diff --git a/main.tex b/main.tex
index f1c8d2efa4679bb0b88058a285c700b1fd17d2ed..4c6899f6a2088c6ebef87e67caa6b9d46d3811ef 100644
--- a/main.tex
+++ b/main.tex
@@ -24,6 +24,8 @@ sorting=ynt
 \newcommand{\NN}{\mathbb{N}}
 \newcommand{\PP}{\mathbb{P}}
 \newcommand{\chern}{\operatorname{ch}}
+\newcommand{\coh}{\operatorname{coh}}
+\newcommand{\homol}{\mathcal{H}}
 \newcommand{\lcm}{\operatorname{lcm}}
 \newcommand{\firsttilt}[1]{\mathcal{B}^{#1}}
 \newcommand{\bddderived}{\mathcal{D}^{b}}
@@ -32,11 +34,13 @@ sorting=ynt
 
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{corrolary}{Corrolary}[section]
-\newtheorem{lemmadfn}{Lemma/Definition}[section]
-\newtheorem{dfn}{Definition}[section]
 \newtheorem{lemma}{Lemma}[section]
 \newtheorem{fact}{Fact}[section]
 \newtheorem{example}{Example}[section]
+\newtheorem{lemmadfn}{Lemma/Definition}[section]
+
+\theoremstyle{definition}
+\newtheorem{definition}{Definition}[section]
 \newtheorem{problem}{Problem Statement}
 
 \begin{document}
@@ -161,12 +165,12 @@ Throughout this article, as noted in the introduction, we will be exclusively
 working over one of the following two surfaces: principally polarized abelian
 surfaces and $\PP^2$.
 
-\begin{dfn}[Pseudo-semistabilizers]
+\begin{definition}[Pseudo-semistabilizers]
 \label{dfn:pseudo-semistabilizer}
 % NOTE: SURFACE SPECIALIZATION
 	Given a Chern Character $v$, and a given stability
 	condition $\sigma_{\alpha,\beta}$,
-	a pseudo-semistabilizing $u$ is a `potential' Chern character:
+	a \textit{pseudo-semistabilizing} $u$ is a `potential' Chern character:
 	\[
 		u = \left(r, c\ell, d \frac{1}{2} \ell^2\right)
 	\]
@@ -182,13 +186,23 @@ surfaces and $\PP^2$.
 
 	Note $u$ does not need to be a Chern character of an actual sub-object of some
 	object in the stability condition's heart with Chern character $v$.
-\end{dfn}
+\end{definition}
 
 At this point, and in this document, we do not care about whether
 pseudo-semistabilizers are even Chern characters of actual elements of
 $\bddderived(X)$, some other sources may have this extra restriction too.
 
-\begin{lemma}[ Sanity check for Pseudo-semistabilizers ]
+\begin{definition}[Pseudo-walls]
+\label{dfn:pseudo-wall}
+	Let $u$ be a pseudo-semistabilizer of $v$, for some stability condition.
+	Then the \textit{pseudo-wall} associated to $u$ is the set of all stablity
+	conditions where $u$ is a pseudo-semistabilizer of $v$.
+\end{definition}
+
+% TODO possibly reference forwards to Bertram's nested wall theorem section to 
+% cover that being a pseudo-semistabilizer somewhere implies also on whole circle
+
+\begin{lemma}[Sanity check for Pseudo-semistabilizers]
 % NOTE: SURFACE SPECIALIZATION
 	Given a stability
 	condition $\sigma_{\alpha,\beta}$,
@@ -198,7 +212,31 @@ $\bddderived(X)$, some other sources may have this extra restriction too.
 \end{lemma}
 
 \begin{proof}
-	q.e.d. (TODO)
+	Suppose $E\hookrightarrow F\twoheadrightarrow G$ is a semistabilizing
+	sequence with respect to a stability condition $\sigma_{\alpha,\beta}$.
+	\begin{equation*}
+		\chern(E) = -\chern(\homol^{-1}_{\coh}(E)) + \chern(\homol^{0}_{\coh}(E))
+	\end{equation*}
+	Therefore, $\chern(E)$ is of the form $(r,c\ell,d\frac{1}{2}\ell^2)$
+	provided that this is true for any coherent sheaf.
+	For any coherent sheaf $H$, we have the following:
+	\begin{equation*}
+		\chern(H) = \left(c_0(H), c_1(H), - c_2(H) + \frac{1}{2} {c_1(H)}^2\right)
+	\end{equation*}
+	Given that $\ell$ generates the Neron-Severi group, $c_1(H)$ can then be
+	written as a multiple of $\ell$.
+	Furthermore, for $\PP^2$ and principally polarized abelian surfaces,
+	$\ell^2=1$ or $2$.
+	This fact along with $c_2$ being an integer on surfaces implies that
+	$\chern(H)$ (and hence $\chern(E)$ too) is of the required form
+	$(r,c\ell,d\frac{1}{2}\ell^2)$ for some $r,c,d \in \ZZ$.
+	
+
+	Since all the objects in the sequence are in $\firsttilt\beta$, we have
+	$\chern_1^{\beta} \geq 0$ for each of them. Due to additivity
+	($\chern(F) = \chern(E) + \chern(G)$), we can deduce
+	$0 \leq \chern(E) \leq \chern(F)$.
+	% TODO unfinished
 \end{proof}
 
 
@@ -214,7 +252,7 @@ These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$, and
 are illustrated in Fig \ref{fig:charact_curves_vis}
 (dotted line for $i=1$, solid for $i=2$).
 
-\begin{dfn}[Characteristic Curves $V_v$ and $\Theta_v$]
+\begin{definition}[Characteristic Curves $V_v$ and $\Theta_v$]
 Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
 define two characteristic curves on the $(\alpha, \beta)$-plane:
 
@@ -222,7 +260,7 @@ define two characteristic curves on the $(\alpha, \beta)$-plane:
 	V_v &\colon \chern_1^{\alpha, \beta}(v) = 0 \\
 	\Theta_v &\colon \chern_2^{\alpha, \beta}(v) = 0
 \end{align*}
-\end{dfn}
+\end{definition}
 
 \begin{fact}[Geometry of Characteristic Curves]
 The following facts can be deduced from the formulae for $\chern_i^{\alpha, \beta}(v)$
@@ -615,7 +653,7 @@ however this will also be proved again in passing in this article.
 \subsection{Loose Bounds on $\chern_0(E)$ for Semistabilizers Along Fixed
 $\beta\in\QQ$}
 
-\begin{dfn}[Twisted Chern Character]
+\begin{definition}[Twisted Chern Character]
 \label{sec:twisted-chern}
 For a given $\beta$, define the twisted Chern character as follows.
 \[\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)\]
@@ -631,7 +669,7 @@ Component-wise, this is:
 where $\chern_i$ is the coefficient of $\ell^i$ in $\chern$.
 
 % TODO I think this^ needs adjusting for general Surface with $\ell$
-\end{dfn}
+\end{definition}
 
 $\chern^\beta_1(E)$ is the imaginary component of the central charge
 $\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$