Vector Space with Standard Affine Structure is Affine Space

Theorem
Let $E$ be a vector space.

Let $\left({\mathcal E, E, +, -}\right)$ be the standard affine structure on $E$.

Then with this structure, $\mathcal E$ is an affine space.

Proof
We are required to show that:

By the definition of the standard affine structure, the addition and subtraction operations are simply those in the vector space $E$.

That is, we want to show that:

By definition the addition operation on a vector space is commutative and associative.

But all three axioms are immediate consequences of commutativity and associativity.

This concludes the proof.