Ordering of Reciprocals

Theorem
Let $x, y \in \R$ be real numbers such that $x > 0, y > 0$ or $x < 0$ and $y < 0$.

Then:
 * $x \le y \iff \dfrac 1 y \le \dfrac 1 x$

Proof
By Reciprocal Function Strictly Decreasing, the reciprocal function is strictly decreasing.

By Mapping from Toset to Poset is Dual Order Embedding iff Strictly Decreasing, the reciprocal function is a dual order embedding.

That is:
 * $x \le y \iff \dfrac 1 y \le \dfrac 1 x$