# Duals of Isomorphic Ordered Sets are Isomorphic

## Theorem

Let $\struct {S, \preccurlyeq_1}$ and $\struct {T, \preccurlyeq_2}$ be ordered sets.

Let $\struct {S, \succcurlyeq_1}$ and $\struct {T, \succcurlyeq_2}$ be the dual ordered sets of $\struct {S, \preccurlyeq_1}$ and $\struct {T, \preccurlyeq_2}$ respectively.

Let $f: \struct {S, \preccurlyeq_1} \to \struct {T, \preccurlyeq_2}$ be an order isomorphism.

Then $f: \struct {S, \succcurlyeq_1} \to {T, \succcurlyeq_2}$ is also an order isomorphism.

## Proof

 $\displaystyle \forall x, y \in S: \ \$ $\displaystyle x$ $\succcurlyeq_1$ $\displaystyle y$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle y$ $\preccurlyeq_1$ $\displaystyle x$ Definition of Dual Ordering $\displaystyle \leadstoandfrom \ \$ $\displaystyle \map f y$ $\preccurlyeq_2$ $\displaystyle \map f x$ Definition of Order Isomorphism $\displaystyle \leadstoandfrom \ \$ $\displaystyle \map f x$ $\succcurlyeq_2$ $\displaystyle \map f y$ Definition of Dual Ordering

$\blacksquare$