# Properties of Affine Spaces

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## 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 $u, v \in V$:

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

## Proof

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

We have:

\(\ds \paren {p - p} + \paren {q - p}\) | \(=\) | \(\ds \paren {p + \paren {q - p} } - p\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds q - p\) |

From Zero Element is Unique:

- $p - p = 0$

$\Box$

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

Using $(1)$ we see that:

\(\ds p + 0\) | \(=\) | \(\ds p + \paren {p - p}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds p\) |

$\Box$

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

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:

\(\ds p + u\) | \(=\) | \(\ds p + v\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \paren {p + u} - p\) | \(=\) | \(\ds \paren {p + v} - p\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \paren {p - p} + u\) | \(=\) | \(\ds \paren {p - p} + v\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds u\) | \(=\) | \(\ds v\) | by $(1)$ |

$\Box$

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

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:

\(\ds q - p\) | \(=\) | \(\ds r - p\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds p + \paren {q - p}\) | \(=\) | \(\ds p + \paren {r - p}\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds q\) | \(=\) | \(\ds r\) | by hypothesis $q - p = r - p$ |

$\blacksquare$