Woset is Isomorphic to Set of its Initial Segments

Theorem
Let $\left({S, \preceq}\right)$ be a woset.

Let $A = \left\{{S_a: a \in S}\right\}$ where $S_a$ is the segment of $S$ determined by $a$.

Then:
 * $\left({S, \preceq}\right) \cong \left({A, \subseteq}\right)$

where $\cong$ denotes order isomorphism.

Proof
Define $f: S \to A$ as:
 * $\forall a \in S: f \left({a}\right) = S_a$

From the definition of a segment of a woset, $s_1 \prec s_2 \implies f \left({s_1}\right) \subset f \left({s_2}\right)$.

The result follows from Ordering Equivalent to a Subset Relation.