ion $P$, which corresponds to a top dimensional maximal simplex, as shown in Figure \[Fig7\] (i). We see that $a$ is adjacent to only $b$ w.r.t. $P$. Let $a' \in \lambda([a])$. Let $P'$ be a set of pairwise disjoint elements of $\lambda([P])$ containing $a'$. Let $b' \in \lambda([b]) \cap P'$, $c' \in \lambda([c]) \cap P'$, $x_i' \in \lambda([x_i]) \cap P'$ for $i = 1, 2, \cdots, n-2$. Since $a'$ is separating and $a'$ should be adjacent to only one curve w.r.t $P'$, we see that there is a subsurface $T \subseteq N$ such that it is homeomorphic to sphere with four boundary components where three of the boundary components of $T$ are the boundary components of $N$, $b'$ is a boundary component of $T$, and $a'$ divides $T$ into two pair of pants as shown in Figure \[Fig7\] (ii). Since $b$ is adjacent to $c$ and $x_1$, $b'$ is adjacent to $c'$ and $x_1'$. Since $c$ is not adjacent to any other curve, $c'$ will not be adjacent to any other curve. This implies that $c'$ is either a 1-sided curve or a separating curve as shown in Figures \[Fig7\] (ii), \[Fig8\] (ii).

Suppose $c'$ is as shown in Figures \[Fig7\] (ii). By using the adjacency relations between $x_i$’s we get $P'$ as shown in Figure \[Fig7\] (ii).

Let $z, t$ be as shown in Figure \[Fig7\] (i). We see that $[z] \neq [t]$, each of $z$ and $t$ intersects $x_{n-2}$ essentially and each of them is disjoint from all the other curves in $P$. Let $z' \in \lambda([z]), t' \in \lambda([t])$ such that each of $z'$ and $t'$ intersects each curve in $P'$ minimally. Since $\lambda$ is injective by Lemma \[inj\], $[z'] \neq [t']$. We see that $z', t'$ are not in $P'$, and they both are disjoint from all the curves in $P'$ except for $x_{n-2}'$. We get a contradiction again similar to case (a), by using Scharlemann’s result about complex of curves of projective plane with two boundary components.

Similarly, if $c'$ is as shown in Figures \[Fig8\] (ii), by using the adjacency relations between $x_i$’s we get $P'$ as shown in \[Fig8\] (ii). By choosing curves $w, y