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(cos(#)cos,h).x = (1/2)*(cos(2*x)-cos(2*(x-h)))
proof
  let x;
  bD(cos(#)cos,h).x = (cos(#)cos).x -(cos(#)cos).(x-h) by DIFF_1:4
    .= (cos.x)*(cos.x)-(cos(#)cos).(x-h) by VALUED_1:5
    .= cos(x)*cos(x) -cos(x-h)*cos(x-h) by VALUED_1:5
    .= (1/2)*(cos(x+x)+cos(x-x))-cos(x-h)*cos(x-h) by SIN_COS4:32
    .= (1/2)*(cos(x+x)+cos(x-x)) -(1/2)*(cos((x-h)+(x-h))+cos((x-h)-(x-h)))
  by SIN_COS4:32
    .= (1/2)*cos(2*x)-(1/2)*cos(2*(x-h));
  hence thesis;
end;
