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 Th15:
  n <> 0 & e1 in OpenHypercube(e,r) implies Sum sqr (e1-e) < n*r^2
  proof
    assume that
A1: n <> 0 and
A2: e1 in OpenHypercube(e,r);
    set R1 = sqr(e1-e);
    set R2 = n|->(r^2);
A6: now
      let i;
      assume
A7:   i in Seg n;
A8:   dom e1 = Seg n & dom e = Seg n by FINSEQ_1:89;
      dom(e1-e) = dom e1 /\ dom e by VALUED_1:12;
      then
A9:   (e1-e).i = e1.i-e.i by A7,A8,VALUED_1:13;
A10:   R1.i = ((e1-e).i)^2 by VALUED_1:11;
A11:   R2.i = r^2 by A7,FINSEQ_2:57;
A12:   |.e1.i-e.i.| < r by A1,A2,Th14;
      ((e1-e).i)^2 = |.(e1-e).i.|^2 by COMPLEX1:75;
      hence R1.i < R2.i by A9,A10,A11,A12,SQUARE_1:16;
    end;
    then
A13: for i st i in Seg n holds R1.i <= R2.i;
    ex i st i in Seg n & R1.i < R2.i
    proof
      consider i being object such that
A14:   i in Seg n by A1,XBOOLE_0:def 1;
      reconsider i as Nat by A14;
      take i;
      thus thesis by A14,A6;
    end;
    then Sum R1 < Sum R2 by A13,RVSUM_1:83;
    hence thesis by RVSUM_1:80;
  end;
