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 Th28:
  2*(m+1) <= n implies card Domin_0(n,m+1) = (n choose (m+1)) - (n choose m)
proof
  set CH=Choose(n,m+1,1,0);
  set Z=Domin_0(n,m+1);
  set W=Choose(n,m,1,0);
A1: card CH = (card n) choose (m+1) & card n=n by CARD_FIN:16;
  card (CH \ Z) = card CH - card Z by Th21,CARD_2:44;
  then
A2: card Z=card CH - card (CH \ Z);
  assume 2 * (m+1) <= n;
  then card Z=card CH-card W by A2,Th27;
  hence thesis by A1,CARD_FIN:16;
end;
