Dual of Lattice Ordering is Lattice Ordering

Theorem
Let $\left({S, \preccurlyeq}\right)$ be a lattice.

Let $\preccurlyeq$ be the lattice ordering on a lattice $\left({S, \preccurlyeq}\right)$.

Then its dual ordering $\succcurlyeq$ is also a lattice ordering.

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

It is to be shown that:
 * for all $x, y \in S$, the ordered set $\left({\left\{{x, y}\right\}, \succcurlyeq}\right)$ admits both a supremum and an infimum.

Let $x, y \in S$.

Then $\left({\left\{{x, y}\right\}, \preccurlyeq}\right)$ admits both a supremum and an infimum.

Let $c = \sup \left({\left\{{x, y}\right\}, \preccurlyeq}\right)$.

Then by definition of supremum:
 * $\forall s \in \left\{{x, y}\right\}: s \preccurlyeq c$
 * $\forall d \in S: c \preccurlyeq d$

where $d$ is an upper bound of $\left({\left\{{x, y}\right\}, \preccurlyeq}\right) \subseteq S$.

Hence by definition of dual ordering:
 * $\forall s \in \left\{{x, y}\right\}: c \succcurlyeq s$
 * $\forall d \in S: d \succcurlyeq c$

where $d$ is an upper bound of $\left({\left\{{x, y}\right\}, \preccurlyeq}\right) \subseteq S$.

By Upper Bound is Lower Bound for Inverse Ordering, $d$ is a lower bound of $\left({\left\{{x, y}\right\}, \succcurlyeq}\right) \subseteq S$.

So by definition of infimum:
 * $c = \inf \left({\left\{{x, y}\right\}, \succcurlyeq}\right)$

That is, $\left({\left\{{x, y}\right\}, \succcurlyeq}\right)$ admits an infimum.

Let $c = \inf \left({\left\{{x, y}\right\}, \preccurlyeq}\right)$.

Then by definition of infimum:
 * $\forall s \in \left\{{x, y}\right\}: c \preccurlyeq s$
 * $\forall d \in S: d \preccurlyeq c$

where $d$ is a lower bound of $\left({\left\{{x, y}\right\}, \preccurlyeq}\right) \subseteq S$.

Hence by definition of dual ordering:
 * $\forall s \in \left\{{x, y}\right\}: s \succcurlyeq c$
 * $\forall d \in S: c \succcurlyeq d$

where $d$ is a lower bound of $\left({\left\{{x, y}\right\}, \preccurlyeq}\right) \subseteq S$.

By Lower Bound is Upper Bound for Inverse Ordering, $d$ is an upper bound of $\left({\left\{{x, y}\right\}, \succcurlyeq}\right) \subseteq S$.

So by definition of supremum:
 * $c = \sup \left({\left\{{x, y}\right\}, \succcurlyeq}\right)$

That is, $\left({\left\{{x, y}\right\}, \succcurlyeq}\right)$ admits a supremum.

Hence $\left({\left\{{x, y}\right\}, \succcurlyeq}\right)$ admits both a supremum and an infimum.

That is, $\succcurlyeq$ is a lattice ordering.