deletions | additions       diff --git a/This_example_shows_that_direction__.tex b/This_example_shows_that_direction__.tex  index c4cf554..a8a1b5a 100644  --- a/This_example_shows_that_direction__.tex  +++ b/This_example_shows_that_direction__.tex 
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\begin{proof}  Assume that for all $p\in \Atoms$, for all $C\subseteq\N$ such that $C$ is a cycle in $G_{p}$, for all $i,j\in C$: $\O_i(p)=\O_j(p)$.  Consider an arbitrary $p\in \Atoms$ \Atoms$,  and an arbitrary $i\in \N$. There Let $k$ be the distance from $i$ to $l$, where $l$  is some the closest  agent$l$  in a cycle $C\subseteq\N$ of $G_p$, such that $l$ is at some distance $k$ from $i$. $G_p$.  We show that for any suchdistance  $k\in\mathbb{N}$, there exists an $n \in \mathbb{N}$, such that $\O^{n}_ i(p)$ is stable. \begin{itemize}  \item If $k=0$: $i\in C$, hence $\O_{i}(p)$ is stable.   \item if $k=n+1$: Assume that for agent $j$ at distance $n$ from agent $l$,