reserve n for Nat;

theorem Th8:
  for S being Subset of TOP-REAL n holds S is non bounded iff
   for r being Real st r > 0
    ex x, y being Point of Euclid n st x in S & y in S & dist (x, y) > r
proof
  let S be Subset of TOP-REAL n;
  reconsider S9 = S as Subset of Euclid n by TOPREAL3:8;
  hereby
    assume S is non bounded;
    then S9 is non bounded by JORDAN2C:11;
    hence
    for r being Real st r > 0
      ex x, y being Point of Euclid n st x in S & y in S & dist (x, y) > r
      by TBSP_1:def 7;
  end;
  assume
A1: for r being Real st r > 0
  ex x, y being Point of Euclid n st x in S
  & y in S & dist (x, y) > r;
  S is non bounded
  proof
    assume S is bounded;
    then S is bounded Subset of Euclid n by JORDAN2C:11;
    hence thesis by A1,TBSP_1:def 7;
  end;
  hence thesis;
end;
