theorem
  for M being non empty MetrStruct holds M is bounded iff for X being
  Subset of M holds X is bounded
proof
  let M be non empty MetrStruct;
  hereby
    assume
A1: M is bounded;
    let X be Subset of M;
    [#]M is bounded by A1;
    hence X is bounded by TBSP_1:14;
  end;
  assume for X being Subset of M holds X is bounded;
  then [#]M is bounded;
  hence thesis by TBSP_1:18;
end;
