Suppose $v,x$ are vectors in $V$ and $U,W$ are subspaces of $V$ such that $v+U=x+W$. Prove that $U = W$.

We add $-x$ to both sides to get $(v-x) + U = W$ which, since $(v-x)+U$ is a subspace it must contain zero implying $(x-v) \in U$ and (since $U$ contains inverses) $(v-x) \in U$ finally giving $(v-x)+U=U=W$.

If you're uncomfortable with adding $-x$ to both sides feel free to rewrite it in terms of components.