Lemma 2

Let M be a semisimple left A-pseudomodule, and let $x$ be an element of M. There exists an element $a\in A$ such that $ax=x$.

Let N be the left A-pseudomodule $\widetilde{A}x/Ax$. It satisfies AN $=\{0\}$. By Corollary 3 of VIII, p. 56 applied to the $\widetilde{A}$-modules M and $\widetilde{A}x$, the A-pseudomodule N is semisimple. By definition, every simple sub-pseudomodule S of N satisfies AS $\not=\{0\}$. Consequently, the pseudomodule N is zero, and we obtain that $x\in Ax$.