Here is a proof I am reading for class:

Let be a local ring, and finitely generated -modules. If , then or .

Proof. Let be a the maximal ideal of and be the residue field. We may assume that and prove that By Nakayama's lemma, we see that . Therefore is naturally a -vector space with rank . Hence by Nakayama's Lemma.

I don't understand the part in red. Can somebody explain why ?