Lemma 1

Let J be a finite subset of I, r an integer such that Card $J \geq r$ and fa symmetric formal power series which is homogeneous of degree r in the $X_i$ ($i \in I$). Let $\bar{f}$ be the formal power series obtained by substituting 0 for $X_i$ for each i in $I - J$. If $\bar{f} = 0$, then we have $f = 0$.

Let us put $f = \sum_{|\alpha| = r} a_\alpha X^\alpha$ (where $|\alpha|$ is the length $\sum_{i \in I} \alpha_i$ of the multiindex $a = (\alpha_i)_{i \in I}$). If $a$ is a multiindex of length $r$, and $J'$ is the support of $a$ (the set of $i \in I$ such that $\alpha_i \neq 0$), then Card $J' \leq r$, hence there exists a permutation $\sigma \in S_I$ such that $\sigma(J') \subset J$. Put $\beta_i = \alpha_{\sigma^{-1}(i)}$ for $i \in I$, then the monomial $X^\beta = \prod_{i \in I} X_{\sigma(i)}^{\alpha_i}$ depends only on the indeterminates $X_j$ ($j \in J$), whence $a_\beta = 0$ by the hypothesis $\overline{f} = 0$. Since $f$ is symmetric, we have $a_\alpha = a_\beta$ and since $\alpha$ was arbitrary, we conclude that $f = 0$.