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

## Theorem

$\forall x, y \in \R: x > y \iff \left({-x}\right) < \left({-y}\right)$

## Proof 1

Let $x > y$.

 $\displaystyle x$ $>$ $\displaystyle y$ $\displaystyle \leadsto \ \$ $\displaystyle x + \paren {-x}$ $>$ $\displaystyle y + \paren {-x}$ Real Number Axioms: $\R O1$: compatibility with addition $\displaystyle \leadsto \ \$ $\displaystyle 0$ $>$ $\displaystyle y + \paren {-x}$ Real Number Axioms: $\R A4$: Inverses $\displaystyle \leadsto \ \$ $\displaystyle 0 + \paren {-y}$ $>$ $\displaystyle y + \paren {-x} + \paren {-y}$ Real Number Axioms: $\R O1$: compatibility with addition $\displaystyle \leadsto \ \$ $\displaystyle 0 + \paren {-y}$ $>$ $\displaystyle \paren {y + \paren {-y} } + \paren {-x}$ Real Number Axioms: $\R A1$: Associativity and $\R A2$: Commuativity $\displaystyle \leadsto \ \$ $\displaystyle 0 + \paren {-y}$ $>$ $\displaystyle 0 + \paren {-x}$ Real Number Axioms: $\R A4$: Inverses $\displaystyle \leadsto \ \$ $\displaystyle \paren {-y}$ $>$ $\displaystyle \paren {-x}$ Real Number Axioms: $\R A3$: Identity $\displaystyle \leadsto \ \$ $\displaystyle \paren {-x}$ $<$ $\displaystyle \paren {-y}$ Definition of Dual Ordering

$\Box$

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

 $\displaystyle \paren {-x}$ $<$ $\displaystyle \paren {-y}$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {-x} + x$ $<$ $\displaystyle \paren {-y} + x$ Real Number Axioms: $\R O1$: compatibility with addition $\displaystyle \leadsto \ \$ $\displaystyle 0$ $<$ $\displaystyle \paren {-y} + x$ Real Number Axioms: $\R A4$: Inverses $\displaystyle \leadsto \ \$ $\displaystyle 0 + y$ $<$ $\displaystyle \paren {-y} + x + y$ Real Number Axioms: $\R O1$: compatibility with addition $\displaystyle \leadsto \ \$ $\displaystyle 0 + y$ $<$ $\displaystyle \paren {\paren {-y} + y} + x$ Real Number Axioms: $\R A1$: Associativity and $\R A2$: Commuativity $\displaystyle \leadsto \ \$ $\displaystyle 0 + y$ $<$ $\displaystyle 0 + x$ Real Number Axioms: $\R A4$: Inverses $\displaystyle \leadsto \ \$ $\displaystyle y$ $<$ $\displaystyle x$ Real Number Axioms: $\R A3$: Identity $\displaystyle \leadsto \ \$ $\displaystyle x$ $>$ $\displaystyle y$ Definition of Dual Ordering

$\blacksquare$

## Proof 2

 $\displaystyle x < y$ $\iff$ $\displaystyle y - x > 0$ Inequality iff Difference is Positive $\displaystyle$ $\iff$ $\displaystyle -x + y > 0$ Real Number Axioms: $\R A2$: Commutativity of Addition $\displaystyle$ $\iff$ $\displaystyle -x + -\left({ -y }\right) > 0$ Negative of Negative Real Number $\displaystyle$ $\iff$ $\displaystyle -x - \left({ -y }\right) > 0$ Definition of subtraction $\displaystyle$ $\iff$ $\displaystyle -y < -x$ Inequality iff Difference is Positive

$\blacksquare$