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
  cl_Ball(p,r) = ClosedHyperInterval(@p - (n|-> r), @p + (n|-> r))
  proof
    reconsider q = p as Point of TOP-REAL n by EUCLID:22;
    ex a be Element of REAL n st a = p &
      ClosedHypercube(q,n |-> r)
        = ClosedHyperInterval(a - (n |-> r),a + (n |-> r)) by Th52;
    hence thesis by Th70;
  end;
