Product of Affine Spaces is Affine Space

Theorem
Let $\EE, \FF$ be affine spaces.

Let $\GG = \EE \times \FF$ be the product of $\EE$ and $\FF$.

Then $\GG$ is an affine space.

Proof
Let $G = \vec \GG$ be the difference space of $\GG$.

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

Proof of $(1)$:

Let $p = \tuple {p', p}, q = \tuple {q', q} \in \GG$.

We have:

Proof of $(2)$:

Let $p = \tuple {p', p''} \in \GG$.

Let $u = \tuple {u', u}, v = \tuple {v', v} \in G$.

We have:

Proof of $(3)$:

Let $p = \tuple {p', p}, q = \tuple {q', q} \in \GG$.

Let $u = \tuple {u', u''} \in G$.

We have: