
theorem
  for a be light positive Real holds max(a,1-a) >= 1/2 & min(a,1-a) <= 1/2
  proof
    let a be light positive Real;
    per cases;
    suppose
      B1: a >= 1-a; then
      a + a >= (1-a) + a & a + (1-a) >= (1-a) + (1-a) by XREAL_1:6; then
      (2*a)/2 >= 1/2 & (2*(1-a))/2 <= 1/2 by XREAL_1:72;
      hence thesis by B1,XXREAL_0:def 9, XXREAL_0:def 10;
    end;
    suppose
      B1: a < 1-a; then
      a + a < (1-a) + a & a + (1-a) < (1-a) + (1-a) by XREAL_1:6; then
      (2*a)/2 < 1/2 & (2*(1-a))/2 > 1/2 by XREAL_1:74;
      hence thesis by B1,XXREAL_0:def 9, XXREAL_0:def 10;
    end;
  end;
