deletions | additions       diff --git a/This_example_shows_that_direction__.tex b/This_example_shows_that_direction__.tex  index 38230cf..c171421 100644  --- a/This_example_shows_that_direction__.tex  +++ b/This_example_shows_that_direction__.tex 
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\item if $k=n$: Assume that it holds for $k=n-1$: 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 three cases:   \begin{itemize}  \item if there is no inconsistency (between influencers of i's opinion and the given constraints), then O^{m+1}_i(p)=O^m_j(p), and i's opinion on p is stable at stage m+1.   \item if there is some inconsistency, but O^{m}_i(p)=O^m_j(p), $O^{m}_i(p)=O^m_j(p)$,  then i's $i$'s  opinion on p $p$  is stable at stage m. $m$.  \item if there is some inconsistency, but O^{m}_i(p) $O^{m}_i(p)  \neq O^m_j(p), O^m_j(p)$,  then O^{m+1}_j(p) $O^{m+1}_j(p)  = O^{m}_j(p) O^{m}_j(p)$  : i $i$  does not change his opinion but nothing guarantees that this won't change later on (that's the interesting case). Consider the two possible subcases: \begin{itemize}  \item a) If consistency occurs at a later stage (say stage m+t $m+t$  for some t), $t$),  then O^{m+t+1}_i(p) $O^{m+t+1}_i(p)  = O^{m}_j(p), O^{m}_j(p)$,  and i's $i$'s  opinion is stable at stage m+t+1, $m+t+1$,  and \item b) If consistency never occurs, then i's $i$'s  opinion on p $p$  is stable already at stage m. $m$.  \end{itemize}  \end{itemize}  \end{itemize}