\subsection{Proof}
\begin{theorem}
Every 4-connected graph $G$ is 3-resilient.
\end{theorem}
\begin{proof}
We prove the statement by induction on the number of edges.


First of all, if the graph has a nontrivial 4-cut, then we use the nontrivial 4-cut reduction.
If $G$ contains an edge $e=(u,v)$ such that $\deg(u) > 4$ and $\deg(v) > 4$ we use the useless edge reduction.
Between two vertices are at most two edges. 
If there is more then there is a 3-cut around these two vertices.
If there is two edges between two vertices we use multi-edge reduction.
If $G$ is $K_5$ it has four independent spanning trees as you can see~\ref{K5}.
If $G$ has no useless edge and it is not $K_5$ it has vertex with degree four and we can use vertex reduction.

\begin{figure}[h]
\centering
\includegraphics{imgs/K5.eps}
\caption{Four independent spanning trees for $K_5$.}
\label{K5}
\end{figure}



\end{proof}


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%\section{Acknowledgment}

 
\end{document}
