User:Keith.U/Sandbox/Equivalence of Definitions of Less Than

Theorem
Let $x, y \in \R$.

Then the following are equivalent:


 * $(1): \quad x < y$
 * $(2): \quad y - x > 0$

Proof
Hence the result.

Note
If the notion of an ordering on $\R$ has not already been defined rigorously, this is often taken to be the definition of $x < y$.