deletions | additions       diff --git a/This_example_shows_that_direction__.tex b/This_example_shows_that_direction__.tex  index a01e5bc..eec3d33 100644  --- a/This_example_shows_that_direction__.tex  +++ b/This_example_shows_that_direction__.tex 
...
\item If $k=0$: $i$ is in the cycle $C$, hence $\O_{i}(p)$ is already stable.   \item if $k=n$: Assume that for agent $j$ at distance $n-1$ from agent $l$, $j$'s opinion on $p$ stabilizes, that is, for some $m \in \mathbb{N}$, $O^{m}_ j(p)$ is stable. We need to consider the following cases:   \begin{itemize}  \item if $\bigwedge_{p \in \Atoms} \O^{m}_{R_p(i)}(p) \wedge \bigwedge_{\varphi \in \C}$ is consistent, then $\O^{m+1}_i(p)=\O^m_j(p)$, and $\O^{m+1}_i(p)$ is stable. \item if $\bigwedge_{p \in \Atoms} \O^{m}_{R_p(i)}(p) \wedge \bigwedge_{\varphi \in \C}$ is not consistent consistent, then  \begin{itemize}  \item if $\O^{m}_i(p)=\O^m_j(p)$,then  $\O^{m}_i(p)$ is stable. \item if $\O^{m}_i(p) \neq O^m_j(p)$,then  $\O^{m+1}_j(p) = \O^{m}_j(p)$. $i$ does not change his opinion at stage $m+1$  but nothing guarantees that this won't change later on. Consider the two possible subcases: \begin{itemize}  \item If there is a $t\in\mathbb{N}$ such that $\bigwedge_{p \in \Atoms} \O^{m}_{R_p(i)}(p) \wedge \bigwedge_{\varphi \in \C}$ is consistent, then $\O^{m+t+1}_i(p) = \O^{m}_j(p)$, and $\O^{m+t+1}_i(p)$ is stable.  \item If there is no such $t$, then $i$'s opinion on $p$ will never change: $\O^{m}_i(p)$ is stable.