I've been reading through Lang's Algebra chapter about semisimple modules and semisimple rings, and after Lang proves a main structure theorem about semisimple rings he proves the following proposition:
Let $k$ be a field and $E$ a finite $k$-vector space. Let $R$ be a $k$-subalgebra of $\operatorname{End}_k E$.
Then $R$ is semisimple if and only if $E$ is semisimple as an $R$-module.
Proof goes like this:
If $R$ is semisimple, every $R$-module is semisimple, including $E$. Conversely, if $E$ is a semisimple $R$-module, then it looks like a finite direct sum of simple $R$-modules: $$ E = \bigoplus_{i = 1}^n E_i$$
For each $i$ there exists $v_i \in E_i$ such that $R.v_i = E_i$. The map $$ r \to (r.v_1, \ldots, r.v_n)$$ is a $R$-homomorphism from $R$ into $E$, and is an injection since $R$ is contained in $\operatorname{End}_k E$. Then, $R$ is isomorphic to a submodule of $E$. The result follows from the fact that a submodule of a semisimple submodule is again semisimple.
I get everything except the bold part. Can someone explain that part to me?