Proof of Real Powers with Positive Exponent Are Weakly Monotone

Let $a>0$ and $0\le s\le t$ (so $s^{a}$ and $t^{a}$ are real numbers, by Real Powers (i) and Real Powers (ii)).

If $s=t$, then $s^{a}=t^{a}$.

Otherwise $s\ne t$, and together with $s\le t$ this gives $0\le s<t$. By the strict monotonicity of real powers, Properties of Real Powers (vi), we get $s^{a}<t^{a}$, and in particular $s^{a}\le t^{a}$. This proves (i).