Product of Affine Spaces is Affine Space

Theorem
Let $\mathcal E, \mathcal F$ be affine spaces.

Let $\mathcal G = \mathcal E \times \mathcal F$ be the product of $\mathcal E$ and $\mathcal F$.

Then $\mathcal G$ is an affine space.

Proof
Let $G = \vec{\mathcal G}$ be the difference space of $\mathcal G$.

We are required to show that the following axioms are satisfied:

Proof of $(1)$:

Let $p = \left({p', p}\right), q = \left({q', q}\right) \in \mathcal G$.

We have:

Proof of $(2)$:

Let $p = \left({p', p''}\right) \in \mathcal G$.

Let $u = \left({u', u}\right), v = \left({v', v}\right) \in G$.

We have:

Proof of $(3)$:

Let $p = \left({p', p}\right), q = \left({q', q}\right) \in \mathcal G$.

Let $u = \left({u', u''}\right) \in G$.

We have: