$A/J$ is a $(d-1)$-dimensional Cohen–Macaulay ring, $$\begin{aligned}
\label{B}
\dim_K \left(A/ \left( J+ (\overline \Theta) \right) \right)_k = h_k(\partial P)=\partial h_k\end{aligned}$$ for $k = 0,1,\dots,d-1$. In particular, since $\dim_K(A/(J+(\overline{\Theta})))_n=\partial h_n =0$ by the assumption, we have $\theta_d^n \in J+ (\overline {\Theta}).$ Then, since $J \supset I_P$, we have $J+(\overline\Theta) \supset I_P + (\overline\Theta)+(\theta_d^n)$. Thus $$\begin{aligned}
\dim_K \big( A/ \big( I_P + (\overline {\Theta})+ (\theta_d^n ) \big) \big)_ k \geq \dim_K \left( A/ \left( J+(\overline {\Theta}) \right) \right)_k=\partial h_k\end{aligned}$$ for all $k =0,1,\dots,d-1$. Then the statement follows from Lemma \[trivial\].

Proof of (5), (6) and (7)
-------------------------

Let $P$ be a $(d-1)$-dimensional simplicial poset. A rank $1$ element of $P$ is called a *vertex of $P$* and a maximal element (w.r.t. a partial order) of $P$ is called a *facet of $P$*. The *cone of $P$* is the product of the posets $$C(P)=P \times \{1,2\}=\{(\sigma,i): \sigma \in P,\ i=1,2\},$$ where the ordering is defined by $(\sigma,i) \geq (\tau,j)$ if $\sigma \geq \tau$ and $i \geq j$. Then $C(P)$ is simplicial and $\Gamma(C(P))$ is homeomorphic to the (topological) cone of $\Gamma(P)$. Also, a straightforward computation shows $h_i(P)=h_i(C(P))$ for all $i=0,1,\dots,d$ and $h_{d+1}(C(P))=0$.

Let $P$ be a simplicial cell ball. We define the simplicial 