Double of Antiperiodic Element is Periodic

Theorem
Let $f: X \to X$ be a function, where $X$ is either $\R$ or $\C$.

Let $L \in X_{\ne 0}$ be an anti-periodic element of $f$.

Then $2L$ is a periodic element of $f$.

In other words, every anti-periodic function is also periodic.

Proof
Let $X = \C$.

By Non-Zero Complex Numbers Closed under Multiplication we have that $2L \in X_{\ne 0}$.

Then:

The proof for when $X = \R$ is nearly identical.