In particular this is done in the context of vector spaces (so I'm using abelian notation).
Let $S$ and $T$ be subspaces of $V$. I am trying to show that $(S + T)/T \cong S/(S \cap T)$.
Then we define $\tau: S + T \rightarrow S/(S \cap T)$ s.t. $\tau(s + t) = s + (S \cap T)$.
I am trying to remind myself why $\tau$ is well-defined.
So let let $v \in S+T$ s.t. $v = s_1 + t_1 = s_2 + t_2$.
Then $\tau(s_1 + t_1) = s_1 + (S \cap T)$
and $\tau(s_2 + t_2) = s_2 + (S \cap T)$
Then why must these two be equal?