Properties of Affine Spaces

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

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

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


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

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