to an element in $$U'=\{u_{k,l}, \bar{u}_{k,l}\mid 1 \leq k \leq n\wedge 1 \leq l \leq 3\},$$ and each element $u_k$ appears exactly twice negated and twice unnegated in ${{\mathcal C}}$, it follows that there is a partition of $B\setminus B_2$ into $B_3$ and $B_4$ so that all pairs in the resulting two sets are pairwise disjoint. Setting $C_3 = C_4 = \emptyset$ completes the construction of an instance of [4-Disjoint-Intermezzo]{}. Noting that it is straightforward to compute the partition $$B \cup C = \bigcup_{1 \leq \ell \leq 4} \left ( B_\ell \cup C_\ell \right )$$ in polynomial time and that we did not modify the construction described by Guttmann and Maucher [@guttmann06 Lemma 1] itself, it follows from the same proof that $I$ has a satisfying truth assignment if and only if $\bigcup_{1 \leq \ell \leq 4} \left ( B_\ell \cup C_\ell \right )$ has a [4-Disjoint-Intermezzo]{} ordering.

[**Remark.**]{} By the construction of an instance of [4-Disjoint-Intermezzo]{} in the proof of Theorem \[t:4-DI\], we note that no pair or triple occurs twice and that, for each $\ell\in\{1,2,3,4\}$, we have $B_\ell\cup C_\ell\ne\emptyset$. We will freely use these facts throughout the remainder of the paper.

Hardness of [CPS-Existence]{} {#sec:cps}
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In this 