Lemma 2

Let $f$ be a symmetric formal power series of degree $r$ in the $X_i (i \in I)$. There exists a unique polynomial $P \in B \cap A[S]$, isobaric of weight $r$, such that $f = \psi_1(P)$.

The case when $I$ is finite follows from Th. 1 (IV, p. 62).

Assume that $I$ is infinite and choose a finite subset $J$ of $I$, containing $r$ elements. We keep the notation of Lemma 1; we remark that every isobaric polynomial of weight $r$ in the $S_n \ (n \geq 1)$ depends only on $S_1, \ldots, S_r$ and that $\overline{s}_1, \ldots, \overline{s}_r$ are the elementary symmetric polynomials in the $r$ indeterminates $X_j \ (j \in J)$. If $P$ is an isobaric polynomial of weight $r$ in the $S_n$ and $h = f - \psi_1(P)$, then $\overline{h} = \overline{f} - P(\overline{s}_1, \ldots, S_r)$ and Lemma 1 shows the relation $f = \psi_1(P)$ to be equivalent to $\overline{f} = P(\overline{s}_1, \ldots, \overline{s}_r)$. By Th. 1 (IV, p. 62) there exists a unique polynomial $P \in A[S]$, isobaric of weight $r$ and such that $\overline{f} = P(\overline{s}_1, \ldots, \overline{s}_r)$, whence the result.