Duals of Isomorphic Ordered Sets are Isomorphic

Theorem
Let $\left({S, \mathop{\preccurlyeq_1} }\right)$ and $\left({T, \mathop{\preccurlyeq_2} }\right)$ be ordered sets.

Let $\left({S, \mathop{\succcurlyeq_1} }\right)$ and $\left({T, \mathop{\succcurlyeq_2} }\right)$ be the dual ordered sets of $\left({S, \mathop{\preccurlyeq_1} }\right)$ and $\left({T, \mathop{\preccurlyeq_2} }\right)$ respectively.

Let $f: \left({S, \mathop{\preccurlyeq_1} }\right) \to \left({T, \mathop{\preccurlyeq_2} }\right)$ be an order isomorphism.

Then $f: \left({S, \mathop{\succcurlyeq_1} }\right) \to \left({T, \mathop{\succcurlyeq_2} }\right)$ is also an order isomorphism.