
theorem Th21:
  for M being finite-degree Matroid for C being finite Subset of M
  holds C is independent iff card C = Rnk C
proof
  let M be finite-degree Matroid;
  let C be finite Subset of M;
  set X = {card A where A is independent Subset of M: A c= C};
  hereby
    assume C is independent;
    then card C in X;
    then Segm card C c= Segm Rnk C by ZFMISC_1:74;
    then
A1: card C <= Rnk C by NAT_1:39;
    Rnk C <= card C by Th20;
    hence card C = Rnk C by A1,XXREAL_0:1;
  end;
  ex A being independent Subset of M st A c= C & card A = Rnk C by Th18;
  hence thesis by CARD_2:102;
end;
