Composition of Affine Transformations is Affine Transformation

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

Let $\LL: \EE \to \FF$ and $\MM: \FF \to \GG$ be affine transformations.

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

Then the composition $\MM \circ \LL: \EE \to \FF$ is an affine transformation with tangent map $M \circ L$.

Proof
Let $\NN = \MM \circ \LL : \EE \to \GG$ be the composition.

We want to show that for any $p, q \in \EE$
 * $\map \GG Q = \map \GG p + \map {M \circ L} {\vec {p q} }$

We find that:

Now let:
 * $p' = \map \LL p$

and:
 * $q' = \map \LL p + \map L {\vec {p q} }$

so:
 * $\vec {p' q'} = \map L {\vec {p q} }$

Then:

as required.