theorem
  T is totally_bounded implies T is bounded
proof
  assume T is totally_bounded;
  then consider Y such that
A1: Y is finite & the carrier of T = union Y and
A2: for P st P in Y ex x being Element of T st P = Ball(x,1);
  for P being Subset of T st P in Y holds P is bounded
  proof
    let P be Subset of T;
    assume P in Y;
    then ex x being Element of T st P = Ball(x,1) by A2;
    hence thesis;
  end;
  then [#]T is bounded by A1,Th17;
  hence thesis by Th18;
end;
