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
  f is A,n,B,k-dominated-election & g is A,i,B,j-dominated-election implies
    f^g is A,n+i,B,k+j-dominated-election
proof
  assume that
A1:   f is A,n,B,k-dominated-election
    and
A2:   g is A,i,B,j-dominated-election;
A3: A<>B by A1,Th13;
  reconsider F=f as Element of (n+k)-tuples_on {A,B} by A1;
  reconsider G=g as Element of (i+j)-tuples_on {A,B} by A2;
A4: len F = n+k by CARD_1:def 7;
A5: len (F^G) = (n+k)+(i+j) by CARD_1:def 7;
  then reconsider FG=F^G as Element of ((n+k)+(i+j))-tuples_on {A,B}
    by FINSEQ_2:92;
   card (FG"{A}) = card (F"{A}) + card (G"{A}) by FINSEQ_3:57
                .= n + card (G"{A}) by A1,Def1
                .= n + i by A2,Def1;
  hence f^g in Election(A,n+i,B,k+j) by Def1;
  let h be Nat such that
A6: h>0;
  per cases;
    suppose h <= n+k;
      then FG|h = F|h by A4,FINSEQ_5:22;
      hence thesis by A1,A6;
    end;
    suppose
A7:     h > n+k & h <= n+k+i+j;
      then reconsider h1=h-(n+k) as Nat by NAT_1:21;
A8:     h1 <>0 by A7;
      h1+len F = h by A4;
      then
A9:     FG| h = F^(G|h1) by FINSEQ_6:14;
      then
A10:    card ((FG|h)"{A}) = card (F"{A}) + card ((G|h1)"{A}) by FINSEQ_3:57
                         .= n+card ((G|h1)"{A}) by A1,Def1;
A11:  card ((FG|h)"{B}) = card (F"{B}) + card ((G|h1)"{B}) by A9,FINSEQ_3:57
                       .= k+card ((G|h1)"{B}) by A3,A1,Th11;
      card ((G|h1)"{A}) >  card ((G|h1)"{B}) by A8,A2;
      hence thesis by A10,A11,Th14,A1,XREAL_1:8;
    end;
    suppose h > n+k+i+j;
      then
A12:    FG|h=FG by FINSEQ_1:58,A5;
A13:  card (FG"{A}) = card (F"{A}) +card (G"{A}) by FINSEQ_3:57
                   .= n+ card (G"{A}) by A1,Def1
                   .= n+i by A2,Def1;
A14:  card (FG"{B}) = card (F"{B}) +card (G"{B}) by FINSEQ_3:57
                   .= k+card (G"{B}) by A3,A1,Th11
                   .= k+j by A3,A2,Th11;
      n > k by Th14,A1;
      hence thesis by Th14,A2,A13,A14,XREAL_1:8,A12;
    end;
end;
