Proposition 3

— a) If I is a finite set with n elements and $n! \cdot 1$ is invertible in A, then $\lambda_I$ induces a bicontinuous isomorphism of $A[[P_1, \ldots, P_n]]$ onto $A[[X]]^{\mathrm{sym}}$. b) If I is infinite and A is a Q-algebra, then $\lambda_I$ is a bicontinuous isomorphism of $A[[P]]$ onto $A[[X]]^{\mathrm{sym}}$. We shall treat the case a) only, the case b) being quite similar.

By Th. 2 (IV, p. 68) we can identify $\mathbf{A}[[\mathbf{X}]]^{\text{sym}}$ with the algebra of formal power series $\mathbf{A}[[S_1, \ldots, S_n]]$, with $S_k$ corresponding to $s_k$. By Lemma 4 of IV, p. 70, there exist formal power series $g_1, \ldots, g_s$ of order $\geq 2$ in the indeterminates $s_1, \ldots, s_n$ such that

Since $k! . 1$ is invertible in $\mathbf{A}$, Lemma 2 of IV, p. 35 proves the existence of an automorphism $T$ of the topological $\mathbf{A}$-algebra $\mathbf{A}[[\mathbf{X}]]^{\text{sym}}$ which maps $s_k$ to $p_k$ for $l \leq k \leq n$. Now Prop. 3, a) is an immediate consequence.