the other hand, assume we get $G$ from a product $\cG = \prod G_i$ of proper interval graphs $G_i$ in min-max form by removing vertex intervals ${{[{{\alpha}_{[i]}}, {{\beta}^{[j]}}]}}$. The ordering on the $G_i$ is a chain lattice $P_i$, so induces on $V(\cG)$ a lattice $\cP = \prod P_i$. By a result in [@Ri74], the subset induced by removing sets of the form ${{[{{\alpha}_{[i]}}, {{\beta}^{[j]}}]}}$ is a sublattice. By Fact \[fact:sublat\] it is therefore compatible with the subgraph $G$ of $\cG$ that it induces.

We start with some results from the literature that will help us prove Theorem \[thm:embedding\].

Setup for the proof of Theorem \[thm:embedding\]
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The first statement of Theorem \[thm:embedding\] is acually from [@Si15]. We explain, as we will have to build on this. Generalising the notion of a [*chain decomposition*]{} in which the subchains must be disjoint, a [*chain cover*]{} of a poset $P$ is family $\cC = \{C_1, \dots, C_d\}$ of subchains of $P$ such that every element of $P$ is in one chain. Given $\cC$, label the elements of $P$ so that the subchain $C_i$ is ${{1}^{(i)}} \prec\dots \prec {{{n_i}}^{(i)}}$ for some $n_i$; if an element is in more than one chain, it gets more 