Equivalence of Definitions of Affine Space

Associativity Axioms implies Weyl's Axioms
Assume the axioms $(A1)$, $(A2)$, $(A3)$. Then for any $p, q \in \mathcal E$ we have:

Therefore by Identity is Unique applied to the vector space $V$ we have:

Now let $p \in \mathcal E$, $v \in V$ as in $(W1)$.

We must show there exists a unique $q \in \mathcal E$ such that $v = q - p$.

Let $q = p + v$. Then:

Now let $r \in \mathcal E$ be any other element such that $v = r - p$. Then:

This shows that $q$ is unique and establishes $(W1)$.

Now let $p, q, r \in \mathcal E$ as in $(W2)$. Then:

This establishes $(W2)$.

Weyl's Axioms implies Group Action
Assume the axioms $(W1)$, $(W2)$.

Let $\phi: \mathcal E \times V \to \mathcal E$ be the group action defined by:


 * $\forall \tuple {p, v} \in \mathcal E \times V: p + v := \map \phi {p, v} = q$

where $q \in \mathcal E$ is the unique point such that $v = q - p$ given by $(W1)$.