 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 Prop1a: :: Proposition 1 a)
  kappa (X,Y) = 1 iff X c= Y
  proof
    per cases;
    suppose
A1: X <> {};
    thus kappa (X,Y) = 1 implies X c= Y
    proof
      assume kappa (X,Y) = 1; then
      card (X /\ Y) / card X = 1 by KappaDef,A1; then
      card (X /\ Y) = card X by XCMPLX_1:58; then
      X /\ Y = X by CARD_2:102,XBOOLE_1:17;
      hence thesis by XBOOLE_1:17;
    end;
    assume X c= Y; then
    X /\ Y = X by XBOOLE_1:28; then
    kappa (X,Y) = card X / card X by A1,KappaDef;
    hence thesis by XCMPLX_1:60,A1;
    end;
    suppose
      X = {};
      hence thesis by KappaDef;
    end;
  end;
