Linearity of Function defined using Function with Translation Property/Lemma

Lemma
Let $f$ be a real function.

Let $f$ have the translation property.

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

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

Then:
 * $\forall n \in \Z: \map {f_x} {n l} = n \map {f_x} l$

Proof
Let $n$ be an integer.

We have: