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 Th22:
  Domin_0(n,m) is empty iff 2*m > n
proof
  thus Domin_0(n,m) is empty implies 2 * m > n
  proof
    set q=m-->1;
    assume
A1: Domin_0(n,m) is empty;
    assume
A2: 2*m <= n;
    m<=m+m by NAT_1:12;
    then reconsider nm=n-m as Nat by A2,NAT_1:21,XXREAL_0:2;
    set p=nm-->0;
    2*m-m <= nm by A2,XREAL_1:9;
    then
A3: p^q is dominated_by_0 by Th5;
    dom (p^q)=len p+len q & dom p=nm by AFINSQ_1:def 3;
    then
A4: dom (p^q)=nm+m;
A5: Sum (p^q)=Sum p+ Sum q by AFINSQ_2:55;
    Sum p=0*nm & Sum q =1*m by AFINSQ_2:58;
    hence thesis by A1,A5,A4,A3,Def2;
  end;
  assume
A6: 2 * m > n;
  assume Domin_0(n,m) is non empty;
  then consider x being object such that
A7: x in Domin_0(n,m);
  consider p such that
  p = x and
A8: p is dominated_by_0 and
A9: dom p = n and
A10: Sum p = m by A7,Def2;
  p|n=p by A9;
  hence thesis by A6,A8,A10,Th2;
end;
