Lemma 2

There exists an $\alpha$-mapping $\sigma$ of $g$ into $\mathscr{L}_{K}(P)$ such that:

(1) $\sigma(x_{\lambda})z_{M} = z_{\lambda}z_{M}$ for $\lambda \leq M$; (2) $\sigma(x_{\lambda})z_{M} \equiv z_{\lambda}z_{M}$ (mod. $P_{p}$) if $M$ has $p$ elements.

By Lemma 1 there exists a homomorphism $f$ of the K-module $g \otimes_{K} P_{p}$ into $P$ satisfying, for all $p$, conditions (A$_{p}$), (B$_{p}$), (C$_{p}$) (where $f_{p}$ is replaced by $f$). This homomorphism defines a homomorphism $\sigma$ of the K-module $g$ into the K-module $\mathscr{L}_{K}(P)$ and $\sigma$ is an $\alpha$-mapping because of condition (C$_{p}$). Finally, $\sigma$ satisfies properties (1) and (2) of the lemma because of conditions (A$_{p}$) and (B$_{p}$).