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.