Let $E$ be an affine space over the vector space $V$, and let $U \subseteq V$ be a vector subspace. We define the equivalence relation $$P \sim Q := \exists v \in U \text{ such that } P = Q + v$$ on $E$.
I have to show that $E / \sim$ is an affine space over the quotient space $V/U$.
Therefore, I would simply check the properties of an affine space, for example $$P + 0 = P.$$ Since we work on $E / \sim$, and since we defined $\sim$ on $E$, its elements have to be equivalence classes. Therefore, I actually have to show something like $$[P] + 0 = [P].$$ Now I'm wondering which rules of addition I have to apply. We've already proven that on a quotient space such as $V/U$ I'm allowed to write something like $$[v] + [w] = [v + w].$$ Is there something similar for the addition on affine spaces?