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 Th16:
  n <> 0 & e1 in OpenHypercube(e,r) implies dist(e1,e) < r * sqrt(n)
  proof
    assume that
A1: n <> 0 and
A2: e1 in OpenHypercube(e,r);
A3: dist(e1,e) = |.e1-e.| by EUCLID:def 6;
    0 <= Sum sqr (e1-e) by RVSUM_1:86;
    then
A4: sqrt Sum sqr (e1-e) < sqrt(n*r^2) by A1,A2,Th15,SQUARE_1:27;
    0 <= r by A1,A2;
    then sqrt(r^2) = r by SQUARE_1:22;
    hence thesis by A3,A4,SQUARE_1:29;
  end;
