\begin{definition}[Definition:Order Embedding/Definition 4]
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: S \to T$ be a mapping.
Let $T' = \Img S$ be the image of $S$ under $\phi$.
$\phi$ is an '''order embedding of $S$ into $T$''' {{iff}}:
:the restriction of $\phi$ to $S \times T'$ is an order isomorphism between $\struct {S, \preceq_1}$ and $\struct {T', \preceq_2 \restriction_{T' \times T'} }$.
\end{definition}