Proposition 7

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 without constant term of series in $A[[X_1, \ldots, X_n]]$ such that $J_f(0)$ is invertible in $A$. Then, for all $x \in m^{x^n}$, $f(x) \in m^{x^n}$ and $x \mapsto f(x)$ is a bijection of $m^{x^n}$ onto itself, the inverse bijection being $x \mapsto g(x)$, where $g$ is given by relation (16) in no. 4.

The fact that $f(x) \in m^{x^n}$ is obvious when theft are polynomials and follows in the general case from Proposition 6 and the fact that $m$ is closed in $A$. The other assertions of the proposition are then immediate consequences of (16), (17) and (18).