
theorem Th19:
  for M being finite-degree Matroid for C being Subset of M for A
being independent Subset of M holds A is_maximal_independent_in C iff A c= C &
  card A = Rnk C
proof
  let M be finite-degree Matroid;
  let C be Subset of M;
  set X = {card A where A is independent Subset of M: A c= C};
  let A be independent Subset of M;
  consider B being independent Subset of M such that
A1: B c= C and
A2: card B = Rnk C by Th18;
A3: now
    let A be independent Subset of M;
    assume that
A4: A c= C and
A5: card A = Rnk C;
    thus A is_maximal_independent_in C
    proof
      thus A is independent & A c= C by A4;
      let B be Subset of M;
      assume B is independent;
      then reconsider B9 = B as independent Subset of M;
      assume B c= C;
      then card B9 in X;
      then
A6:   card B9 c= Rnk C by ZFMISC_1:74;
      assume
A7:   A c= B;
      then card A c= card B9 by CARD_1:11;
      then card A = card B9 by A5,A6;
      hence thesis by A7,CARD_2:102;
    end;
  end;
  hereby
    assume
A8: A is_maximal_independent_in C;
    hence A c= C;
    B is_maximal_independent_in C by A3,A1,A2;
    hence card A = Rnk C by A2,A8,Th16;
  end;
  thus thesis by A3;
end;
