Properties of Affine Spaces

Theorem
Let $\mathcal E$ be an affine space with difference space $V$.

Then for all $p,q,r \in \mathcal E$ and all $u,v \in V$ we have
 * 1) $p - p = 0$
 * 2) $p + 0 = p$
 * 3) $q - p = r - p$ if and only if $q = r$
 * 4) $p + u = p + v$ if and only if $u = v$