% % \providecommand{\Gp}{G'} \providecommand{\bGM}{\beta_{\mathrm{GM}}} \providecommand{\mt}{\tilde m} \providecommand{\eps}{\varepsilon} \providecommand{\Du}{D_{\mathcal U}} \providecommand{\Phiu}{\Phi_{\mathcal U}} \providecommand{\epsE}{\varepsilon_{E}} \providecommand{\lPV}{\lambda_{\mathrm{PV}}} \providecommand{\Kcell}{K_{\mathrm{cell}}} \providecommand{\Floc}{F_{1}^{\mathrm{loc}}} \providecommand{\Funit}{F_{\mathrm{unit}}} \providecommand{\supp}{\operatorname{supp}} \providecommand{\Vol}{\operatorname{Vol}} \providecommand{\cK}{\mathcal K} \providecommand{\cN}{\mathcal N} \providecommand{\cU}{\mathcal U} \providecommand{\cP}{\mathcal P} \providecommand{\sat}{\operatorname{sat}} \providecommand{\bC}{\mathbf C} \providecommand{\Dep}{\mathfrak{D}} \providecommand{\bD}{b_{D}} \providecommand{\NN}{\mathbb{N}} \providecommand{\RR}{\mathbb{R}} \section{Brydges\textendash Federbush \S9.1 on slit domains, and a counting step} \label{app:A} In this appendix we prove Lemmas~\ref{lem:7.39} and~\ref{lem:7.53}, carrying \cite[\S9.1]{BF80} with \cite[\S5]{B78} over to $\Gp$. The arguments are given here in condensed form. The complete proofs are in Section~\ref{S4-S:S4.8} of the supplement, and Table~\ref{tab:C.1} lists the structural facts about the expansion that the arguments use.\footnote{Of these facts, S6 is used in the proof of Lemma~\ref{lem:A.1}, S5 in the paragraph on link data and S13 in item~(4) of the proof of Proposition~\ref{prop:A.8}.} Proposition~\ref{prop:A.5} counts the sets with the tree weight $w(T)$ kept outermost, which cancels the moments $d_j!$ that the count produces. \emph{Conventions.} We assume that $(U,k)$ is admissible and that $\sigma=(Z,N_Z)$ is a seed configuration in $(U,k)$ (Section~\ref{sec:7.6}). By Proposition~\ref{prop:7.long}, at $k=0$ the configuration sets may be those of long wall fields. We write $|Y|$ for the number of $\cU$-cells of $Y$ and set $\dist(A,B):=\mt\,d_{\Gp}(A,B)$. The $\cU$-cell graph joins two $\cU$-cells when a bond of $\Gp$ joins them, and $D_{\cU}$ bounds its degree; by (R7) we may take $D_{\cU}:=\frac{4\pi}3(2C_{\cU}+2)^3/c_u$, a constant, since the $\cU$-cells joined to a given one lie in the ball of radius $2C_{\cU}\mt^{-1}+2$ about one of its sites and each has more than $c_u\mt^{-3}$ sites ($\mt\le1$). The quantity $\cK^{(Z)}_{N_X}(X)$ is the sum over $\bar y=(Y_1,\dots,Y_n)$, $Y_1=Z$, of the terms of Proposition~\ref{prop:7.25}. For $2\le j\le n$, \emph{link $j$} is the operator $\kappa(j-1)[Y_j]$ of \eqref{eq:C.E20}, which creates $Y_j$, differentiates $s_{j-1}$, and is a sum of elementary terms of the five types listed after \eqref{eq:C.E16}. We also write $X_j:=Y_1\cup\dots\cup Y_j$. A \emph{tree} is a map $T:\{2,\dots,n\}\to\NN$ with $T(j)i$ and a cube $a_\beta\subset\bigcup_{\ell\le i}Y_\ell$. Under this rule the expansion is an identity, by the proof of Proposition~\ref{prop:7.25}, and $\partial_{s_{j-1}}w_s(\tau)=\prod_{k=T(j)}^{j-2}s_k$, as among $k\le j-1$ the set $\supp\tau$ meets both $X_k$ and its complement exactly for $k\ge T(j)$. The rule stated in \cite[p.