Order Isomorphism is Symmetric

Theorem
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.

Let $\struct {S_1, \preccurlyeq_1}$ be isomorphic to $\struct {S_2, \preccurlyeq_2}$.

Then $\struct {S_2, \preccurlyeq_2}$ is isomorphic to $\struct {S_1, \preccurlyeq_1}$.

Proof
Let $\phi: S_1 \to S_2$ be an order isomorphism from $\struct {S_1, \preccurlyeq_1}$ to $\struct {S_2, \preccurlyeq_2}$.

From Inverse of Order Isomorphism is Order Isomorphism, $\phi^{-1}: S_2 \to S_1$ is an order isomorphism from $\struct {S_2, \preccurlyeq_2}$ to $\struct {S_1, \preccurlyeq_1}$.

The result follows.