Theorem: Let $S$ and $T$ be submodules of $M$, and let $S+T=\{s+t,s\in S,t \in T\}$. Then $S+T$ and $S∩T$ are submodules of M, and:
$S/(S∩T)≃(S+T)/T$.
I saw the proof but don't we have to check also $S \cap T$ is submodule of $S$ and $T$ is submodule of $S+T$? As I remember these terms have to be satisfied due to definition. Or maybe it is obvious for everyone?
It does need to be checked. It's rather easy though: They are subsets and they are modules, therefore they are submodules.