diff --git a/eulerg/eulerg.tex b/eulerg/eulerg.tex
index c6d65c5..dcae125 100644
--- a/eulerg/eulerg.tex
+++ b/eulerg/eulerg.tex
@@ -43,29 +43,7 @@ and write down set theory equations.
 The interest here though, is to define relationships between the contours, that allow
 processing and parsing of the diagram without resorting to extra area operations in the concerete plane.
 
-\section{Defining `pair-wise intersection' and `enclosure'}
-
-%\begin{figure}[htp]
-%  \begin{center}
-%    \subfigure[Original image]{\label{fig:edge-a}\includegraphics[scale=0.75]{toucan.eps}}
-%    \subfigure[After Laplace edge detection]{\label{fig:edge-b}\includegraphics[scale=0.75]{laplace_toucan.eps}} \\
-%    \subfigure[After Sobel edge detection]{\label{fig:edge-c}\includegraphics[scale=0.75]{sobel_toucan.eps}}
-%  \end{center}
-%  \caption{Various edge detection algorithms}
-%  \label{fig:edge}
-%\end{figure}
-
-%
-%\begin{figure}[h]
-% \centering
-% \includegraphics[width=200pt,keepaspectratio=true]{./eulerg/eulerg1.jpg}
-% % eulerg1.jpg: 513x215 pixel, 72dpi, 18.10x7.58 cm, bb=0 0 513 215
-% \caption{An Euler Diagram showing enclosure and Pair-wise Intersection}
-% \label{fig:eulerg1}
-%\end{figure}
-%
-%
-%
+\section{Defining `pair-wise intersection' \\ and `enclosure'}
 
 \begin{figure}[ht]
 \centering
@@ -165,6 +143,8 @@ $$ B {\enc} A \wedge A {\enc} C \implies B {\enc} C $$
 Enlcosure relationships are transitive.
 \end{definition}
 
+
+\pagebreak[1]
 \section{Representing Euler Diagrams \\ as sets of relationships}
 
 The diagram in figure \ref{fig:eulerg1} can be represented by the following relationships.
@@ -185,7 +165,7 @@ and {\em enclosure} is transitive and {\pic} is not, we can represent
 an {\em enclosure} relationship as a directed vertice and 
 {\pic} as non-directed.
 
-\pagebreak[0]
+\pagebreak[1]
 \section{The {\pic}}
 In graph theory a node is said to be reachable from another node
 if you can start at the one node, travel via the edges
@@ -271,7 +251,10 @@ Figure \ref{fig:eulerg_pic_g} only shows the {\pic}, but does not show the conto
 enclosing $PIC1$. Figure \ref{fig:eulerg_pic_g_a}
 shows contour A enclosing all elements in $PIC1$
 
+\pagebreak[1]
 \subsection{Enclosure and pair-wise \\ intersection in the graph}
+Because enclose is a directed relationship and {\em pair-wise intersection} is non-directed
+we can represent them both on the same graph, see figure \ref{fig:eulerg_pic_g_a}.
 \begin{figure}[h]
  \centering
  \includegraphics[width=200pt,bb=0 0 330 162,keepaspectratio=true]{./eulerg/eulerg_pic_g_a.jpg}
@@ -282,9 +265,11 @@ shows contour A enclosing all elements in $PIC1$
 
 \pagebreak[1]
 \subsection{Reducing clutter in the graph}
-Because we know that a contour enclosing a contour within a {\pic} but not belonging
-to it, encloses all elements of the {\pic}, see definition \ref{def:encpic}, we can draw this in a less cluttered way
+Contour A encloses the pure intersection chain $PIC1$;
+using definition \ref{def:encpic}, we can draw this in a less cluttered way
 see figure \ref{fig:eulerg_pic_g_a_unc}.
+We only need to show contour A enclosing one member of the {\pic} $PIC1$
+in order to show that contour A encloses all contours in $PIC1$.
 
 
 
@@ -297,7 +282,7 @@ see figure \ref{fig:eulerg_pic_g_a_unc}.
 \end{figure}
 
 
-\pagebreak[0]
+\pagebreak[1]
 \section{Reduction of searches \\ for available zones}
 
 Another property of any {\pic} $P$, is that
diff --git a/thesis.tex b/thesis.tex
index 85b1cb7..6e9bd3e 100644
--- a/thesis.tex
+++ b/thesis.tex
@@ -139,7 +139,7 @@ reference the MSC document and describe the Java extension classes.
 Software documentation for fmmd tool.
 
 \typeout{ ----------------  Euler Diagrams represented as graphs}
-\chapter {Euler Diagrams Represented as graphs}
+\chapter {Euler Diagrams \\ Represented as graphs}
 \input{eulerg/eulerg}
 
 \chapter{Fast Zone Discrimination Algorithm}