diff --git a/eulerg/eulerg.tex b/eulerg/eulerg.tex
index 04b9ca6..94dc128 100644
--- a/eulerg/eulerg.tex
+++ b/eulerg/eulerg.tex
@@ -104,9 +104,11 @@ $$ D \cap E \neq \emptyset $$
 
 Clearly though these intersections are different, because
 in the $A$, $B$ case 
-$$ A \backslash B = \emptyset \wedge B \backslash A \neq \emptyset $$
+$ A \backslash B = \emptyset \wedge B \backslash A \neq \emptyset $
 This is not the case for $D$, $E$ where:
-$$ D \backslash E \neq \emptyset \wedge E \backslash D \neq \emptyset $$
+$ D \backslash E \neq \emptyset \wedge E \backslash D \neq \emptyset $.
+Another way of expressing this is that $A \cap B \neq \emptyset$ and 
+$ D \subset E$.
 
 \paragraph{Enclosure}
 To distinguish between these we can term the $A$, $B$ case to be 
@@ -194,15 +196,17 @@ Figure \ref{fig:eulerg_pic} shows a {\pic} consisting of contours $M,N,O,P$ and
  \label{fig:eulerg_pic}
 \end{figure}
 
-\textbf{rule:}
+%\textbf{rule:}
 \begin{definition}
 If any contour in a {\pic} is enclosed by any contour not belonging to the chain, 
 all the countours within the 
-{\pic} will be enclosed by it. This is because a contour 
-enclosing which bisects(????) another contour in a {\pic}
-becomes part of the pure~intersection~chain. Hmmmm thats true but a better way to say it ????
+{\pic} will be enclosed by it. 
 \end{definition}
 
+This is because a contour 
+enclosing which bisects another contour in a {\pic}
+becomes part of the pure~intersection~chain.
+% Hmmmm thats true but a better way to say it ????
 %The diagram in figure \ref{fig:eulerg_enc} can be represented by the following relationships.
 
 
@@ -284,9 +288,8 @@ see figure \ref{fig:eulerg_pic_g_a_unc}.
 \end{figure}
 
 
-\pagebreak[4]
-\section{reduction of searches for available zones}
-
+\pagebreak[0]
+\section{Reduction of searches for available zones}
 
 Another property of any {\pic} $P$, is that
 the maximum number of euler zones within it is 
@@ -300,12 +303,28 @@ a contour, we change the nature of the zones within
 the {\pic}, but the number of zones contributed by the {\pic}
 stays the same.
 \begin{definition}
-A pure intersection chain has a maximum number of possible Euler zones, and exists as independent entities in the diagram. This
-allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones.
+\label{picreduction}
+The number of available zones within a {\pic} $P$ does not change 
+when other contours are added or removed from the diagram
+that are not, or would not become members of the {\pic} $P$.
 \end{definition}
-This is to say, the the number of zones within a {\pic} is not affected by changes in the diagram 
+his is to say, the the number of zones within a {\pic} is not affected by changes in the diagram 
 that do not alter the {\pic}.
+This allows us to analyses {\pic}s separately, thus reducing the $2^N$ overhead of analysing an Euler diagram for available zones.
 
+\subsection{Available Zone Searching}
+
+The available zones in an Euler diagram represent set theoretic combinations
+that can be used in the diagram. 
+%For FMMD analyis, the test~cases
+Searching for an available zone involves finding out if the intersection exists, and then determining whether it is covered up
+by any other contours. A brute force search for available zones using area operations
+is therefore of the order $N.2^N$ (where N is the number of contours in the diagram).
+By using the result in definition \ref{picreduction}, we can break the diagram into small segments
+(the {\pic}s) which have an order $|P|.2^{|P|}$.
+The order of area operations is generally\footnote{In the case where the diagram is not comprised of just one {\pic}, which has no enclosing contours} 
+reduced  by requiring several  $|P|.2^{|P|}$
+instead of $N.2^N$ as $P \leq N$.
 
 \vspace{40pt}