Lemma 3

Let M be an A-module that is the direct sum of a family $(M_i)_{i\in I}$ of primordial submodules, and let $p$ be a nonzero projector of M. There exists an index $i\in I$ such that $p$ induces an isomorphism from $M_i$ to a direct factor submodule of $p(M)$.

Since $p$ is nonzero, $1_M-p$ is not injective. By Lemma 2, there exists an index $i\in I$ such that $p$ induces an isomorphism from $M_i$ to a direct factor submodule of M. Every projector of M with image $p(M_i)$ defines, by restriction, a projector of $p(M)$ with image $p(M_i)$, so $p(M_i)$ is a direct factor submodule of $p(M)$.