~212]{BF80} has $Y_{i+1}$ in place of $\bigcup_{j>i}Y_j$. Under that rule a term meeting $Y_1$ and $Y_3$ but not $Y_2$ carries no factor $s_1$, and no tree weight dominates it. Finally, link $j$ differentiates $s_{j-1}$ and no other $s_k$, so each $s_k$ is differentiated once, in link $k+1$. \qed \end{proof} As the bounds of Section~\ref{sec:7.6} and Appendix~\ref{app:B} on $R_\theta(s)$ are uniform in $s$ (Table~\ref{tab:C.1}, S1), the weights lie in $[0,1]$ and $\int q(T,s)\,ds=w(T)$, grouping the elementary terms by their link data $\lambda=(\lambda_j)_{j\ge2}$ (defined below; they determine the tree $T=T_\lambda$) gives \begin{equation}\label{eq:A.treesum} |\cK^{(Z)}_{N_X}(X)|\le\sum_{\bar y}\sum_{\lambda}w(T_\lambda)\,\sup_{s}\sum_{\theta:\,\lambda(\theta)=\lambda} \Bigl|\int d\mu_s\,e^{E(X,s)}R_\theta(s)\,Q^\natural_{N_X}(X)\Bigr|_0 , \end{equation} with the supremum outside the sum over the terms with one link datum, which Proposition~\ref{prop:B.7} bounds together. \begin{lemma}\label{lem:A.2} For $n\ge2$ and $y_1,\dots,y_n\ge0$, \begin{equation}\label{eq:A.vol} \sum_T\int_{[0,1]^{n-1}}q(T,s)\prod_{j=2}^ny_{T(j)}\,ds\le e^{y_1+\dots+y_{n-1}} . \end{equation} In particular $\sum_Tw(T)\le e^{n-1}$. \end{lemma} \begin{proof} This is \cite[(5.5)]{B78} and \cite[Lemma~5.5]{BF77}, for real weights. The sum over $T$ is $\prod_{m=1}^{n-1}(y_m+P_{m-1}(s))$ with $P_m:=s_m(y_m+P_{m-1})$, $P_0:=0$, and we integrate over $s_{n-2},\dots,s_1$ in turn using $\int_0^1x\,e^{sx}\,ds\le e^x$.\footnote{For $n\le7$ we have also evaluated the left side of the lemma's inequality exactly, and it is at most the right side (Section~\ref{ssec:E.5}).} \qed \end{proof} \begin{lemma}\label{lem:A.3} For $n\ge2$, $y_1\ge0$ and $b_2,\dots,b_{n-1}\in[0,1)$, \begin{equation}\label{eq:A.mom} \sum_Tw(T)\,y_1^{d_1(T)}\prod_{i=2}^{n-1}d_i(T)!\,b_i^{d_i(T)}\le e^{y_1}\prod_{i=2}^{n-1}(1-b_i)^{-1}. \end{equation} \end{lemma} \begin{proof} We use the identity $d!\,b^d=\int_0^\infty(bt)^de^{-t}\,dt$ for each $i=2,\dots,n-1$. Since $\prod_jy_{T(j)}=\prod_iy_i^{d_i(T)}$ and $d_n(T)=0$, Tonelli's theorem and \eqref{eq:A.vol} with $y_i:=b_it_i$ ($2\le i\le n-1$) bound the left side by $\int_{\RR_+^{n-2}}e^{y_1+\sum_ib_it_i}e^{-\sum_it_i}\,dt=e^{y_1}\prod_{i=2}^{n-1}(1-b_i)^{-1}$. \qed \end{proof} For a finite set $A$ of $\cU$-cells, $\mathfrak Y(A)$ is the set of nonempty unions $Y$ of $\cU$-cells each of whose components in the $\cU$-cell graph meets $A$. For $a=\log(2eD_{\cU})+a'+a''$ with $a'\ge4$, $a''\ge0$, and all integers $d\ge0$, \begin{equation}\label{eq:A.rooted} \textstyle\sum_{Y\in\mathfrak Y(A)}|Y|^d\,e^{-a|Y|}\le e^{-a''}\,2^{|A|}\,d!\,2^{-d}: \end{equation} if $W$ contains each $\cU$-cell independently with probability $p\in[0,1]$, then $p^{|Y|}(1-p)^{|\partial Y|}$, $\partial Y$ the set of cells outside $Y$ adjacent to $Y$, is the probability that $W\supseteq Y$ and $W\cap\partial Y=\emptyset$; for each $W$ at most $2^{|A|}$ sets $Y\in\mathfrak Y(A)$ have this property, since such a $Y$ is the union of the components of $W$ that meet $A\cap Y$. Hence $\sum_{Y\in\mathfrak Y(A)}p^{|Y|}(1-p)^{|\partial Y|}\le2^{|A|}$, and with $|\partial Y|\le D_{\cU}|Y|$ and $p=1/(D_{\cU}+1)$, at most $2^{|A|}(e(D_{\cU}+1))^N\le2^{|A|}(2eD_{\cU})^N$ sets $Y\in\mathfrak Y(A)$ have $|Y|=N$. We conclude with $N^de^{-a'N/2}\le d!