can suppose that $e_i^{(n)} \to e_i$ and $\varphi_n e_i^{(n)} \to \tilde{e}_i$ for each $i \in \{0, \ldots, m\}$. In particular, $\tilde{e}_0 = \tilde{d}$ and $\tilde{e}_m = \tilde{c}$. Since $\tilde{c}$ is not opposite $\tilde{d}$, the gallery of vertices $\tilde{d} = \tilde{e}_0, \tilde{e}_1, \ldots, \tilde{e}_m = \tilde{c}$ must stammer. This means that there exists $k \in \{0, \ldots, m-2\}$ such that $\tilde{e}_k = \tilde{e}_{k+2}$ (if there is more than one such $k$, then we choose the smallest one). In other words, since $e_k \neq e_{k+2}$, the two sequences $e_k^{(n)} \to e_k$ and $e_{k+2}^{(n)} \to e_{k+2}$ are centered at $e_{k+1}^{(n)} \to e_{k+1}$ and collapsed by $(\varphi_n)$. Now take $c'$ a vertex of $A$ such that $\operatorname{\mathrm{D}}(c',d) = m-k-1$ and $c'_n$ the vertex of $A_n$ (for $n \in \operatorname{\mathbf{N}}$) such that $c'_n \to c'$. Thanks to this choice, $e_{k+1}$ is opposite $c'$ and $\tilde{e}_{k+1}$ is opposite the limit $\tilde{c}'$ of $(\varphi_n c'_n)$ (in the particular case where $\tilde{e}_1 \in \tilde{A}$, we choose $c'$ so that the minimal gallery from $\tilde{d}$ to $\tilde{c}'$ does not contain $\tilde{e}_1$). We can therefore apply Lemma \[lemma:opposite\] to the sequences $e_k^{(n)} \to e_k$ and $e_{k+2}^{(n)} \to e_{k+2}$ centered at $e_{k+1}^{(n)} \to e_{k+1}$ and the opposite sequence $c'_n \to c'$.

