Corollary 3

*With the notation & Corollary 2, let $a \in \mathbf{A}^n$; let us write $e = J_r^{(n-r)}(\mathbf{a})$ (not necessarily invertible in $\mathbf{A}$) and suppose that $f(\mathbf{a}) \equiv 0$ (mod. $e^{2m^{\times(n-r)}}$). Then there exist $n - r$ formal power series without constant term $\phi_i$ ($r + 1 \leq i \leq n$) in $\mathbf{A}[[X_1, \ldots, X_r]]$ such that, for all $t = (t_1, \ldots, t_r) \in m^{\times r}$,

for $r + 1 \leq i \leq n$.

*For $1 \leq i \leq r$, let $f_i(X) = X_i - a$, and let $u = (f_1, \ldots, f_n)$; then $J_u(a) = e$ and Theorem 2 may be applied to the system $u$. With the notation of Theorem 2 it follows from the above definitions that $g_i(X) = X$, for $1 \leq i \leq r$, whence $h_i(X) = X_i$ for $1 \leq i \leq r$; moreover, if $M' \in \mathbf{M}_n(\mathbf{A})$ is such that $M_u(a) . M' = e I_n$, $M'$ is of the form

Replacing $y$ by $M'.z$ (where $z = (z_1, \ldots, z_n) \in m^{\times n}$) in formula (20), we obtain

By hypothesis, $f_j(a) = e^2 b_j$, where $b_j \in m$ for $r + 1 \leq j \leq n$. Let us write $\psi_j(X_1, \ldots, X_r) = h_j(M'.X)$ and

for $r + 1 \leq j \leq n$. For $r + 1 \leq i \leq n$, substituting $t_j$ for $z_j$ for $1 \leq j \leq r$ and $b_j$ for $z_j$ for $r + 1 \leq j \leq n$ in (26), we obtain relations (25) for all $t \in m^{\times r}$.