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Theorem(Nilradical lemma). Let $K$ be a field of characteristic $p > 2$ and $A_{\infty} \colon= K[[X_1,\ldots,X_{\infty}]]$ be an infinitely-many-variable formal power series ring over $K$. We shall define the closed ideal $I$ of $A_{\infty}$ as follows$\colon$
\begin{equation*}
I \colon= (X_1^p,...