Example 3

Let $A$ be a complete regular noetherian local ring of dimension $r$. The ring $A \{ X_1, ..., X_n \}$ of restricted formal power series is a regular noetherian local ring of dimension $r + n$ (§ 3, No. 4, Remark 1). If $(a_1, ..., a_r)$ is a system of coordinates of $A$, then $(a_1, ..., a_r, X_1, ..., X_n)$ is a system of coordinates of $A \{ X_1, ..., X_n \}$.