# Order of Real Numbers is Dual of Order of their Negatives

## Theorem

$\forall x, y \in \R: x > y \iff \paren {-x}< \paren {-y}$

## Proof 1

Let $x > y$.

 $\ds x$ $>$ $\ds y$ $\ds \leadsto \ \$ $\ds x + \paren {-x}$ $>$ $\ds y + \paren {-x}$ Real Number Axioms: $\R \text O 1$: compatibility with addition $\ds \leadsto \ \$ $\ds 0$ $>$ $\ds y + \paren {-x}$ Real Number Axioms: $\R \text A 4$: Inverses $\ds \leadsto \ \$ $\ds 0 + \paren {-y}$ $>$ $\ds y + \paren {-x} + \paren {-y}$ Real Number Axioms: $\R \text O 1$: compatibility with addition $\ds \leadsto \ \$ $\ds 0 + \paren {-y}$ $>$ $\ds \paren {y + \paren {-y} } + \paren {-x}$ Real Number Axioms: $\R \text A 1$: Associativity and $\R \text A 2$: Commuativity $\ds \leadsto \ \$ $\ds 0 + \paren {-y}$ $>$ $\ds 0 + \paren {-x}$ Real Number Axioms: $\R \text A 4$: Inverses $\ds \leadsto \ \$ $\ds \paren {-y}$ $>$ $\ds \paren {-x}$ Real Number Axioms: $\R \text A 3$: Identity $\ds \leadsto \ \$ $\ds \paren {-x}$ $<$ $\ds \paren {-y}$ Definition of Dual Ordering

$\Box$

Let $\paren {-x} < \paren {-y}$.

 $\ds \paren {-x}$ $<$ $\ds \paren {-y}$ $\ds \leadsto \ \$ $\ds \paren {-x} + x$ $<$ $\ds \paren {-y} + x$ Real Number Axioms: $\R \text O 1$: compatibility with addition $\ds \leadsto \ \$ $\ds 0$ $<$ $\ds \paren {-y} + x$ Real Number Axioms: $\R \text A 4$: Inverses $\ds \leadsto \ \$ $\ds 0 + y$ $<$ $\ds \paren {-y} + x + y$ Real Number Axioms: $\R \text O 1$: compatibility with addition $\ds \leadsto \ \$ $\ds 0 + y$ $<$ $\ds \paren {\paren {-y} + y} + x$ Real Number Axioms: $\R \text A 1$: Associativity and $\R \text A 2$: Commuativity $\ds \leadsto \ \$ $\ds 0 + y$ $<$ $\ds 0 + x$ Real Number Axioms: $\R \text A 4$: Inverses $\ds \leadsto \ \$ $\ds y$ $<$ $\ds x$ Real Number Axioms: $\R \text A 3$: Identity $\ds \leadsto \ \$ $\ds x$ $>$ $\ds y$ Definition of Dual Ordering

$\blacksquare$

## Proof 2

 $\ds x$ $<$ $\ds y$ $\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

$\blacksquare$