Dual of Preordered Set is Preordered Set

Theorem
Let $P = \left({S, \preceq}\right)$ be a preordered set.

Then dual of $P$, $P^{-1} = \left({S, \succeq}\right)$ is also a preordered set.

Proof
By Inverse of Reflexive Relation is Reflexive:
 * $\succeq$ is reflexive.

By Inverse of Transitive Relation is Transitive:
 * $\succeq$ is transitive.

Hence $\succeq$ is a preordering.