Dual of Total Ordering is Total Ordering

Theorem
Let $\preccurlyeq$ be a total ordering.

Then its inverse $\preccurlyeq^{-1}$ or $\succcurlyeq$ is also a total ordering.

Proof
Let $\left({S, \preccurlyeq}\right)$ be a totally ordered set.

From Inverse of Ordering is Ordering we have that $\succcurlyeq$ is an ordering.

Let $x, y \in S$.

Then $x \preccurlyeq y$ or $y \preccurlyeq x$.

But by definition of inverse relation, $y \succcurlyeq x$ or $x \succcurlyeq y$.

Hence the result by definition of total ordering.