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
  for n,m st 2*m <= n ex CardF be XFinSequence of NAT st card Domin_0(n,
  m) = Sum CardF + card Domin_0(n-'1,m) & dom CardF = m & for j st j < m holds
  CardF.j = card Domin_0(2*j,j) * card Domin_0(n-'2*(j+1),m-'(j+1))
proof
  let n,m such that
A1: 2*m <=n;
  set Z=Domin_0(n,m);
  set Zne={pN: pN in Domin_0(n,m) & {N: 2*Sum(pN|N)=N & N>0}<>{}};
A2: Zne c= Z
  proof
    let x be object;
    assume x in Zne;
    then ex pN st x=pN & pN in Domin_0(n,m) & {N: 2*Sum(pN|N)=N & N>0}<>{};
    hence thesis;
  end;
  set Ze={pN: pN in Domin_0(n,m) & {N: 2*Sum(pN|N)=N & N>0}={}};
A3: Ze c= Z
  proof
    let x be object;
    assume x in Ze;
    then ex pN st x=pN & pN in Domin_0(n,m) & {N: 2*Sum(pN|N)=N & N>0}={};
    hence thesis;
  end;
  reconsider Zne as finite set by A2;
  consider C be XFinSequence of NAT such that
A4: card Zne = Sum C and
A5: dom C = m and
A6: for j st j < m holds C.j=card Domin_0(2*j,j)*card Domin_0(n-'2*(j+1
  ),m-'(j+1)) by A1,Th37;
  reconsider Ze as finite set by A3;
  take C;
A7: Ze misses Zne
  proof
    assume Ze meets Zne;
    then consider x being object such that
A8: x in Ze and
A9: x in Zne by XBOOLE_0:3;
A10: ex qN st qN=x & qN in Z & {N: 2*Sum(qN|N)=N & N>0}={} by A8;
    ex pN st pN=x & pN in Z & {N: 2*Sum(pN|N)=N & N>0}<>{} by A9;
    hence thesis by A10;
  end;
A11: Z c= Ze \/ Zne
  proof
    let x be object such that
A12: x in Z;
    consider p be XFinSequence of NAT such that
A13: p = x and
    p is dominated_by_0 and
    dom p = n and
    Sum p = m by A12,Def2;
    reconsider p as Element of NAT^omega by AFINSQ_1:def 7;
    set I={N: 2*Sum(p|N)=N & N>0};
    now
      per cases;
      suppose
        I={};
        then p in Ze by A12,A13;
        hence thesis by A13,XBOOLE_0:def 3;
      end;
      suppose
        I<>{};
        then p in Zne by A12,A13;
        hence thesis by A13,XBOOLE_0:def 3;
      end;
    end;
    hence thesis;
  end;
  Ze\/ Zne c= Z by A3,A2,XBOOLE_1:8;
  then
A14: Ze\/Zne=Z by A11;
  now
    per cases;
    suppose
A15:  n=0;
      then 2*m=0 by A1;
      then C={} by A5;
then A16:Sum C= 0;
      n-1<1-1 by A15;
      hence card Z = Sum C + card Domin_0(n-'1,m) by A15,A16,XREAL_0:def 2;
    end;
    suppose
A17:  n>0;
      then reconsider n1=n-1 as Nat by NAT_1:20;
      n=n1+1;
      then
A18:  card Ze = card Domin_0(n1,m) by Th40;
      n1 = n-'1 by A17,NAT_1:14,XREAL_1:233;
      hence card Z = Sum C+card Domin_0(n-'1,m) by A7,A14,A4,A18,CARD_2:40;
    end;
  end;
  hence thesis by A5,A6;
end;
