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

theorem
  A is cycle & e in A implies e is_dependent_on A\{e}
proof
  assume that
A1: A is cycle and
A2: e in A;
  reconsider Ae = A\{e} as independent Subset of M by A1,A2;
  Ae is_maximal_independent_in A by A1,A2,Th38;
  then Rnk A = card Ae by Th19;
  hence Rnk((A\{e})\/{e}) = card Ae by A2,ZFMISC_1:116
    .= Rnk (A\{e}) by Th21;
end;
