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
  Ball(p,r) = OpenHyperInterval(@p - (n|-> r), @p + (n|-> r))
  proof
    reconsider e = p as Point of Euclid n;
    ex a being Element of REAL n st a = e &
      OpenHypercube(e,r) = OpenHyperInterval(a - (n|-> r),
                             a + (n|-> r)) by Th51;
    hence thesis by Th68;
  end;
