reserve x, x1, x2, y, X, D for set,
  i, j, k, l, m, n, N for Nat,
  p, q for XFinSequence of NAT,
  q9 for XFinSequence,
  pd, qd for XFinSequence of D;

theorem Th6:
  p is dominated_by_0 implies p|n is dominated_by_0
proof
  assume
A1: p is dominated_by_0;
A2: for k st k<=dom (p|n) holds 2*Sum((p|n)|k)<=k
  proof
    let k;
    assume k <= dom (p|n);
    then
A3: Segm k c= Segm len(p|n) by NAT_1:39;
    dom (p|n) = dom p/\n by RELAT_1:61;
    then (p|n)|k=p|k by A3,RELAT_1:74,XBOOLE_1:18;
    hence thesis by A1,Th2;
  end;
  rng (p|n) c= rng p & rng p c= {0,1} by A1;
  then rng(p|n)c={0,1};
  hence thesis by A2;
end;
