Corollary

Let $A$ be a ring satisfying Hensel's condition and $x = (x_1, \ldots, x_n)$ a family of topologically nilpotent elements of $A$. Let $g = (g_1, \ldots, g_q)$ be a system without constant term of series in $A[[X_1, \ldots, X_s]]$ and $f = (f_1, \ldots, f_p)$ a system of formal power series in $A[[X_1, \ldots, X_q]]$. Then $g(x) = (g_1(x), \ldots, g_q(x))$ is a family of topologically nilpotent elements of $A$ and

The fact that the $g_i(x)$ are topologically nilpotent follows immediately from Proposition 6 and the fact that in $A$ the ideal of topologically nilpotent elements is closed. Relation (18) is obvious when the $f_j$ are polynomials; on the other hand, if $m$ and $m'$ are the ideals of series without constant term in

respectively, clearly the relation $f \in {m'}^k$ implies $f(g_1, \ldots, g_q) \in {m'}^k$. The two sides of (18) are therefore continuous functions of $f$ to $(A[[X_1, \ldots, X_q]])^p$ if $A[[X_1, \ldots, X_q]]$ is given the $m$-adic topology, by virtue of the above remark and Proposition 6; whence relation (18).

In what follows, for a ring $A$ and an ideal $m$ of $A$ we shall denote by $m^{x^n}$ the product set $\prod_{i=1}^n m$, in $A^n$, where $m_i = m$ for $1 \leq i \leq n$, to avoid ambiguity.