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 Th23:
  f is A,n,B,k-dominated-election implies f.1 = A
proof
  set f1=f|1;
  assume
A1:f is A,n,B,k-dominated-election;
  then n > k by Th14;
  then len f >= 1+0 by A1,NAT_1:13;
  then
A2: len f1 = 1 by FINSEQ_1:59;
  card (f1"{A}) > card (f1"{B}) by A1;
  then
A3: f1"{A} is non empty;
  dom f1 = Seg 1 by FINSEQ_1:def 3,A2;
  then
A4: f1"{A} = {1} by RELAT_1:132,FINSEQ_1:2,A3,ZFMISC_1:33;
  1 in {1} by FINSEQ_1:2;
  then f1.1 in {A} by A4,FUNCT_1:def 7;
  then A=f1.1 by TARSKI:def 1;
  hence thesis by FINSEQ_3:112;
end;
