Corollary 2

Let $\mathbf{A}$ be a ring and $\mathfrak{m}$ an ideal of $\mathbf{A}$ such that the ordered pair $(\mathbf{A}, m)$ satisfies Hensel's conditions. Let $r, n$ be integers such that $0 \leq r < n$ and $\mathbf{f} = (f_{r+1}, \ldots, f_n)$ is a system of $n - r$ elements of $\mathbf{A}\{\mathbf{X}_1, \ldots, \mathbf{X}_n\}$; let $J^{(n-r)}_{\mathbf{f}}(\mathbf{X})$ denote the minor of $M_{\mathbf{f}}(\mathbf{X})$ consisting of the columns of index $j$ such that $r + 1 \leq j \leq n$. Let $a \in \mathbf{A}^n$ be such that $J^{(n-r)}_{\mathbf{f}}(a)$ is invertible in $\mathbf{A}$ and $\mathbf{f}(a) \equiv 0 \pmod{\mathfrak{m}^{(n-r)}}$. Then there exists a unique $\mathbf{x} = (x_1, \ldots, x_n) \in \mathbf{A}^n$ such that $x_k = a_k$ for $1 \leq k \leq r$, $\mathbf{x} \equiv a \pmod{\mathfrak{m}^n}$ and $\mathbf{f}(\mathbf{x}) = 0$.

Substituting $a$, for $X_k$ for $1 \leq k \leq r$ in the $f_i$ (no. 2, Remark 3 ), we see immediately that we may restrict our attention to the case where $r = 0$ to prove the corollary. Theorem 1 and Proposition 7 show then that $\mathbf{f}$ defines a bijection of $a + \mathfrak{m}^n$ onto $\mathbf{f}(a) + \mathfrak{m}^n = \mathfrak{m}^n$; the corollary follows from the fact that $0 \in \mathfrak{m}^n$.