
theorem Def2:
  for r being Real holds r is liouville iff
   for n being non zero Nat ex p being Integer, q being Nat st 1 < q &
    0 < |. r - p/q .| < 1 / (q |^ n)
  proof
    let r be Real;
    thus r is liouville implies
     for n being non zero Nat ex p being Integer, q being Nat st 1 < q &
      0 < |. r - p/q .| < 1 / (q |^ n);
    assume
Z1: for n being non zero Nat
     ex p being Integer, q being Nat st
      1 < q & 0 < |. r - p/q .| < 1 / (q |^ n);
    let n be Nat;
    per cases;
    suppose n is non zero;
      hence thesis by Z1;
    end;
    suppose
A1:   n is zero;
      consider p being Integer, q being Nat such that
A2:   1 < q and
A3:   0 < |. r - p/q .| and
A4:   |. r - p/q .| < 1 / (q |^ 1) by Z1;
      take p,q;
      thus 1 < q & 0 < |. r - p/q .| by A2,A3;
A5:   q |^ 0 = 1 by NEWTON:4;
      1 / q < 1 / 1 by A2,XREAL_1:76;
      hence thesis by A1,A4,A5,XXREAL_0:2;
    end;
  end;
