 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 4 e)
  kappa_1 (X,Y) = kappa (X \/ Y,Y)
  proof
    per cases;
    suppose
A1:   X \/ Y <> {}; then
      kappa (X \/ Y,Y) = card ((X \/ Y) /\ Y) / card (X \/ Y) by KappaDef
         .= card Y / card (X \/ Y) by XBOOLE_1:21;
      hence thesis by A1,Kappa1;
    end;
    suppose
A1:   X = {}; then
AA:   X c= Y;
      kappa (X \/ Y, Y) = 1 by Prop1a,A1;
      hence thesis by Prop11a,AA;
    end;
  end;
