Corollary 3

Let M be a semisimple module and N a submodule of M. The modules N and $M/N$ are semisimple. More precisely, if M is the direct sum of a family $(S_i)_{i\in I}$ of simple modules, then there exists a subset J of I such that $M/N$ is isomorphic to $\bigoplus_{j\in J}S_j$ and N to $\bigoplus_{i\in I-J}S_i$.

Choose J as in Theorem 1. The module $N'=\bigoplus_{j\in J}S_j$ is supplementary to N in M; it is therefore isomorphic to $M/N$. Moreover, the submodules N and $\bigoplus_{i\in I-J}S_i$ of M are both supplementary to $N'$ and therefore isomorphic to $M/N'$.