 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 Prop1b: :: Proposition 1 b)
  Y c= Z implies kappa (X,Y) <= kappa (X,Z)
  proof
    assume
A0: Y c= Z;
    per cases;
    suppose
A1:   X <> {};
      card (X /\ Y) <= card (X /\ Z) by NAT_1:43,A0,XBOOLE_1:26; then
      card (X /\ Y) / card X <= card (X /\ Z) / card X by XREAL_1:72; then
      kappa (X,Y) <= card (X /\ Z) / card X by A1,KappaDef;
      hence thesis by A1,KappaDef;
    end;
    suppose X = {}; then
      kappa (X,Y) = 1 & kappa (X,Z) = 1 by KappaDef;
      hence thesis;
    end;
  end;
