Properties of Affine Spaces

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

Let $0$ denote the zero element of $V$.

Then the following hold for all $p,q,r \in \mathcal E$ and all $u,v \in V$:


 * $(1): \quad p - p = 0$
 * $(2): \quad p + 0 = p$
 * $(3): \quad p + u = p + v \iff u = v$
 * $(4): \quad q - p = r - p \iff q = r$

Proof

 * $(1): \quad p - p = 0$:

We have:

Now since the Zero Element is Unique we must have $p - p = 0$.


 * $(2): \quad p + 0 = p$

Using $(1)$ we see that


 * $(3): \quad p + u = p + v \iff u = v$

If $u = v$ then we must have $p + u = p + v$ since by definition a mapping has a unique image point on a given element.

Suppose now that $p + u = p + v$. We have


 * $(4): \quad q - p = r - p \iff q = r$:

If $q = r$ then we must have $q - p = r - p$ since by definition a mapping has a unique image point on a given element.

Suppose that $q - p = r - p \in V$. Then