) -cos(x-h/2)*cos(x-h/2) by VALUED_1:5
    .= (1/2)*(cos((x+h/2)+(x+h/2))+cos((x+h/2)-(x+h/2))) -cos(x-h/2)*cos(x-h
  /2) by SIN_COS4:32
    .= (1/2)*(cos((x+h/2)+(x+h/2))+cos((x+h/2)-(x+h/2))) -(1/2)*(cos((x-h/2)
  +(x-h/2)) +cos((x-h/2)-(x-h/2))) by SIN_COS4:32
    .= (1/2)*cos(2*(x+h/2))-(1/2)*cos(2*(x-h/2));
  hence thesis;
end;
