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

theorem Th41:
  A is cycle & B is cycle & A c= B implies A = B
proof
  assume that
A1: A is dependent and
  for e st e in A holds A \ {e} is independent and
  B is dependent and
A2: for e st e in B holds B \ {e} is independent;
  assume that
A3: A c= B and
A4: A <> B;
  consider x being object such that
A5: not (x in A iff x in B) by A4,TARSKI:2;
  reconsider x as Element of M by A5;
A6: A c= B\{x} by A3,A5,ZFMISC_1:34;
  B\{x} is independent by A2,A3,A5;
  hence contradiction by A1,A6,Th3;
end;
