 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 Prop11b: :: rif_1 for kappa_2
  kappa_2 (X,Y) = 1 iff X c= Y
  proof
    thus kappa_2 (X,Y) = 1 implies X c= Y
    proof
      assume kappa_2 (X,Y) = 1; then
      card (X` \/ Y) = card ([#]R) by XCMPLX_1:58;
      hence thesis by LemmaSet,LemmaCard2;
    end;
    assume X c= Y; then
    X` \/ Y = [#]R by LemmaSet;
    hence thesis by XCMPLX_1:60;
  end;
