Power Function on Base between Zero and One is Strictly Decreasing/Rational Number

Theorem
Let $a \in \R$ be a real number such that $0 < a < 1$.

Let $f: \Q \to \R$ be the real-valued function defined as:
 * $\map f q = a^q$

where $a^q$ denotes $a$ to the power of $q$.

Then $f$ is strictly decreasing.

Proof
Let $\dfrac r s, \dfrac t u \in \Q$, where $r, t \in \Z, s, u \in \Z_{>0}$.

Let $\dfrac r s < \dfrac t u$.

Then: