
theorem Th18:
  for M being finite-degree Matroid for C being Subset of M ex A
  being independent Subset of M st A c= C & card A = Rnk C
proof
  let M be finite-degree Matroid;
  let C be Subset of M;
  defpred P[Nat] means for A being independent Subset of M st A c= C holds
  card A <= $1;
  defpred Q[Nat] means ex A being independent Subset of M st A c= C & $1 =
  card A;
  set X = {card A where A is independent Subset of M: A c= C};
A1: {} M c= C;
  card {} = card {};
  then
A2: ex n being Nat st Q[n] by A1;
  consider n being Nat such that
A3: for A being finite Subset of M st A is independent holds card A <= n
  by Def6;
A4: ex ne being Nat st P[ne]
  proof
    take n;
    thus thesis by A3;
  end;
  consider n0 being Nat such that
A5: P[n0] & for m being Nat st P[m] holds n0 <= m from NAT_1:sch 5(A4);
  now
    let a be set;
    assume a in X;
    then consider A being independent Subset of M such that
A6: a = card A and
A7: A c= C;
    card A <= n0 by A5,A7;
     then Segm card A c= Segm n0 by NAT_1:39;
    hence a c= Segm n0 by A6;
  end;
  then
A8: Rnk C c= n0 by ZFMISC_1:76;
A9: for k being Nat st Q[k] holds k <= n0 by A5;
  consider n being Nat such that
A10: Q[n] & for m being Nat st Q[m] holds m <= n from NAT_1:sch 6(A9,
  A2);
  P[n] by A10;
  then
A11: n0 <= n by A5;
  consider A being independent Subset of M such that
A12: A c= C and
A13: n = card A by A10;
  take A;
  n <= n0 by A5,A10;
  then
A14: n = n0 by A11,XXREAL_0:1;
  then n0 in X by A12,A13;
  then n0 c= Rnk C by ZFMISC_1:74;
  hence thesis by A8,A12,A13,A14;
end;
