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