 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 d)
  kappa(X,Y) <= kappa_1(X,Y) <= kappa_2(X,Y)
  proof
    per cases;
    suppose
A0:   X <> {}; then
WA:   kappa_1(X,Y) = card Y / card (X \/ Y) by Kappa1;
U1:   card (X /\ Y) <= card X by NAT_1:43,XBOOLE_1:17;
J0:   card Y <= card (X \/ Y) by NAT_1:43,XBOOLE_1:7;
      X \/ Y = X \/ (Y \ X) by XBOOLE_1:39; then
D3:   card (X \/ Y) = card X + card (Y \ X) by CARD_2:40,XBOOLE_1:85;
d4:   Y = (X /\ Y) \/ (Y \ X) by XBOOLE_1:51;
     (X /\ Y) misses (Y \ X) by XBOOLE_1:85,17; then
D4:  card Y = card (X /\ Y) + card (Y \ X) by d4,CARD_2:40;
     (X \/ Y)` = X` /\ Y` by XBOOLE_1:53; then
     (X \/ Y)` \/ Y = (X` \/ Y) /\ (Y` \/ Y) by XBOOLE_1:24
                   .= (X` \/ Y) /\ [#]R by PRE_TOPC:2
                   .= (X` \ Y) \/ Y by XBOOLE_1:39; then
d5:   X` \/ Y = (X` \ Y) \/ Y = (X \/ Y)` \/ Y by XBOOLE_1:39;
      (X \/ Y)` c= Y` by SUBSET_1:12,XBOOLE_1:7; then
      (X \/ Y)` misses Y by XBOOLE_1:63,79; then
D5:   card (X` \/ Y) = card ((X \/ Y)`) + card Y by d5,CARD_2:40;
      card (X /\ Y) / card X <= (card (X /\ Y) + card (Y \ X)) /
         (card X + card (Y \ X)) by Lemacik,A0,U1; then
      kappa (X,Y) <= (card (X /\ Y) + card (Y \ X)) /
        card (X \/ Y) by A0,KappaDef,D3;
      hence kappa(X,Y) <= kappa_1(X,Y) by A0,Kappa1,D4;
      (X \/ Y)` \/ (X \/ Y) = [#]R by PRE_TOPC:2; then
      card ((X \/ Y)`) + card (X \/ Y) = card [#]R by TDLAT_1:2,CARD_2:40;
      hence thesis by D5,A0,J0,Lemacik,WA;
    end;
    suppose
      X = {}; then
b3:   X = {}R; then
      kappa_1(X,Y) = 1 by Lemma2;
      hence thesis by b3,Lemma3,KappaDef;
    end;
  end;
