Linearity of Function defined using Function with Translation Property

Theorem
Let $f$ be a real function.

Let $f$ have the translation property.

Let $x$ and $l$ be a real numbers.

Define:
 * $\map {f_x} l = \map f {x + l} - \map f x$

Then:
 * $\forall q \in \Q: \map {f_x} {q l} = q \map {f_x} l$

Lemma
Let $q$ be a rational number.

Choose integers $n$, $m$ such that:
 * $\dfrac n m = q$

We need to prove that:
 * $\map {f_x} {q l} = q \map {f_x} l$

We have: