reserve M for finite-degree Matroid,
  A,B,C for Subset of M,
  e,f for Element of M;

theorem Th33:
  Rnk Span A = Rnk A
proof
  consider Ca being independent Subset of M such that
A1: Ca c= A and
A2: card Ca = Rnk A by Th18;
  A c= Span A by Th31;
  then Ca c= Span A by A1;
  then consider C being independent Subset of M such that
A3: Ca c= C and
A4: C is_maximal_independent_in Span A by Th14;
  now
    assume C c/= Ca;
    then consider x being object such that
A5: x in C and
A6: x nin Ca;
    C c= Span A by A4;
    then x in Span A by A5;
    then consider e being Element of M such that
A7: x = e and
A8: e is_dependent_on A;
    {e} c= C by A5,A7,ZFMISC_1:31;
    then Ca\/{e} c= C by A3,XBOOLE_1:8;
    then reconsider Ce = Ca\/{e} as independent Subset of M by Th3;
    Ce c= A\/{e} by A1,XBOOLE_1:9;
    then consider D being independent Subset of M such that
A9: Ce c= D and
A10: D is_maximal_independent_in A\/{e} by Th14;
    card Ca = Rnk (A\/{e}) by A2,A8
      .= card D by A10,Th19;
    then
A11: card Ce <= card Ca by A9,NAT_1:43;
    card Ca <= card Ce by NAT_1:43,XBOOLE_1:7;
    then card Ca = card Ce by A11,XXREAL_0:1;
    then Ca = Ce by CARD_2:102,XBOOLE_1:7;
    then e nin {e} by A6,A7,XBOOLE_0:def 3;
    hence contradiction by TARSKI:def 1;
  end;
  then C = Ca by A3;
  hence thesis by A2,A4,Th19;
end;
