\begin{definition}[Definition:Affine Space/Weyl's Axioms]
Let $K$ be a field.
Let $\struct{V, +_V, \circ}$ be a vector space over $K$.
Let $\EE$ be a set on which a mapping is defined:
:$- : \EE \times \EE \to V$
satisfying the following associativity conditions:
{{begin-axiom}}
{{axiom | n = \text W 1
        | q = \forall p \in \EE: \forall v \in V: \exists ! q \in \EE
        | m = v = q - p
}}
{{axiom | n = \text W 2
        | q = \forall p, q, r \in \EE
        | m = \paren{r - q} +_V \paren{q - p} = r - p
}}
{{end-axiom}}
Then the ordered pair $\tuple {\EE, -}$ is an '''affine space'''.
\end{definition}