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 Th11:
  0 < r implies e in OpenHypercube(e,r)
  proof
    assume
A1: 0 < r;
    set f = Intervals(e,r);
A2: dom f = dom e by Def3;
    now
      let x be object;
      assume x in dom f;
      then
A3:   f.x = ].e.x-r,e.x+r.[ by A2,Def3;
A4:   e.x-r < e.x-0 by A1,XREAL_1:10;
      e.x+0 < e.x+r by A1,XREAL_1:8;
      hence e.x in f.x by A3,A4,XXREAL_1:4;
    end;
    hence thesis by A2,CARD_3:9;
  end;
