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.