\,(2/a')^d$. \emph{Link data.} The \emph{link datum} $\lambda_j$ of an elementary term $\theta_j$ consists of its type, of the number of points and the set of $\cU$-cells containing them for each of its $E$-terms (at most two), and of the cells $\Delta'_j$, $\Delta''_j$ of Lemma~\ref{lem:A.1}. The set $A(\lambda_j)$ consists of $\Delta''_j$ and the $\cU$-cells of its $E$-terms, and $\varphi\ge0$ is a weight on link data with \begin{equation}\label{eq:A.Cphi} C_\varphi:=\sup_{\Delta_0}\sum_{\lambda:\,\Delta'=\Delta_0}\varphi(\lambda)\,2^{|A(\lambda)|}<\infty . \end{equation} Each $Y_j$ ($j\ge2$) of a term with $\theta_j\ne0$ lies in $\mathfrak Y(A(\lambda_j))$. Indeed, each element is connected in the $\cU$-cell graph, since a wall component is a component of $\hat\Sigma$ and, by Definition~\ref{def:7.23}, a closure adds bond neighbours and linked closures share a $\cU$-cell. Each element of $Y_j$ also meets $S(\theta_j)\setminus X_{j-1}$, because link $j$ creates $Y_j$ as the union of the elements that meet this set. This set lies in the cells of $A(\lambda_j)$, as the point $y$ of a covariance line lies in $X_{j-1}$. \begin{proposition}\label{prop:A.5} Let $a$ be as in \eqref{eq:A.rooted}, with $2C_\varphi e^{-a''}\le\frac12$. For every seed configuration $\sigma=(Z,N_Z)$, \begin{equation}\label{eq:A.comb} \sum_{n\ge1}\ \sum_T\ \sum_{(Y_j,\lambda_j)_{j=2}^n}w(T)\prod_{j=2}^n\varphi(\lambda_j)\,e^{-a|Y_j|}\le2e^{|Z|}, \end{equation} where $Y_1:=Z$, the sum runs over all trees $T$, all link data with $\Delta'_j\subseteq Y_{T(j)}$, and all $Y_j\in\mathfrak Y(A(\lambda_j))$ (the $Y_j$ may overlap, and $Y_j\subseteq U$ need not hold), and the term $n=1$ equals $1$. \end{proposition} \begin{proof} We keep the tree outermost with its weight and count the sets backward from the last one, each set rooted at its own link and each link rooted in an earlier set. The roots in earlier sets produce the moments of Lemma~\ref{lem:A.3}. We fix $n\ge2$ and $T$ and sum in the order $(Y_n,\lambda_n),\dots,(Y_2,\lambda_2)$, innermost first, and inside stage $j$ first over $Y_j$, then over $\lambda_j$ with $\Delta'_j$ fixed, then over $\Delta'_j\subseteq Y_{T(j)}$. In this order every constraint ties a variable only to variables summed further out. We claim that at stage $j$ the summand depends on $Y_j$ only through $|Y_j|^{d_j(T)}e^{-a|Y_j|}$ and the constraint $Y_j\in\mathfrak Y(A(\lambda_j))$. At stage $j$, \eqref{eq:A.rooted} gives $\sum_{Y_j\in\mathfrak Y(A(\lambda_j))}|Y_j|^{d_j}e^{-a|Y_j|}\le e^{-a''}2^{|A(\lambda_j)|}d_j!\,2^{-d_j}$. The sum over $\lambda_j$ with $\Delta'_j$ fixed then gives at most $C_\varphi e^{-a''}d_j!