Size of minimal generating set in Noetherian local ring is unique

Statement

Let $R$ be a Noetherian local ring and ${\mathfrak {m}}$ be its unique maximal ideal. Call a minimal generating set for ${\mathfrak {m}}$ a generating set for ${\mathcal {m}}$ as an ideal, such that no proper subset of it generates ${\mathcal {m}}$ as an ideal. Then, the size of any two minimal generating sets for ${\mathfrak {m}}$ is the same. In fact, the size of any minimal generating set equals the dimension of ${\mathcal {m}}/{\mathcal {m}}^{2}$ as a $R/{\mathfrak {m}}$-vector space.

Proof

The assertion follows from Nakayama's lemma.