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 Th22:
  Ball(e,r) c= OpenHypercube(e,r)
  proof
    let g be object;
    assume
A1: g in Ball(e,r);
    then reconsider g as Point of Euclid n;
A2: dom Intervals(e,r) = dom e by Def3;
A3: dom g = Seg n & dom e = Seg n by FINSEQ_1:89;
    now
      let x be object;
      reconsider u = e.x, v = g.x as Real;
      assume
A4:   x in dom Intervals(e,r);
      then
A5:   Intervals(e,r).x = ].u-r,u+r.[ by A2,Def3;
      dom(g-e) = dom g /\ dom e by VALUED_1:12;
      then
A6:   (g-e).x = v-u by A2,A3,A4,VALUED_1:13;
A7:   v = u + (v-u);
      |.(g-e).x.| < r by A1,Th10;
      hence g.x in Intervals(e,r).x by A6,A5,A7,FCONT_3:3;
    end;
    hence thesis by A2,A3,CARD_3:9;
  end;
