Equivalence of Definitions of Generalized Ordered Space/Definition 2 implies Definition 1

Theorem
Let $(X,\preceq)$ be a totally ordered set.

Let $\tau$ be a topology over $X$.

Let $(X',\preceq',\tau')$ be a linearly ordered space.

Let $\phi:X \to X'$ be an order embedding and a topological embedding.

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