
theorem
  for a be non integer Real holds frac a = frac -a iff 2*a is odd Integer
  proof
    let a be non integer Real;
    2*a is odd Integer implies frac a = frac -a
    proof
      assume 2*a is odd Integer; then
      reconsider n = (2*a - 1)/2 as Integer;
      a = n + 1/2 & -a = -(n+1) + 1/2 & 1/(1+1) is light;
      hence thesis;
    end;
    hence thesis by EFR;
  end;
