Order Types of Duals of Isomorphic Sets are Equal

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

Let:
 * $\map \ot {S_1, \preccurlyeq_1} = \map \ot {S_2, \preccurlyeq_2}$

where $\ot$ denotes the order type operator.

Let $\struct {S_1, \succcurlyeq_1}$ and $\struct {S_2, \succcurlyeq_2}$ denote the dual ordered sets of $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$.

Then:
 * $\map \ot {S_1, \succcurlyeq_1} = \map \ot {S_2, \succcurlyeq_2}$