reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

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;
