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 Th11:
  A <> B implies (v in Election(A,n,B,k) iff card (v"{B}) = k)
proof
  assume
A1: A<>B;
A2: rng v c= {A,B};
A3: len v = n+k by CARD_1:def 7;
A4: card (v"{A}) + card (v"{B}) = len v by Th6,A2,A1;
  thus v in Election(A,n,B,k) implies card (v"{B}) = k
  proof
    assume v in Election(A,n,B,k);
    then card (v"{A})=n by Def1;
    hence thesis by A4,A3;
  end;
  assume card (v"{B}) = k;
  hence v in Election(A,n,B,k) by A4,A3,Def1;
end;
