reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem
  for a, o being Point of TOP-REAL n holds
  n <> 0 & a in Ball(o,r) implies |.Sum(a-o).| < n*r
  proof
    let a, o be Point of TOP-REAL n;
    set R1 = a-o, R2 = n|->r;
    assume that
A1: n <> 0 and
A2: a in Ball(o,r);
A3: Sum R2 = n*r by RVSUM_1:80;
A5: for j being Nat st j in Seg n holds (abs R1).j < R2.j
    proof
      let j be Nat;
      assume j in Seg n;
      then
A6:   R2.j = r by FINSEQ_2:57;
      |.R1.j.| < r by A2,EUCLID_9:9;
      hence thesis by A6,VALUED_1:18;
    end;
    then
A7: for j being Nat st j in Seg n holds (abs R1).j <= R2.j;
    ex j being Nat st j in Seg n & (abs R1).j < R2.j
    proof
      take 1;
      1 <= n by A1,NAT_1:14;
      hence 1 in Seg n;
      hence thesis by A5;
    end;
    then
A8: Sum abs R1 < Sum R2 by A7,RVSUM_1:83;
    |.Sum R1.| <= Sum abs R1 by Th21;
    hence thesis by A3,A8,XXREAL_0:2;
  end;
