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 Th12:
  p is dominated_by_0 & 2*Sum (p|k) = k & p = (p|k)^q implies q is
  dominated_by_0
proof
  assume that
A1: p is dominated_by_0 and
A2: 2 * Sum (p|k)=k and
A3: p = (p|k)^q;
A4: len (p|k)=k by A1,A2,Th11;
  rng q c= rng p & rng p c= {0,1} by A1,A3,AFINSQ_1:25;
  hence rng q c= {0,1};
  let n such that
  n <= dom q;
  p|(len (p|k)+n)=(p|k)^(q|n) by A3,AFINSQ_1:59;
  then
A5: Sum (p|(len (p|k)+n))= Sum (p|k)+ Sum (q|n) by AFINSQ_2:55;
  2* Sum (p|(len (p|k)+n))<= len (p|k)+n by A1,Th2;
  then k+ 2 * Sum (q|n) <= len (p|k)+n by A2,A5;
  hence thesis by A4,XREAL_1:6;
end;
