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

theorem Th38:
  A is cycle iff A is non empty & for e st e in A holds A\{e}
  is_maximal_independent_in A
proof
  thus A is cycle implies A is non empty & for e st e in A holds A\{e}
  is_maximal_independent_in A
  proof
    assume that
A1: A is dependent and
A2: for e being Element of M st e in A holds A \ {e} is independent;
    thus A is non empty by A1;
    let e;
    set Ae = A\{e};
    assume
A3: e in A;
    hence Ae is independent & Ae c= A by A2,XBOOLE_1:36;
    let B;
    assume that
A4: B is independent and
A5: B c= A and
A6: Ae c= B;
    A = Ae\/{e} by A3,ZFMISC_1:116;
    hence thesis by A1,A4,A5,A6,ZFMISC_1:138;
  end;
  set a = the Element of A;
  assume that
A7: A is non empty and
A8: for e st e in A holds A\{e} is_maximal_independent_in A;
  a in A by A7;
  then reconsider a as Element of M;
  set Ae = A\{a};
A9: Ae is_maximal_independent_in A by A7,A8;
  hereby
    assume A is independent;
    then A = Ae by A9;
    then a nin {a} by A7,XBOOLE_0:def 5;
    hence contradiction by TARSKI:def 1;
  end;
  let e;
  assume e in A;
  then A\{e} is_maximal_independent_in A by A8;
  hence thesis;
end;
