reserve D,D1,D2 for non empty set,
        d,d1,d2 for XFinSequence of D,
        n,k,i,j for Nat;
reserve A,B for object,
        v for Element of (n+k)-tuples_on {A,B},
        f,g for FinSequence;

theorem Th24:
  for d be XFinSequence of NAT holds
    d in Domin_0(n+k,k) iff <*0*>^(XFS2FS d) in DominatedElection(0,n+1,1,k)
proof
  let d be XFinSequence of NAT;
  set Xd=XFS2FS d,Z=<*0*>,ZXd=Z^Xd;
  reconsider D=d as XFinSequence of REAL;
A1:  len Z = 1 by FINSEQ_1:39;
  rng Z ={0} by FINSEQ_1:39;
  then
A2: rng Z c= {0,1} by ZFMISC_1:7;
  thus d in Domin_0(n+k,k) implies ZXd in DominatedElection(0,n+1,1,k)
    proof
      assume
A4:     d in Domin_0(n+k,k);
      then
A5:     d is dominated_by_0 by CATALAN2:20;
A6:   Sum d = k by A4,CATALAN2:20;
      len d = n+k by A4,CATALAN2:20;
      then
A7:     len Xd = n+k by AFINSQ_1:def 9;
A8:   rng Xd c= {0,1} by A5,Th2;
      rng (Z^Xd) = rng Z\/rng Xd by FINSEQ_1:31;
      then reconsider ZX=ZXd as FinSequence of {0,1}
        by XBOOLE_1:8,A8,A2,FINSEQ_1:def 4;
      len ZX = n+k+1 by A1,A7,FINSEQ_1:22;
      then
A9:     ZX is Tuple of n+1+k,{0,1} by CARD_1:def 7;
A10:  Sum ZXd = 0 + Sum Xd by RVSUM_1:76
             .= 0+addreal $$ XFS2FS D by RVSUM_1:def 12
             .= addreal "**" D by AFINSQ_2:44
             .= k by A6,AFINSQ_2:48;
      for i st i >0 holds 2* Sum (ZXd|i) < i
      proof
        let i such that
A11:      i>0;
        reconsider i1=i-1 as Nat by A11,NAT_1:14,NAT_1:21;
        ZXd| i = ZXd|Seg (i1+1)
              .= Z ^ (Xd|i1) by A1,FINSEQ_6:14;
        then
A12:      Sum (ZXd| i) = 0 + Sum (Xd|i1) by RVSUM_1:76
              .= 0 + Sum XFS2FS (d|i1) by Th1
              .= Sum XFS2FS (D|i1)
              .=addreal $$ XFS2FS (D|i1) by RVSUM_1:def 12
              .= addreal "**" (D|i1) by AFINSQ_2:44
              .= Sum (d|i1) by AFINSQ_2:48;
        2* Sum (d|i1) < i1+1 by A5,CATALAN2:2,NAT_1:13;
        hence thesis by A12;
      end;
      then ZXd is 0,n+1,1,k-dominated-election by A9,Th22,A10;
      hence thesis by Def3;
  end;
  assume ZXd in DominatedElection(0,n+1,1,k);
  then
A13: ZXd is 0,n+1,1,k-dominated-election by Def3;
  then ZXd is Tuple of n+1+k,{0,1} by Th22;
  then
A14: rng ZXd c= {0,1} by FINSEQ_1:def 4;
  rng Xd c= rng ZXd by FINSEQ_1:30;
  then
A15: rng Xd c= {0,1} by A14;
A16: len ZXd = n+1+k by A13,CARD_1:def 7;
A17: len ZXd = 1 + len Xd by A1,FINSEQ_1:22;
A18: len Xd = len d by AFINSQ_1:def 9;
  for k st k <= dom d holds 2*Sum (d|k) <= k
  proof
    let k such that k <= dom d;
    ZXd| (k+1) = Z ^ (Xd|k) by A1,FINSEQ_6:14;
    then Sum (ZXd| (k+1)) = 0 + Sum (Xd|k) by RVSUM_1:76
         .= 0 + Sum XFS2FS (d|k) by Th1
         .= Sum XFS2FS (D|k)
         .= addreal $$ XFS2FS (D|k) by RVSUM_1:def 12
         .= addreal "**" (D|k) by AFINSQ_2:44
         .= Sum (d|k) by AFINSQ_2:48;
    then  2* Sum (d|k) < k+1 by A13,Th22;
    hence thesis by NAT_1:13;
  end;
  then A19:d is dominated_by_0 by A15, Th2;
  Sum (ZXd) = 0 + Sum (Xd) by RVSUM_1:76
           .= Sum XFS2FS D
           .= addreal $$ XFS2FS (D) by RVSUM_1:def 12
           .= addreal "**" (D) by AFINSQ_2:44
           .= Sum (d) by AFINSQ_2:48;
  then Sum d = k by A13,Th22;
  hence d in Domin_0(n+k,k) by A19,A16,A17,A18,CATALAN2:def 2;
end;
