 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 Prop11a: :: rif_1 for kappa_1
  kappa_1 (X,Y) = 1 iff X c= Y
  proof
    per cases;
    suppose
A1:   X \/ Y <> {} & Y <> {}; then
B1:   kappa_1 (X,Y) = card Y / card (X \/ Y) by Kappa1;
      thus kappa_1 (X,Y) = 1 implies X c= Y
      proof
        assume kappa_1 (X,Y) = 1; then
        card Y = card (X \/ Y) by XCMPLX_1:58,B1;
        hence thesis by LemmaCard;
      end;
      assume X c= Y; then
      card (X \/ Y) = card Y by XBOOLE_1:12;
      hence thesis by A1,XCMPLX_1:60,B1;
    end;
    suppose
A1:   X \/ Y <> {} & Y = {}; then
      kappa_1 (X,Y) = card Y / card (X \/ Y) by Kappa1
        .= 0 by A1;
      hence thesis by A1;
    end;
    suppose
      X \/ Y = {};
      hence thesis by Kappa1;
    end;
  end;
