
theorem
  for a be positive Real holds 1/2 > frac a implies frac (2*a) = 2*(frac a)
  proof
    let a be positive Real;
    assume
    A1: 1/2 > frac a;
    per cases;
    suppose
      frac a is zero; then
      reconsider a as Integer;
      frac (2*a) = 0;
      hence thesis;
    end;
    suppose frac a is positive; then
      reconsider a as non integer Real;
      reconsider b = frac a as light positive Real;
      2*(1/2) > 2*(frac a) by A1,XREAL_1:68; then
      reconsider c = 2*b as light positive Real by COMPLEX3:1;
      c = frac (2*(frac a));
      hence thesis by FR3;
    end;
  end;
