# 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:

 $\displaystyle \left({p - p}\right) + \left({q - p}\right)$ $=$ $\displaystyle \left({p + \left({q - p}\right)}\right) - p$ $\displaystyle$ $=$ $\displaystyle q - p$
$p - p = 0$

• $(2): \quad p + 0 = p$

Using $(1)$ we see that:

 $\displaystyle p + 0$ $=$ $\displaystyle p + \left({p - p}\right)$ $\displaystyle$ $=$ $\displaystyle p$

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

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:

 $\displaystyle p + u$ $=$ $\displaystyle p + v$ $\displaystyle \implies \ \$ $\displaystyle \left({p + u}\right) - p$ $=$ $\displaystyle \left({p + v}\right) - p$ $\displaystyle \implies \ \$ $\displaystyle \left({p - p}\right) + u$ $=$ $\displaystyle \left({p - p}\right) + v$ $\displaystyle \implies \ \$ $\displaystyle u$ $=$ $\displaystyle v$ by $(1)$

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

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

 $\displaystyle q - p$ $=$ $\displaystyle r - p$ $\displaystyle \implies \ \$ $\displaystyle p + \left({q - p}\right)$ $=$ $\displaystyle p + \left({r - p}\right)$ $\displaystyle \implies \ \$ $\displaystyle q$ $=$ $\displaystyle r$ By the hypothesis $q - p = r - p$

$\blacksquare$