reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;
reserve n for non zero Nat;
reserve n for non zero Nat;
reserve n for Nat,
        X for set,
        S for Subset-Family of X;
reserve n for Nat,
        S for Subset-Family of REAL;
reserve n       for Nat,
        a,b,c,d for Element of REAL n;
reserve n for non zero Nat;
reserve n     for non zero Nat,
        x,y,z for Element of REAL n;
reserve p for Element of EMINFTY n;

theorem
  for e being Point of Euclid n st e = p holds Ball(p,r) = OpenHypercube(e,r)
  proof
    let e be Point of Euclid n;
    assume
A1: e = p;
    Ball(p,r) = product Intervals(@p,r) by Th68;
    hence thesis by A1;
  end;
