Corollary

Let $q$ be a closed ideal of $A$ contained in $m$. Then the relation $x \equiv x' \pmod{q^{x^n}}$ is equivalent to $f(x) \equiv f(x') \pmod{q^{x^n}}$.

For every formal power series $f \in A[[X_1, \ldots, X_n]]$,

where the $h_i$ belong to $A[[X_1, \ldots, X_n, Y_1, \ldots, Y_n]]$ (Algebra, Chapter IV, § 5, no. 8, Proposition 9); it follows immediately that the relation $x \equiv x' \pmod{q^{x^n}}$ implies $f(x) \equiv f(x') \pmod{q^{x^n}}$. The converse is obtained by replacing $f$ by its "inverse" $g$.