Power of Positive Real Number is Positive/Integer

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

Let $n \in \Z$ be an integer.

Then:
 * $x^n > 0$

where $x^n$ denotes the $n$th power of $x$.

Proof
By Power of Positive Real Number is Positive: Natural Number, the theorem is already proven for non-negative integers.

Suppose $n \in \Z_{< 0}$.

When $n < 0$, by Real Number Ordering is Compatible with Multiplication: Negative Factor:
 * $-n > 0$

Then, by Power of Positive Real Number is Positive: Natural Number:
 * $x^{-n} > 0$

Therefore, by Reciprocal of Strictly Positive Real Number is Strictly Positive:
 * $x^n = \dfrac 1 {x^{-n} } > 0$