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 Th12:
  q in OpenHypercube(p,r/2) implies OpenHypercube(q,r/2) c= OpenHypercube(p,r)
  proof
    assume
A1: q in OpenHypercube(p,r/2);
      let x be object;
      assume
A2:   x in OpenHypercube(q,r/2);
      then reconsider x1 = x as Point of TOP-REAL n;
      now
        let i be Nat;
        assume
A3:     i in Seg n;
        then x1.i in ].q.i-r/2,q.i+r/2.[ by A2,TIETZE_2:4; then
A4:     q.i-r/2 < x1.i & x1.i < q.i+r/2 by XXREAL_1:4;
        q.i in ].p.i-r/2,p.i+r/2.[ by A1,A3,TIETZE_2:4;
        then p.i - r/2 < q.i & q.i < p.i + r/2 by XXREAL_1:4;
        then p.i - r/2 - r/2 < q.i - r/2 & q.i + r/2 < p.i + r/2 + r/2
          by XREAL_1:8;
        then p.i - r < x1.i & x1.i < p.i + r by A4,XXREAL_0:2;
        hence x1.i in ].p.i-r,p.i+r.[ by XXREAL_1:4;
      end;
      hence x in OpenHypercube(p,r) by TIETZE_2:4;
  end;
