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 Th14:
  (n <> 0 or 0 < r) & e1 in OpenHypercube(e,r) implies
  for x being set holds |.(e1-e).x.| < r & |.e1.x-e.x.| < r
  proof
    assume that
A1: n <> 0 or 0 < r and
A2: e1 in OpenHypercube(e,r);
A3: dom e1 = Seg n & dom e = Seg n by FINSEQ_1:89;
A4: dom(e1-e) = dom e1 /\ dom e by VALUED_1:12;
    let x be set;
    per cases;
    suppose
A5:   x in dom e1;
      then
A6:   (e1-e).x = e1.x-e.x by A3,A4,VALUED_1:13;
      dom e = dom Intervals(e,r) by Def3;
      then
A7:   e1.x in Intervals(e,r).x by A5,A3,A2,CARD_3:9;
      Intervals(e,r).x = ].e.x-r,e.x+r.[ by A3,A5,Def3;
      hence thesis by A6,A7,FCONT_3:1;
    end;
    suppose
A8:   not x in dom e1;
      then not x in dom abs(e1-e) by A4,A3,VALUED_1:def 11;
      then abs(e1-e).x = 0 by FUNCT_1:def 2;
      then
A9:   |.(e1-e).x.| = 0 by VALUED_1:18;
      e1.x = 0 & e.x = 0 by A3,A8,FUNCT_1:def 2;
      hence thesis by A9,A1,A2;
    end;
  end;
