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 $p,q \in \mathcal E$
 * $\displaystyle\mathcal G\left(q\right) = \mathcal G\left(p\right) + M \circ L \left( \vec{ pq } \right)$

We find that:

Now let $p' = \mathcal L\left(p\right)$ and $q' = \mathcal L\left(p\right) + L \left( \vec{ pq } \right)$, so
 * $\vec{ p'q' } = L \left( \vec{ pq } \right)$

and we have that

as required.