deletions | additions       diff --git a/This_example_shows_that_direction__.tex b/This_example_shows_that_direction__.tex  index e3eb402..f6ae6d2 100644  --- a/This_example_shows_that_direction__.tex  +++ b/This_example_shows_that_direction__.tex 
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\end{theorem}  \begin{proof}  Assume that for all $p\in \Atoms$, for all $S\subseteq\N$ such that $S$ is a cycle in $G_{p}$, for all $i,j\in C$: S$:  $O_i(p)=O_j(p)$. Consider an arbitrary $p\in \Atoms$, and an arbitrary $i\in \N$. Let $k$ be the distance from $i$ to $l$, where $l$ is the closest agent in a cycle $C\subseteq\N$ $S\subseteq\N$  of $G_p$. We show that for any such $k\in\mathbb{N}$, there exists an $n \in \mathbb{N}$, such that $O^{n}_ i(p)$ is stable. \begin{itemize}  \item If $k=0$: $i\in C$, S$,  hence $O_{i}(p)$ is stable. \item if $k=n+1$: Assume that for agent $j$ at distance $n$ fromagent  $l$, for some $m \in \mathbb{N}$, $O^{m}_ j(p)$ is stable. We need to consider the following cases:  \begin{enumerate}  \item If $\bigwedge_{p \in \Atoms} O^{m}_{R_p(i)}(p) \wedge \bigwedge_{\varphi \in \C}$ is consistent,