Ordering of Reciprocals/Proof 1

Theorem
Let $x, y \in \R$ be real numbers such that $x, y \in \left({0 \,.\,.\, \to}\right)$ or $x, y \in \left({\gets \,.\,.\, 0}\right)$

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

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

By Mapping from Totally Ordered Set is Dual Order Embedding iff Strictly Decreasing, the reciprocal function is a dual order embedding.

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