Dual Distributive Lattice is Distributive

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice.

Then
 * $L$ is a distributive lattice


 * $L^{-1}$ is a distributive lattice

where $L^{-1} = \left({S, \succeq}\right)$ denotes the dual of $L$.

Sufficient Condition
Let $L$ be a distributive lattice.

By Dual of Lattice Ordering is Lattice Ordering:
 * $L^{-1}$ is lattice.

Let $x, y, z \in S$.

$\vee'$ and $\wedge'$ denotes join and meet in $L^{-1}$.

Thus

Necessary Condition
This follows by mutatis mutandis.