Duals of Isomorphic Ordered Sets are Isomorphic

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

Let $\succcurlyeq_1$ and $\succcurlyeq_2$ be the duals of $\preccurlyeq_1$ and $\preccurlyeq_2$ respectively.

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

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