reserve
  x, y for object,
  i, n for Nat,
  r, s for Real,
  f1, f2 for n-element real-valued FinSequence;
reserve e, e1 for Point of Euclid n;

theorem Th13:
  r <= s implies OpenHypercube(e,r) c= OpenHypercube(e,s)
  proof
    assume
A1: r <= s;
A2: dom Intervals(e,s) = dom e by Def3;
A3: dom Intervals(e,r) = dom e by Def3;
    now
      let x be object;
      assume
A4:   x in dom Intervals(e,r);
      reconsider u = e.x as Real;
A5:   Intervals(e,r).x = ].u-r,u+r.[ & Intervals(e,s).x = ].u-s,u+s.[
      by A4,A3,Def3;
      u-s <= u-r & u+r <= u+s by A1,XREAL_1:6,10;
      hence Intervals(e,r).x c= Intervals(e,s).x by A5,XXREAL_1:46;
    end;
    hence thesis by A2,A3,CARD_3:27;
  end;
