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 Th23:
  Domin_0(n,0) = {n-->0}
proof
  set p=n-->0;
A1: Domin_0(n,0) c= {p}
  proof
    let x be object;
    assume x in Domin_0(n,0);
    then consider q such that
A2: x=q and
    q is dominated_by_0 and
A3: dom q = n and
A4: Sum q =0 by Def2;
    len q=n & q is nonnegative-yielding by A3;
    then q = n --> 0 or (q={} & n=0) by A4,AFINSQ_2:62;
    then q= n-->0;
    hence thesis by A2,TARSKI:def 1;
  end;
  {p} c= Domin_0(n,0)
  proof
A5: p is dominated_by_0 by Lm2;
    dom p=n & Sum p=n*0 by AFINSQ_2:58;
    then
A6: p in Domin_0(n,0) by A5,Def2;
    let x be object;
    assume x in {p};
    hence thesis by A6,TARSKI:def 1;
  end;
  hence thesis by A1;
end;
