we prove our first main lemma. We say that a simplicial poset $P$ contains a simplicial poset $Q$ if there is an order ideal $I \subset P$ which is isomorphic to $Q$.

\[3.A\] Fix integers $1 \leq i \leq m \leq d$ and $1 \leq j \leq d-i$. Let $P$ be a constructible $(d-1)$-ball such that $\partial P$ contains $\Delta_d(m)$ and let $\ell = \min\{m,i+j\}$. Then there is a constructible $(d-1)$-ball $Q$ such that

-   $h(Q)=h(P) + {\mathbf{e}}_i + {\mathbf{e}}_{j}$, and

-   $\partial Q$ contains $\Delta_d(\ell)$.

By the assumption, there is an order ideal $L \subset \partial P$ which is isomorphic to $\Delta_d(m)$. We write $$L=\big\{\mathcal L(F): F \subset [d],\ F \not \supset [m]\big\},$$ where the ordering is defined by $\mathcal L(F)>\mathcal L(G)$ if $F \supset G$. Since $i \leq m$, $$I=\big\{\mathcal L(F): F \subset [d],\ F \not \supset [i]\big\} \cong \Delta_d(i)$$ is an order ideal of $\partial P$. We prepare two Boolean algebras $$A=\big\{{\mathcal{A}}(F): F \subset [d]\big\} \mbox{ and } B=\big\{{\mathcal{B}}(F): F \subset [d] \big\},$$ where the ordering is defined by inclusion on $F$. Consider the order ideal $$J=\big\{ {\mathcal{A}}(F): F \subset [d],\ F \not \supset \{i+1,\dots,i+j\}\big\}
\cong \Delta_d(j).$$ Let $$Q=
(P \cup_I A) \cup_J B
=(P \cup A 