Proposition 5

Let $\mathbf{A}$ be a commutative ring and $\mathbf{f} = (f_1, \ldots, f_n)$ a system without constant term of $n$ series in $\mathbf{A}[[X_1, \ldots, X_n]]$. Suppose that $J_f(0)$ is invertible in $\mathbf{A}$. Then there exists a system without constant term $\mathbf{g} = (g_1, \ldots, g_n)$ of $n$ series in $\mathbf{A}[[X_1, \ldots, X_n]]$ such that

This system is unique and

(17) $g \circ f = 1_1$.

The existence and uniqueness of $g$ follows from Algebra, Chapter IV, § 5, no. 9, Proposition 10, applied to the $n$ formal power series

It follows from (15) and (16) that $J_f(0)J_g(0) = 1$ and hence $J_g(0)$ is also invertible. We conclude that there exists a system $h = (h_1, \ldots, h_n)$ of $n$ series without constant term in $A[[X_1, \ldots, X_n]]$ such that $g \circ h = 1_1$; from this relation and (16) it then follows, with the aid of (10), that

Proposition 5 and formulae (10) and (15) show that under the law of composition $(f, g) \mapsto f \circ g$ the set of systems $f = (f_1, \ldots, f_n)$ of $n$ series without constant term in $A[[X_1, \ldots, X_n]]$ for which $J_f(0)$ is invertible in $A$, is a group.