interval graph.

Assume that ${\bm{\langle} u ]} \subsetneq {\bm{\langle} v ]} \subsetneq {\bm{\langle} w ]} \mbox{ and } {\bm{\langle} u ]} \sim {\bm{\langle} w ]}$ in $G(C,A)$. So $u < v < w$ and $u+1 \to w$ in $A$. As we may assume that $A$ satisfies we have $u+1 \to v$ and $v+1 \to w$ in $A$, and so ${\bm{\langle} u ]} \sim {\bm{\langle} v ]}$ and ${\bm{\langle} v ]} \sim {\bm{\langle} w ]}$.

Reflexive $\operatorname{DL}$-graphs as subgraphs of products of Proper Interval Graphs {#sec:main2}
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In this section we prove Theorem \[thm:embedding\] and give some related results.

\[thm:embedding\] For any $\operatorname{DL}$-pair $(G,L)$, there is an embedding $L {\buildrel {} \over \cong}\cP {-}\cV$, where $\cV$ is a union of intervals of the form ${{[{{\alpha}_{[i]}}, {{\beta}^{[j]}}]}}$, of $L$ into a product $\cP = \prod P_i$ of chains. Further, for each $P_i$ there is a proper interval graph $G_i$ compatible with $P_i$ such that $G$ is the subgraph of $\cG = \prod_{i = 1}^dG_i$ induced by vertices in $L$.

Before proving this, we observe that this gives Corollary \[cor:charPI\].

If $G$ is a $\operatorname{DL}$-graph then Theorem \[thm:embedding\] gives us the necessary embedding of $G$ into a product of proper interval graphs.

On 