\,2^{-d_j}$ by \eqref{eq:A.Cphi}, and the remaining sum contributes $\sum_{\Delta'_j\subseteq Y_{T(j)}}1=|Y_{T(j)}|$, a factor that is handled at stage $T(j)0); \end{equation} where $\one_{\mathrm{line}}(\lambda)=1$ if $\lambda$ is of a type with a covariance line (types (2) to (5), Lemma~\ref{lem:A.1}) and $\one_{\mathrm{line}}(\lambda)=0$ for type (1); $c_\gamma<\infty$, since by (R7) at most $\frac{4\pi}3(r+2C_{\cU}+1)^3/c_u$ $\cU$-cells lie within distance $r$ of a given one. \begin{lemma}[Link sum]\label{lem:A.7} Let $c_{10},\delta_2>0$ and $e^{c''_{10}}\ge16e\max(1,c_{c_{10}/2})$. Then $C_\varphi\le1+8c_{\delta_2}<\infty$. Moreover, for every $E$-term $\mathcal E$ of $\lambda$, with $\mathfrak a$ the set of $\cU$-cells of its points, and every augmented tree $\eta^A$ of \cite[Appendix~1]{BF80} whose marked points lie in the cells of $\mathfrak a$, one in each, the contribution of $\mathcal E$ to $L_0(\lambda)$ is at most $L_{\eta^A}$; so $e^{c_{10}L_0(\lambda)}e^{c''_{10}O_0(\lambda)}$ is dominated, inside $|\cdot|_0$, by replacing each $E$-term $\mathcal E$ by $\sum_{\eta^A}e^{c_{10}L_{\eta^A}}e^{c''_{10}t}\mathcal E_{\eta^A}$, for any decomposition $\mathcal E=\sum_{\eta^A}\mathcal E_{\eta^A}$ (\cite[(9.18), (9.18a)]{BF80}). Which point of $\mathcal E$ lies in which cell of $\mathfrak a$, the positions and the charges are not part of $\lambda$; they are summed inside $|\cdot|_0$. \end{lemma} \begin{proof} The argument is that of 8$\Sigma$ and 10$\Sigma$ of \cite[p.~217]{BF80}, with the choice $B_8B_{10}=e^{c''_{10}O_0}e^{c_{10}L_0}$, together with (9.18) and the lengths \cite[(A1.3)]{BF80}, adapted as follows. (1) Positions are sets of $\cU$-cells. We count a set of $k$ cells once for its $(k-1)!$ orderings, and the tree sum gives $e^kc_{c_{10}/2}^{k-1}$ (by Cayley's formula, $k^{k-2}$ labelled trees on $k$ vertices, and $k^{k-2}/(k-1)!\le e^k$). The terms with the same link datum differ in the assignment of the points of each $E$-term to the cells of its set, in their positions and in their charges, and for $t=2$ also in the form. Proposition~\ref{prop:B.7} bounds their sum through the bound (M4) of Lemma~\ref{lem:B.1}(d), which holds for the sum over all assignments. This is needed because a set of $k$ cells carries $k!\,S(t,k)$ assignments of $t$ points, where $S$ denotes the Stirling numbers of the second kind, and $t!$ assignments if $k=t$. (2) The factor $2^{|A(\lambda)|}$ costs $4^k$ per $E$-term with a marked cell. This raises by $\log2$ the constant $c''_{10}$; \cite[p.~217]{BF80} gives no value for it. The cell $\Delta''$ costs $2c_{\delta_2}$ in each of the four types with a covariance line. Hence $C_\varphi\le1+8c_{\delta_2}$. (3) The length $L_{\eta^A}$ is an infimum over the positions of the marked points in the cells of $\mathfrak a$ and of the auxiliary points, measured with $\dist$, and a tree on given points has at least half the length of a shortest tree spanning them, which dominates one spanning their cells. Hence $L_0\le L_{\eta^A}$, and $e^{c_{10}L_0}|\mathcal E|\le\sum_{\eta^A}e^{c_{10}L_{\eta^A}}|\mathcal E_{\eta^A}|$ as $\mathcal E=\sum_{\eta^A}\mathcal E_{\eta^A}$. Under the conditions $c_{10}\le\frac79r_C$ and $\mu\ge1+2c''_{10}$, whose constants are fixed in the order given in Section~\ref{S:H1}, the proof of Proposition~\ref{prop:B.7} absorbs the factors $e^{c_{10}L_{\eta^A}}e^{c''_{10}t}$ of $\kappa'$ when it bounds the weights of the augmented trees. \qed \end{proof} In the count of \cite[(7.3) to (7.5)]{GM82} the Mayer factors $E(a_1,\dots,a_t)$ keep the ordered tuple of cubes $(a_1,\dots,a_t)$ among the summed parameters, with a weight exponential in the least tree length $L(a_1,\dots,a_t)$ of \cite[(1.11b)]{GM82}. This count is that of \cite[10$\Sigma$]{BF80}. There, on \cite[p.