Minors of 3-Connected Matroids

In [5] it was shown that if M is a 3-connected matroid with a minor isomorphic to U²₄, then every pair of elements of M are in a U²₄ minor. The proof was fairly complicated. Here we derive that theorem from a more general result. We show that to test whether U²₄ (or any other 3-connected matroid N) has the property described above, it is only necessary to test that it works for those matroids M with 5 (or more generally, |E (N)| + 1) elements. This is essentially a lemma which will be used in a subsequent paper.