Equivalence of Definitions of Lower Section

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $U \subseteq S$.

Then the following are equivalent:

where $U^\preceq$ is the lower closure of $U$.

Proof
By the Duality Principle, it suffices to prove that:


 * $(1^*)$, $(2^*)$ and $(3^*)$ are equivalent

where these are the dual statements of $(1)$, $(2)$ and $(3)$, respectively.

By Dual Pairs, it can be seen that these dual statements are as follows:

Their equivalence is proved on Equivalence of Definitions of Upper Set.