diff --git a/main.tex b/main.tex
index 82cc4e04920521810f3d8d73731453a4e9838366..6fe3987026ce362581d7a735af029f842460e09b 100644
--- a/main.tex
+++ b/main.tex
@@ -603,7 +603,7 @@ Note by Tom: So $A^{(\mathcal{P})}$ is the event that the set of all patches is
    	Otherwise if all elements of $S_{><}\setminus P$ are larger than $P_{\max}$ then we view the last summation as $\sum_{f'_{\overline{P}}\in\{0,1'\}^{|S\cap \overline{P}\setminus\{S_{\max}\}|}}\sum_{f''_{\overline{P}}\in\{0,1'\}^{1}}$ and use Lemma~\ref{lemma:probIndep} to conclude the cancellations pairwise regarding the filling of $S_{\max}$, i.e., the value of $f''_{\overline{P}}$. We proceed similarly when 
    	all elements of $S_{><}\setminus P$ are smaller than $P_{\min}$. In the last case we again proceed similarly, but now the cancellations will come from the interplay of $4$ fillings, corresponding to the possible filling of $S_{\min}$ and $S_{\max}$ simultaneously.
    	   
-	I think the same arguments would directly translate to the torus and other translationally invariant objects, so we could go higher dimensional as Mario suggested. Then one would need to replace $|S_{><}|$ by the minimal number $k$ such that there is a $C$ set for which $S\cup C$ is connected.
+	I think the same arguments would translate to the torus and other translationally invariant spaces, so we could go higher dimensional as Mario suggested. Then I think one would need to replace $|S_{><}|$ by the minimal number $k$ such that there is a $C$ set for which $S\cup C$ is connected. I am not entirely sure how to generalise Lemma~\ref{lemma:probIndep} though, which has key importance in the present proof.
     
     Questions:
     \begin{itemize}
