Inverse of Order Isomorphism is Order Isomorphism

Theorem
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi$ be a bijection from $\left({S, \preceq_1}\right)$ to $\left({T, \preceq_2}\right)$.

Then $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ is an order isomorphism iff $\phi^{-1}: \left({T, \preceq_2}\right) \to \left({S, \preceq_1}\right)$ is also an order isomorphism.

Proof
Follows directly from the definition of order isomorphism.