\leq 4$, and let $d_q$, with $q\in\{1,2,\ldots,|A'|\}$, be the unique leaf label of ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$ such that $\{a_i,a_j,d_q\}$ is the leaf set of a pendant subtree of ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$. By construction of ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$, it is easily seen that $d_q$ exists and $a_i<a_j$ in ${{\mathcal O}}$. Hence, $a_i<a_j$ in ${{\mathcal O}}'$. Turning to the triples, let $c_r=(a_i,a_j,a_k)$ be an element of some $C_\ell$ with $1 \leq \ell \leq 4$, and let $d_q$, with $q\in\{1,2,\ldots,|A'|\}$, be the unique leaf label of ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$ such that $\{a_i,a_j,a_k,c_r^1,c_r^2,c_r^3,c_r^4,d_q\}$ is the leaf set of a pendant subtree of ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$. Again, by construction, $d_q$ exists. Let ${{\mathcal S}}_\ell={{\mathcal T}}_\ell|\{a_i,a_j,a_k,c_r^1,c_r^2,c_r^3,c_r^4,d_q\}$ and, similarly, let ${{\mathcal S}}_\ell'={{\mathcal T}}_\ell'|\{a_i,a_j,a_k,c_r^1,c_r^2,c_r^3,c_r^4,d_q\}$. It is straightforward to check that each cherry-picking sequence for ${{\mathcal S}}_\ell$ and ${{\mathcal S}}_\ell'$ satisfies either $$a_i<a_j<a_k,\textnormal{ or }a_j<a_k<a_i.$$ Hence, as ${{\mathcal S}}_\ell$ and ${{\mathcal S}}_\ell'$ are pendant in ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$, respectively, we have $a_i<a_j<a_k$, or $a_j<a_k<a_i$ in ${{\mathcal O}}$ and, consequently, in ${{\mathcal O}}'$. Since the above argument holds for each pair and each triple, it follows that ${{\mathcal O}}'$ is a [4-Disjoint-Intermezzo]{} ordering for $I$ and, so, $I$ 