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

theorem Th37:
  A is cycle implies A is non empty finite
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;
  set e = the Element of A;
  now
    assume
A3: A is non empty set;
    then e in A;
    then reconsider e as Element of M;
    reconsider Ae = A\{e} as independent Subset of M by A2,A3;
    A = Ae\/{e} by A3,ZFMISC_1:116;
    hence thesis;
  end;
  hence thesis;
end;
