theorem Th18:
  N is bounded iff [#]N is bounded
proof
  thus N is bounded implies [#]N is bounded
  proof
    assume N is bounded;
    then consider r such that
A1: 0<r and
A2: for x,y being Point of N holds dist(x,y)<=r;
    for x,y being Point of N st x in [#]N & y in [#]N holds dist(x,y ) <=
    r by A2;
    hence thesis by A1;
  end;
  assume [#]N is bounded;
  then consider r such that
A3: 0<r and
A4: for x,y being Point of N st x in [#]N & y in [#]N holds dist(x,y)<=r;
  take r;
  thus 0<r by A3;
  let x,y be Point of N;
  thus thesis by A4;
end;
