 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 Prop1d: :: binary variant of Proposition 1 d)
  kappa (X, Y \/ Z) <= kappa (X,Y) + kappa (X,Z)
  proof
    per cases;
    suppose
A0:   X <> {}; then
A1:   kappa (X, Y \/ Z) = card (X /\ (Y \/ Z)) / card X by KappaDef;
A2:   kappa (X, Y) = card (X /\ Y) / card X by A0,KappaDef;
A3:   kappa (X, Z) = card (X /\ Z) / card X by A0,KappaDef;
      X /\ (Y \/ Z) = (X /\ Y) \/ (X /\ Z) by XBOOLE_1:23; then
      card (X /\ (Y \/ Z)) / card X <=
        (card (X /\ Y) + card (X /\ Z)) / card X by XREAL_1:72,CARD_2:43;
      hence thesis by A2,A3,A1,XCMPLX_1:62;
    end;
    suppose
A2:   X = {}; then
A3:   kappa (X, Y \/ Z) = 1 by KappaDef;
      kappa (X, Y) = 1 & kappa (X, Z) = 1 by A2,KappaDef;
      hence thesis by A3;
    end;
  end;
