reserve
  x, y for object,
  i, n for Nat,
  r, s for Real,
  f1, f2 for n-element real-valued FinSequence;
reserve e, e1 for Point of Euclid n;

theorem
  for V being Subset of TopSpaceMetr Euclid n st
  for e being Point of Euclid n st e in V
  ex r being Real st r > 0 & OpenHypercube(e,r) c= V
  holds V is open
  proof
    let V be Subset of TopSpaceMetr Euclid n;
    assume
A1: for e being Point of Euclid n st e in V
    ex r being Real st r > 0 & OpenHypercube(e,r) c= V;
    for e be Point of Euclid n st e in V
    ex r being Real st r > 0 & Ball(e,r) c= V
    proof
      let e be Point of Euclid n;
      assume e in V;
      then consider r being Real such that
A2:   r > 0 and
A3:   OpenHypercube(e,r) c= V by A1;
      Ball(e,r) c= OpenHypercube(e,r) by Th22;
      hence thesis by A2,A3,XBOOLE_1:1;
    end;
    hence thesis by TOPMETR:15;
  end;
