
theorem
  for a be positive Real holds 1/2 < frac a implies frac (2*a) < frac a
  proof
    let a be positive Real;
    assume
    A1: 1/2 < frac a; then
    reconsider a as non integer Real;
    A2: frac a < 1 by COMPLEX3:1; then
    2*(1/2) < 2*(frac a) < 2*1 by A1,XREAL_1:68; then
    1 - 1 < 2*frac a - 1 < 2 - 1 by XREAL_1:9; then
    reconsider  y = (2*(frac a)) - 1 as light positive Real by COMPLEX3:1;
    (frac a) + (frac a) < (frac a) + 1 by XREAL_1:6,A2; then
    A5: 2*(frac a) - 1 < ((frac a) + 1) - 1 by XREAL_1:9;
    frac (y + 1) = y;
    hence thesis by A5,FR3;
  end;
