Lemma 3

*Let $x_1, \ldots, x_s$ be elements of $\mathbf{X}$ such that $s \geq 1$ and $x_i \neq x_{i+1}$ for $1 \leq i \leq s-1$; let $n_1, \ldots, n_s$ be non-zero rational integers. Then the element $\prod_{i=1}^s (1+x_i)^{n_i}$ of $\hat{\mathbf{A}}(\mathbf{X})$ is $\neq 1$.

Let $m$ be a maximal ideal of $K$ and $k$ the field $K/m$; let $p : \hat{\mathbf{A}}_K(\mathbf{X}) \to \hat{\mathbf{A}}_k(\mathbf{X})$ be the unique continuous homomorphism of unital $K$-algebras such that $p(x) = x$ for $x \in \mathbf{X}$ (no. 1, Proposition 1). It suffices to prove that $p\left( \prod_{i=1}^s (1+x_i)^{n_i} \right) \neq 1$ and the problem is reduced to the case where $K$ is a field.

In the notation of Lemma 2:

By Lemma 2, there exist integers $a_i > 0$ such that $c_{a_i, n_i} \neq 0$ ($1 \leq i \leq s$). By Algebra, Chapter I, § 7, no. 4, Proposition 6, no monomial $x_1^{b_1} \cdots x_s^{b_s}$ such that $b_i \geq 0$ and $(b_1, \ldots, b_s) \neq (a_1, \ldots, a_s)$ can be equal to $x_1^{a_1} \ldots x_s^{a_s}$. It follows that the coefficient of $x_1^{a_1} \ldots x_s^{a_s}$ in $\prod_{i=1}^s (1 + x_i)^{n_i}$ is $c_{a_1, n_1} \ldots c_{a_s, n_s} \neq 0$, which implies the result.