~214]{BF80}, the positions of the cubes are summed at a fixed augmented tree $\eta^A$, with the constant weights $b_{\eta^A}$ of \cite[(A1.4)]{BF80}. With the least tree length in place of these weights, the counting step needs a coefficient bound for the sum over all positions of the points with their tree length inside the sum. The derivation of \cite[(4.31)]{GM82} provides it through the pinned norm \cite[(4.12)]{GM82}. Theorem~\ref{thm:6.5} states (S2) in this form, and the count above uses it as explained in Lemma~\ref{lem:A.7} and its proof. \begin{proposition}[Lemma~\ref{lem:7.39} on $\Gp$]\label{prop:A.8} Lemma~\ref{lem:7.39} holds, with $C_\varphi$ the constant of Lemma~\ref{lem:A.7}. The same proof gives the bound with $e^{\alpha|X|}$ replaced on the left by $e^{\alpha|X|+\eps\Floc(X,N_X)}$, $\eps\ge0$, and $e^{c_1'\Floc}$ replaced on the right by $e^{(c_1'+\eps)\Floc}$. \end{proposition} \begin{proof} We follow the proof of \cite[Lemma~9.1]{BF80} by the device of \cite[(9.12)]{BF80}, the bound $\sum_v|f(v)|\le(\sum_vB(v)^{-1})\sup_v|B(v)f(v)|$ over the data $v=(n,T,\bar y,(\lambda_j)_{j\ge2},N_X)$ of \eqref{eq:A.treesum}, with $B(v):=e^{a|X\setminus Z|}e^{c_1'\Floc(X,N_X)}\prod_j\varphi(\lambda_j)^{-1}$, $a:=\log(2eD_{\cU})+4+c_4'+a''$, $e^{a''}\ge4C_\varphi$. Terms with the same link data differ only inside $|\cdot|_0$, namely in the maps $l$ of Lemma~\ref{lem:B.2} and in the data of the $E$-terms listed in the proof of Lemma~\ref{lem:A.7}, and their sum is bounded as there. We change the argument at the following points. (1) For the count of the sets we use Proposition~\ref{prop:A.5} with $a'=4$, after summing $N_X$ innermost by Lemma~\ref{lem:7.32}(b) and keeping only $Y_j\in\mathfrak Y(A(\lambda_j))$. Thus $\sum_vw(T)B(v)^{-1}\le e^{(1+c_4'+\log2)|Z|}\le e^{a|Z|}$, the supremum carries $e^{c_2^L|X|}$, $c_2^L:=a+\alpha$, and Lemma~\ref{lem:A.7} turns $\prod_je^{c_{10}L_0(\lambda_j)+c''_{10}O_0(\lambda_j)}$ and $R_\theta$ into $\kappa'$. (2) The seed is fixed and enters through $|Z|$ and $\cN_\sigma(X)$ alone, not through its connectivity. Seeds depending on the configuration are reduced to fixed ones in Proposition~\ref{prop:A.9}. (3) No step refers to the set $X_1$ of Definition~\ref{def:7.23}(c). (4) For $U=X_0^c$ the activity is independent of the region, since it depends only on the sets of the sequence, the block of the covariance on their union and the configuration there. (5) We use only the distance-zero form ($W\subseteq Z$ or $W\supseteq Z$, where the restriction $X\cap W\ne\emptyset$ is vacuous), since on $\Gp$ a $\cU$-cell can have $d_{\Gp}$-diameter of order $L'$ and the clustering form \cite[(9.122)]{BF80} would therefore cost $e^{O(\mt L')}=e^{O(\tau)}$ per non-seed cell. As $\Floc(X,N_X)$ depends neither on $\psi$ nor on $s$, we may move $e^{\eps\Floc}$ into the supremum. \qed \end{proof} For an anchor $S_0$, $\Sigma_{\mathrm{sat}}(S_0)$ is the set of seed configurations $\sigma=(Z,N_Z)$ with $Z\supseteq S_0$ each element of which meets $S_0$. We write $\Floc(\mathcal S,N_{\mathcal S})$ for the sum of $F_1(S_\beta)$ over the wall components of a wall polymer $\mathcal S$, $D_I$ for the largest number of interfaces within Euclidean distance $g$ of a given interface or site, and ($\beta1$), ($\beta3$), ($\beta4$) for the conditions \eqref{eq:A.beta} on $\bGMo$ of Section~\ref{S:H1}. \begin{proposition}[Core estimate]\label{prop:A.9} Let $(U,k)$ be admissible, $\bGM\ge\bGMo$ and $\eps\in[0,c'/2]$. For every seed configuration $\sigma=(Z,N_Z)$ in $(U,k)$, \begin{equation}\label{eq:A.core} \sum_{\substack{X\supseteq Z\\ X\subseteq U}}\ \sum_{N_X\in\cN_\sigma(X)}|\cK^{(Z)}_{N_X}(X)|\,e^{\alpha|X|+\eps\Floc(X,N_X)}\le\cP'\,e^{c_S|Z^o|}\,e^{-(11c'/16-\eps)\Floc(Z,N_Z)}, \end{equation} and the sum restricted to $X\neq Z$ is at most $c_{16}^D$ times the right side. Consequently Lemma~\ref{lem:7.53} holds. \end{proposition} \begin{proof} The proof combines the per-cell bound of \cite[\S9.8]{BF80}, applied after Lemma~\ref{lem:7.39} with the factor $e^{-\Floc}$ divided region by region, with a split of the sum by the first set. (1) \emph{Booking.} Proposition~\ref{prop:A.8}, with $c_1'=c'/4$ and the extraction $e^{\eps\Floc}$, together with Proposition~\ref{prop:7.37}, Lemma~\ref{lem:7.40} and Proposition~\ref{prop:7.41}, leaves $e^{-(3c'/4-\eps)\Floc(X)}$, whose part on $Z$ is constant on $\cN_\sigma(X)$. We divide this factor between the two regions as in Table~\ref{tab:A.1} of the supplement. On $X\setminus Z$ we use $c'/16$ for $|D|$, $|F_2'|$ and $c'/8$ for the wall polymers, and on $Z$ we use $c'/32$ for $|D|$, $|F_2'|$ and $c'/32$ for the cells of $Z^\Sigma$ under ($\beta1$). Then each non-seed cell carries at most $\eps_\Delta\le\frac12c_{16}^D\le\frac12$ by ($\beta4$), lone seed cells carry at most $e^{c_S}$, and the fraction $c'-c'/4-\eps-c'/32-c'/32=11c'/16-\eps$ remains on the seed. This gives \eqref{eq:A.core}. Terms with $X\ne Z$ contain a non-seed cell and gain $\frac12c_{16}^D$. (2) \emph{Splitting.} By Lemma~\ref{lem:7.26}, the map $(X,N_X)\mapsto(\sigma(N_X),X,N_X)$, $\sigma(N_X):=(Z,N_X|_{E(Z)})$ with $Z:=\sat_{N_X}(S_0)$, is a bijection onto the triples with $\sigma\in\Sigma_{\mathrm{sat}}(S_0)$, $X\supseteq Z$, $N_X\in\cN_\sigma(X)$, preserving the summands. (3) \emph{Entropy of the first set.} Section~\ref{S:I2} gives this step in detail. As $11c'/16-\eps\ge c'/8$, the left side of Lemma~\ref{lem:7.53}(a) is at most $\cP'e^{c_S|S_0|}\sum_\sigma\prod_{\mathcal S}e^{-(c'/8)\Floc(\mathcal S,N_{\mathcal S})}$. We assign the wall polymers of $\sigma$ to their first cells in $S_0$ and count them by their wall interfaces and values with Lemmas~\ref{lem:5.12} and~\ref{lem:7.32}(a). Under ($\beta3$) this gives the factor $1+\etaseed$ per cell of $S_0$. From this we obtain (a), then (b), and (c$'$) with $\cP'=1$, $Z^o=\emptyset$. \qed \end{proof}