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 Th20:
  p in Domin_0(n,m) iff p is dominated_by_0 & dom p = n & Sum p = m
proof
  thus p in Domin_0(n,m) implies p is dominated_by_0 & dom p=n & Sum p=m
  proof
    assume p in Domin_0(n,m);
    then ex q st q=p & q is dominated_by_0 & dom q = n & Sum q = m by Def2;
    hence thesis;
  end;
  thus thesis by Def2;
end;
