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

Theorem
Let $\struct {S, \preceq, \tau}$ be a generalized ordered space by Definition 3:

Then $\struct {S, \preceq, \tau}$ is a generalized ordered space by Definition 1:

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

Let $\BB$ be the set of intersections of finite subsets of $\SS$.

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

But $\BB$ is a basis for $\tau$.

Therefore $\tau$ has a basis consisting of convex sets.