According to the wikipedia on Affine Spaces, an Affine Space $A$ is defined as an underlying set $A$ alongside a vector space $\vec{A}$ with right group action (free, transitive) $+$ of $\vec{A}$ on $A$.
After laying out this definition, wikipedia states:
The uniqueness property allows us to define the subtraction of any two $a$ and $b$ of $A$, producing a vector of $$\overrightarrow{A}.$$ This vector, denoted
$$b - a,$$
or
$$ \overrightarrow{ab},$$
is the unique vector in $\overrightarrow{A}$ such that $ a + (b - a) = b$ (equivalently, $a+\overrightarrow{ab}=b$).
This subtraction has the two following properties, called Hermann Weyl's axioms.
- $ \forall a \in A,\; \forall v\in \overrightarrow{A},$ there is a unique point $ b \in A$ such that $ b-a = v$ (equivalently, $\overrightarrow{ab}=v$), and
- $ \forall a,b,c \in A,\; (c - b)+ (b - a) = c - a$ (equivalently, $\overrightarrow{ab} + \overrightarrow{bc} = \overrightarrow{ac}$).
Question: Is the "uniqueness property" an additional axiom that Affine Spaces must satisfy, or does it somehow follow from its definition?
The uniqueness property follows from the fact that for any $a \in A$, the map $\vec{A} \to A$, $v \mapsto a + v$ is a bijection. If $a, b \in A$, the vector $\vec{ab}$ is defined as the inverse image of $b$ under such function.