 reserve R for 1-sorted;
 reserve X,Y for Subset of R;
 reserve R for finite 1-sorted;
 reserve X,Y for Subset of R;
 reserve R for finite Approximation_Space;
 reserve X,Y,Z,W for Subset of R;

theorem :: Proposition 2 a), binary case
  X <> {} & Z \/ W = [#]R & Z misses W implies
    kappa (X,Z) + kappa (X,W) = 1
  proof
    assume
A1: X <> {} & Z \/ W = [#]R & Z misses W;
A3: kappa (X,W) = card (X /\ W) / card X by A1,KappaDef;
A6: X = X /\ (Z \/ W) by A1
     .= X /\ Z \/ X /\ W by XBOOLE_1:23;
A4: card (X /\ Z) + card (X /\ W) = card X by A6,A1,CARD_2:40,XBOOLE_1:76;
    kappa (X,Z) + kappa (X,W) =
         card (X /\ Z) / card X + card (X /\ W) / card X by A1,A3,KappaDef
      .= card X / card X by A4,XCMPLX_1:62;
    hence thesis by XCMPLX_1:60,A1;
  end;
