Composition of Affine Transformations is Affine Transformation

Theorem
Let $\mathcal E$, $\mathcal F$ and $\mathcal G$ be affine spaces with difference spaces $E$, $F$ and $G$ respectively.

Let $\mathcal L: \mathcal E \to \mathcal F$ and $\mathcal M: \mathcal F \to \mathcal G$ be affine transformations.

Let $L$ and $M$ be the tangent maps of $\mathcal L$ and $\mathcal M$ respectively.

Then the composition $\mathcal M \circ \mathcal L: \mathcal E \to \mathcal F$ is an affine transformation with tangent map $M \circ L$.

Proof
Let $\mathcal N = \mathcal M \circ \mathcal L : \mathcal E \to \mathcal G$ be the composition.

We want to show that for any $p, q \in \mathcal E$
 * $\mathcal G \left({q}\right) = \mathcal G \left({p}\right) + M \circ L \left({\vec{p q} }\right)$

We find that:

Now let:
 * $p' = \mathcal L \left({p}\right)$

and:
 * $q' = \mathcal L \left({p}\right) + L \left({\vec{p q} }\right)$

so:
 * $\vec{p' q'} = L \left({\vec{p q} }\right)$

Then:

as required.