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.