3.85 of LADR by Sheldon Axler I'm a little confused on the proof of the implication of $(c)\to(a)$. In his proof, Axler says to take two elements $u_1, u_2$ of $U$ such that $v+u_1=w+u_2$. Whatever follows after that, I understand, but this beginning I don't. How can we just suppose this from $(c)$?
Here is the theorem:
Suppose $U$ is a subspace of $V$ and $v,w\in V$. Then, the following are equivalent.
(a) $v-w\in U$
(b) $v+U=w+U$
(c) $(v+U)\cap(w+U)\neq \varnothing$

Axler is assuming that $(v+U)\cap(w+U)\ne\emptyset$. So, take a vector $z\in(v+U)\cap(w+U)$. Then $z=v+u_1$ for some $u_1\in U$ (since $z\in v+U$) and $z=w+u_2$ for some $u_2\in U$ (since $z\in w+U$). But then $v+u_1=w+u_2$.
