 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 1 c)
  Z c= Y c= X implies kappa (X,Z) <= kappa (Y,Z)
  proof
    assume
AA: Z c= Y c= X;
    per cases;
    suppose
A1:   X <> {} & Y <> {}; then
      kappa (Y,Z) = card (Y /\ Z) / card Y by KappaDef; then
F1:   kappa (Y,Z) = card Z / card Y by AA,XBOOLE_1:28;
      Z c= X by AA; then
e2:   X /\ Z = Z by XBOOLE_1:28;
      kappa (X,Z) = card Z / card X by e2,A1,KappaDef;
      hence thesis by A1,XREAL_1:118,F1,AA,NAT_1:43;
    end;
    suppose X = {} & Y <> {};
      hence thesis by AA;
    end;
    suppose X = {} & Y = {};
      hence thesis;
    end;
    suppose X <> {} & Y = {}; then
      kappa (Y,Z) = 1 by KappaDef;
      hence thesis by XXREAL_1:1;
    end;
  end;
