Power of Positive Real Number is Positive/Real Number

Theorem
Let $x \in \R_{>0}$ be a (strictly) positive real number.

Let $r \in \R$ be a real number.

Then:
 * $x^r > 0$

where $x^r$ denotes the $x$ to the power of $r$.

Proof
From the definition of $x$ to the power of $r$:
 * $x^r = \map \exp {r \ln x}$

The result follows from Exponential of Real Number is Strictly Positive.