 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 Prop2b:
  Y c= Z implies kappa_1 (X,Y) <= kappa_1 (X,Z)
  proof
    assume
A0: Y c= Z; then
B3: card Z = card (Y \/ (Z \ Y)) by XBOOLE_1:45
          .= card Y + card (Z \ Y) by XBOOLE_1:79,CARD_2:40;
ZA: card Y <= card (X \/ Y) by NAT_1:43,XBOOLE_1:7;
ZC: X \/ Z = X \/ (Y \/ (Z \ Y)) by A0,XBOOLE_1:45
          .= X \/ Y \/ (Z \ Y) by XBOOLE_1:4;
    per cases;
    suppose
AA:   X \/ Y <> {}; then
      kappa_1 (X,Y) = card Y / card (X \/ Y) by Kappa1; then
ss:   kappa_1 (X,Y) <= (card Y + card (Z \ Y)) /
        (card (X \/ Y) + card (Z \ Y)) by AA,Lemacik,ZA;
      card Z / (card (X \/ Y) + card (Z \ Y)) <=
        card Z / card ((X \/ Y) \/ (Z \ Y)) by CARD_2:43,AA,XREAL_1:118; then
      kappa_1 (X,Y) <= card Z / card ((X \/ Y) \/ (Z \ Y))
        by ss,B3,XXREAL_0:2;
      hence thesis by Kappa1,AA,ZC;
    end;
    suppose
      X = {} & Z = {};
      hence thesis by A0;
    end;
    suppose
T1:   X \/ Y = {} & Z <> {}; then
T3:   X \/ Z <> {} & X = {};
      kappa_1 (X,Z) = card Z / card (X \/ Z) by Kappa1,T1
         .= 1 by XCMPLX_1:60,T3;
      hence thesis by T1,Kappa1;
    end;
  end;
