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 Th10:
  p is dominated_by_0 & n <= k+len p-2*Sum p implies (k-->0)^p^(n
  -->1) is dominated_by_0
proof
  assume that
A1: p is dominated_by_0 and
A2: n <= k + len p - 2 * Sum p;
  set q=(k-->0);
  dom q=k & len q =dom q;
  then
A3: len (q^p)=k+len p by AFINSQ_1:17;
  Sum q=k*0 by AFINSQ_2:58;
  then
A4: Sum (q^p)=(0 qua Nat)+ Sum p by AFINSQ_2:55;
  q is dominated_by_0 by Lm2;
  then q^p is dominated_by_0 by A1,Th7;
  hence thesis by A2,A3,A4,Th9;
end;
