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;
reserve pN, qN for Element of NAT^omega;

theorem Th38:
  for n st n > 0 holds Domin_0(2*n,n)={pN:pN in Domin_0(2*n,n) & {
N: 2*Sum(pN|N) = N & N > 0}<>{}}
proof
  let n such that
A1: n>0;
  set P={pN: pN in Domin_0(2*n,n) & {N: 2*Sum(pN|N)=N & N>0}<>{}};
  thus Domin_0(2*n,n) c= P
  proof
A2: n+n >(0 qua Nat)+(0 qua Nat) by A1;
    let x be object such that
A3: x in Domin_0(2*n,n);
    consider p such that
A4: x=p and
    p is dominated_by_0 and
A5: dom p = 2*n & Sum p = n by A3,Def2;
A6: p in NAT^omega by AFINSQ_1:def 7;
    2*Sum(p|(2*n))=2*n by A5;
    then 2*n in {N: 2*Sum(p|N)=N & N>0} by A2;
    hence thesis by A3,A4,A6;
  end;
  thus P c= Domin_0(2*n,n)
  proof
    let x be object;
    assume x in P;
    then ex pN st x=pN & pN in Domin_0(2*n,n) & {N: 2*Sum(pN|N)=N & N>0}<>{};
    hence thesis;
  end;
end;
