\begin{definition}[Definition:Affine Space/Associativity Axioms]
Let $K$ be a field.
Let $\struct {V, +_V, \circ}$ be a vector space over $K$.
Let $\EE$ be a set on which two mappings are defined:
:$+ : \EE \times V \to \EE$
:$- : \EE \times \EE \to V$
satisfying the following associativity conditions:
{{begin-axiom}}
{{axiom | n = \text A 1
        | q = \forall p, q \in \EE
        | m = p + \paren {q - p} = q
}}
{{axiom | n = \text A 2
        | q = \forall p \in \EE: \forall u, v \in V
        | m = \paren {p + u} + v = p + \paren {u +_V v}
}}
{{axiom | n = \text A 3
        | q = \forall p, q \in \EE: \forall u \in V
        | m = \paren {p - q} +_V u = \paren {p + u} - q
}}
{{end-axiom}}
Then the ordered triple $\struct {\EE, +, -}$ is an '''affine space'''.
\end{definition}