
theorem
  for a be non integer Real holds -1 < frac a - frac (-a) < 1
  proof
    let a be non integer Real;
    2*0 < 2*(frac a) < 2*1 by XREAL_1:68,INT_1:43; then
    0 - 1 < 2*(frac a) - 1 < 2 - 1 by XREAL_1:9; then
    -1 < 2*(frac a) - (frac a + frac (-a)) < 1 by FRA;
    hence thesis;
  end;
