 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 Prop1e: :: binary variant of Proposition 1 e)
  X <> {} & Y misses Z implies
    kappa (X,Y \/ Z) = kappa (X,Y) + kappa (X,Z)
  proof
    assume
A0: X <> {} & Y misses Z; then
A1: kappa (X,Y \/ Z) = card (X /\ (Y \/ Z)) / card X by KappaDef
       .= card ((X /\ Y) \/ (X /\ Z)) / card X by XBOOLE_1:23;
A2: kappa (X,Y) = card (X /\ Y) / card X &
      kappa (X,Z) = card (X /\ Z) / card X by A0,KappaDef;
    card ((X /\ Y) \/ (X /\ Z)) = card (X /\ Y) + card (X /\ Z)
      by CARD_2:40,A0,XBOOLE_1:76;
    hence thesis by A2,A1,XCMPLX_1:62;
  end;
