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:

From Zero Element is Unique:
 * $p - p = 0$


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

Using $(1)$ we see that:


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

Let $u = v$.

By definition a mapping has a unique image point on a given element.

It follows that:
 * $p + u = p + v$

Let $p + u = p + v$.

We have:


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

Let $q = r$.

By definition a mapping has a unique image point on a given element.

It follows that:
 * $q - p = r - p$

Let $q - p = r - p \in V$.

Then