Remark 3

With the notation of Remark 2, suppose further that $A$ is Hausdorff and complete and let us write every restricted formal power series $f \in A\{X_1, \ldots, X_r\}$ in the form

where the $c_{n_{r+1} \ldots n_p}$ are restricted formal power series. For every system $x = (x_1, \ldots, x_r)$ of elements of $A$, let

It follows immediately from Remark 1 that $\sum_{(n_i)} b_{n_{r+1} \ldots n_p} X_{r+1}^{n_{r+1}} \ldots X_p^{n_p}$ is a restricted formal power series denoted by $f(x_1, \ldots, x_r, X_{r+1}, \ldots, X_p)$; it is said to be obtained by substituting the $x_i$ for the $X_i$ for $1 \leq i \leq r$ in $f$.