On matroid connectivity

If M is a loopless matroid in which M/vbX and M/vbY are connected and X∩Y is non-empty, then one easily shows that M/vb(X∪Y) is connected. Likewise, it is straightforward to show that if G and H are n-connected graphs having at least n common vertices, then G ∪ H is n-connected. The purpose of this note is to prove a matroid connectivity result that is a common generalization of these two observations.