diff --git a/thesis/chapters/stabilizer.tex b/thesis/chapters/stabilizer.tex
index 8e94b5d..7e09cce 100644
--- a/thesis/chapters/stabilizer.tex
+++ b/thesis/chapters/stabilizer.tex
@@ -363,45 +363,87 @@ graph.
     \end{equation}
 \end{lemma}
 \begin{proof}
-    FIXME: This
+    The proof is done using mathematical induction over the edges $\{i,j\} \in
+    E$. A similar proof can be found in \cite{hein_eisert_briegel2008}. 
 
+    \textbf{Base Case:} If $E = \{\}$ the stabilizers are $K_G^{(i)} = X_i$
+    and stabilize the state $\ket{+}$.
+
+    \textbf{Inductive Step:} Let $G' := (V, E \setminus \{\{l,j\}\})$. 
+    By the induction hypothesis the state $\ket{\bar{G}'}$ is stabilized
+    by $K_{G'}^{(i)}$. Applying a $CZ_{l,j}$ to the state $\ket{\bar{G}'}$
+    now transforms the stabilizers to 
 
-    Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before.
-    Note that for any $X_i$: $X_i \ket{+} = +1 \ket{+}$.  In the following
-    discussion the direction $\prod\limits_{\{l,k\} \in E} :=
-    \prod\limits_{\{l,k\} \in E, l < k}$ is introduced as the graph is
-    undirected and edges must not be handled twice.  Set $\ket{\tilde{G}} :=
-    \left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right)\ket{+}$.
 
     \begin{equation}
-        \begin{aligned}
-            K_G^{(i)} \ket{\tilde{G}} 
-            & = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)
-                \left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right) \ket{+} \\
-            & = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,k\} \in E}
-                \left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
-            & = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,k\} \in E}
-                \left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_k \otimes Z_l\right) X_i \ket{+} \\
-        \end{aligned}
+    \begin{aligned}
+        S^{(i)} &= CZ_{l,j} K_{G'}^{(i)} CZ_{l,j}.\\
+    \end{aligned}
     \end{equation}
-    As $X,Z$ anticommute. $X_i$ can now be absorbed into $\ket{+}$. The next
-    step is a bit tricky: A $Z_j$ can be absorbed into a $\ket{0}\bra{0}_j$
-    giving no phase or into a $\ket{1}\bra{1}_j$ yielding a phase of $-1$. If
-    there is no projector on $j$ the $Z_j$ can be commuted to the next
-    projector.  It is guaranteed that a projector on $j$ exists by the
-    definition of $\ket{\tilde{G}}$.
+
+    Note that $CZ_{l,j}$ commutes with $K_{G'}^{(i)}$ for $l \neq i \neq j$.
+    Further $CZ_{l,j} = CZ_{j,l}$. Consider now the case $l = i$:
 
     \begin{equation}
-        \begin{aligned}
-            K_G^{(i)} \ket{\tilde{G}} 
-            & = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l} + \delta_{j,k}}\ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
-			& = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\	
-		    & = +1 \ket{\tilde{G}}	
-        \end{aligned}
+    \begin{aligned}
+        S^{(i)} &= CZ_{i,j} K_{G'}^{(i)} CZ_{i,j} \\
+                &= \left( \ket{1}\bra{1}_j \otimes Z_i + \ket{0}\bra{0}_j \otimes I_i \right)
+                    X_i \prod\limits_{l \in n'_i} Z_l
+                    \left( \ket{1}\bra{1}_j \otimes Z_i + \ket{0}\bra{0}_j \otimes I_i \right)\\
+                &=           \left(\ket{1}\bra{1}_j \otimes Z_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{1}\bra{1}_j \otimes Z_i\right)\\
+                &\mbox{   }+ \left(\ket{1}\bra{1}_j \otimes Z_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{0}\bra{0}_j \otimes I_i\right)\\
+                &\mbox{   }+ \left(\ket{0}\bra{0}_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{1}\bra{1}_j \otimes Z_i\right)\\
+                &\mbox{   }+ \left(\ket{0}\bra{0}_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{0}\bra{0}_j \otimes I_i\right)\\
+                &=           \left(\ket{1}\bra{1}_j \otimes Z_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{1}\bra{1}_j \otimes Z_i\right)\\
+                &\mbox{   }+ \left(\ket{0}\bra{0}_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \left(\ket{0}\bra{0}_j \otimes I_i\right)\\
+                &= \left(\left(-\ket{1}\bra{1}_j + \ket{1}\bra{1}_j\right) \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \\
+                &= \left(Z_j \otimes I_i\right) X_i \prod\limits_{l \in n'_i} Z_l \\
+                &= X_i \prod\limits_{l \in n_i} Z_l \\
+                &= K_G^{(i)}
+    \end{aligned}
     \end{equation}
 
-    The $\delta_{i,l} + \delta_{j,k}$ is either $0$ or $2$ by the definitions
-    of $K_G^{(i)}$ and $\ket{\tilde{G}}$.
+
+
+    %FIXME: This
+
+
+    %Let $\ket{+} := \left(\prod\limits_{l \in V} H_l\right) \ket{0}$ as before.
+    %Note that for any $X_i$: $X_i \ket{+} = +1 \ket{+}$.  In the following
+    %discussion the direction $\prod\limits_{\{l,k\} \in E} :=
+    %\prod\limits_{\{l,k\} \in E, l < k}$ is introduced as the graph is
+    %undirected and edges must not be handled twice.  Set $\ket{\tilde{G}} :=
+    %\left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right)\ket{+}$.
+
+    %\begin{equation}
+    %    \begin{aligned}
+    %        K_G^{(i)} \ket{\tilde{G}} 
+    %        & = X_i \left(\prod\limits_{\{i,j\} \in E} Z_j\right)
+    %            \left(\prod\limits_{\{l,k\} \in E} CZ_{l,k} \right) \ket{+} \\
+    %        & = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)X_i\prod\limits_{\{l,k\} \in E}
+    %            \left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
+    %        & = \left(\prod\limits_{\{i,j\} \in E} Z_j\right)\prod\limits_{\{l,k\} \in E}
+    %            \left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l}}\ket{1}\bra{1}_k \otimes Z_l\right) X_i \ket{+} \\
+    %    \end{aligned}
+    %\end{equation}
+    %As $X,Z$ anticommute. $X_i$ can now be absorbed into $\ket{+}$. The next
+    %step is a bit tricky: A $Z_j$ can be absorbed into a $\ket{0}\bra{0}_j$
+    %giving no phase or into a $\ket{1}\bra{1}_j$ yielding a phase of $-1$. If
+    %there is no projector on $j$ the $Z_j$ can be commuted to the next
+    %projector.  It is guaranteed that a projector on $j$ exists by the
+    %definition of $\ket{\tilde{G}}$.
+
+    %\begin{equation}
+    %    \begin{aligned}
+    %        K_G^{(i)} \ket{\tilde{G}} 
+    %        & = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + (-1)^{\delta_{i,l} + \delta_{j,k}}\ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\
+	%		& = \prod\limits_{\{l,k\} \in E}\left( \ket{0}\bra{0}_k \otimes I_l + \ket{1}\bra{1}_k \otimes Z_l\right) \ket{+} \\	
+	%	    & = +1 \ket{\tilde{G}}	
+    %    \end{aligned}
+    %\end{equation}
+
+    %The $\delta_{i,l} + \delta_{j,k}$ is either $0$ or $2$ by the definitions
+    %of $K_G^{(i)}$ and $\ket{\tilde{G}}$.
 \end{proof}
 
 \subsubsection{Dynamics of the VOP-free Graph States}