Theorem 2

*Let $A$ be a ring and $m$ an ideal of $A$ such that the ordered pair $(A, m)$ satisfies Hensel's conditions. Let $f = (f_1, \ldots, f_n)$ be a system of $n$ elements of

A\{X_1, \ldots, X_n\} and let $a \in A^n$; let us write $J_t(a) = e$. There exists a system $g = (g_1, \ldots, g_n)$ of restricted formal power series without constant term in $A\{X_1, \ldots, X_n\}$ such that (i) $M_g(0) = I$, (unit matrix). (ii) For all $x \in A^n$,

(iii) Let $h = (h_1, \ldots, h_n)$ be the system of formal power series without constant term (not necessarily restricted) such that $g \circ h = 1_n$ (Proposition 5). For all $y \in m^{n \times n}$, For every formal power series $f \in A[[X_1, \ldots, X_n]]$, where the $G_{ij}$ are well determined formal power series in Iff is restricted, so are the elements of $M_f$ and the $G_{ij}$, for these formal power series are polynomials iff is a polynomial and it follows from their uniqueness that for every open ideal $\mathfrak{J}$ of $A$, denoting by $p_\mathfrak{J}: A \to A/\mathfrak{J}$ the canonical mapping, the image of $G_{ij}$ under $p_\mathfrak{J}$ is the coefficient of $Y_i Y_j$ in $p_\mathfrak{J}(F)$ where $F$ is the formal power series $f(X + Y)$ in $A[[X_1, \ldots, X_n, Y_1, \ldots, Y_n]]$; whence our assertion.

This being so, writing formula (21) for each seriesft $(1 \leq i \leq n)$, we obtain for all $x \in A^n$ (no. 2, Proposition 4),

where $r = (r_1, \ldots, r_r)$ is a system of restricted formal power series each of which is of total order $\geq 2$. It follows from formulae (18) of Algebra, Chapter III, § 6, no. 5 that there exists a square matrix $M' \in \mathbf{M}_n(A)$ such that whence using this in (22)

Writing $g = 1, + M' \cdot r$, we see that $g$ satisfies conditions (i) and (ii); then it is sufficient to replace $x$ by $h(y)$ to obtain (iii).