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 holds V is open implies
  for e being Point of Euclid n st e in V
  ex m being non zero Element of NAT st OpenHypercube(e,1/m) c= V
  proof
    let V be Subset of TopSpaceMetr Euclid n;
    assume
A1: V is open;
    let e be Point of Euclid n;
    assume e in V;
    then consider r being Real such that
A2: r > 0 and
A3: Ball(e,r) c= V by A1,TOPMETR:15;
    consider m being non zero Element of NAT such that
A4: OpenHypercube(e,1/m) c= Ball(e,r) by A2,Th19,GOBOARD6:1;
    take m;
    thus thesis by A3,A4;
  end;
