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

Theorem
Let $(X, \preceq, \tau)$ be 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.

Then $(X, \preceq, \tau)$ is a generalized ordered Space by Definition 1.

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

Proof
 Let $\mathcal S$ be a sub-basis for $\tau$ consisting of upper sets and lower sets.

Let $\mathcal B$ be the set of intersections of finite subsets of $\mathcal S$.

By Upper Set is Convex, Lower Set is Convex and Intersection of Convex Sets is Convex Set (Order Theory) :
 * the elements of $\mathcal B$ are convex.

But $\mathcal B$ is a basis for $\tau$, so $\tau$ has a basis consisting of convex sets.