Remark 1

Proposition 4 proves that for every closed ideal $a$ in a ring $A$ which is assumed to be Hausdorff and complete, the relations $b_i \in a$ for $1 \leq i \leq p$ imply $f(b_1, \ldots, b_p) \in a$ for every restricted formal power series $f \in A\{X_1, \ldots, X_p\}$.