Equivalence of Definitions of Generalized Ordered Space/Definition 1 implies Definition 3

Theorem
Let $(X, \preceq, \tau)$ be a generalized ordered Space by Definition 1.

That is, $(X,\tau)$ is a Hausdorff space with a basis consisting of convex sets.

Then $(X, \preceq, \tau)$ is a generalized ordered space by Definition 3.

That is, $(X, \tau)$ is a Hausdorff space with a sub-basis consisting of upper and lower sets.

Proof
Let $\mathcal B$ be a basis for $\tau$ consisting of convex sets.

Let $\mathcal S = \bigl\{ {\uparrow}U: U \in \mathcal B \bigr\} \cup \bigl\{ {\downarrow}U: U \in \mathcal B \bigr\}$.

The elements of $\mathcal S$ are upper and lower sets by Upper Closure is Upper Set and Lower Closure is Lower Set.

We will show that $\mathcal S$ is a sub-basis for $\tau$.

By Upper and Lower Closures of Open Set in GO-Space are Open:
 * $\mathcal S \subseteq \tau$

By Convex Set Characterization (Order Theory), each element of $\mathcal B$ is the intersection of its upper closure with its lower closure, so each element of $\mathcal B$ is generated by $\mathcal S$.

Thus $\mathcal S$ is a sub-basis for $\tau$.