Double of Antiperiodic Element is Periodic

Theorem
Let $f: \R \to \R$ be a real function.

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

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

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

Proof
By Non-Zero Real Numbers Closed under Multiplication we have that $2 L \in \R_{>0}$.

Then: