# Order of Real Numbers is Dual of Order of their Negatives/Proof 2

$\forall x, y \in \R: x > y \iff \paren {-x}< \paren {-y}$
 $\ds x$ $<$ $\ds y$ $\ds \leadstoandfrom \ \$ $\ds y - x$ $>$ $\ds 0$ Inequality iff Difference is Positive $\ds \leadstoandfrom \ \$ $\ds$ $>$ $\ds 0$ Real Number Axioms: $\R \text A 2$: Commutativity of Addition $\ds \leadstoandfrom \ \$ $\ds -x + -\paren {-y}$ $>$ $\ds 0$ Negative of Negative Real Number $\ds \leadstoandfrom \ \$ $\ds -x - \paren {-y}$ $>$ $\ds 0$ Definition of Real Subtraction $\ds \leadstoandfrom \ \$ $\ds -y$ $<$ $\ds -x$ Inequality iff Difference is Positive
$\blacksquare$