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;

theorem Th11:
  q in OpenHypercube(p,r) implies p in OpenHypercube(q,r)
  proof
    assume
A1: q in OpenHypercube(p,r);
    now
      let i be Nat;
      assume i in Seg n;
      then q.i in ].p.i - r,p.i + r.[ by A1,TIETZE_2:4;
      then p.i - r < q.i & q.i < p.i + r by XXREAL_1:4;
      then p.i - r + r < q.i + r & q.i - r < p.i + r - r  by XREAL_1:8;
      hence p.i in ].q.i - r,q.i + r.[ by XXREAL_1:4;
    end;
    hence thesis by TIETZE_2:4;
  end;
