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 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.