# Inequality iff Difference is Positive

## Theorem

Let $x, y \in \R$.

Then the following are equivalent:

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

## Proof

 $\displaystyle x < y$ $\leadstoandfrom$ $\displaystyle y > x$ Definition of Dual Ordering $\displaystyle$ $\leadstoandfrom$ $\displaystyle y + \paren {-x} > x + \paren {-x}$ Real Number Ordering is Compatible with Addition $\displaystyle$ $\leadstoandfrom$ $\displaystyle y + \paren {-x} > 0$ Real Number Axioms: $\R \text A 4$: Inverse Elements $\displaystyle$ $\leadstoandfrom$ $\displaystyle y - x > 0$ Definition of Field Subtraction

Hence the result.

$\blacksquare$

## Note

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