diff --git a/main.tex b/main.tex
index 8367c5b2b179f8c4b5ec5e869c4eaaf5af372e36..85102ab6ab14a318c7a896bf61f893f44697f193 100644
--- a/main.tex
+++ b/main.tex
@@ -1132,47 +1132,26 @@ the constant and linear terms match up with the ones
 for the bound found for $d$ in subsubsection \ref{subsect-d-bound-radiuscond}.
 
 \subsubsection{
+	Semistability of the Quotient:
 	\texorpdfstring{
-		$\Delta(G) \geq 0$
+		$\Delta(v-u) \geq 0$
 	}{
-		Δ(G) ≥ 0
+		Δ(v-u) ≥ 0
 	}
 }
 \label{subsect-d-bound-bgmlv3}
 
-This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
+Expressing $\Delta(v-u)\geq 0$ in term of $q$ and rearranging as a bound on
+$d$ yields:
 
 \begin{sagesilent}
 # Third Bogomolov-Gieseker form expression that must be non-negative:
 bgmlv3 = Δ(v-u)
-\end{sagesilent}
-
-\begin{equation}
-	\sage{0 <= bgmlv3.expand() }
-\end{equation}
-
-
-\noindent
-Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
-we get the following:
-
-\begin{sagesilent}
 bgmlv3_with_q = (
 	bgmlv3
 	.expand()
 	.subs(c == c_in_terms_of_q)
 )
-\end{sagesilent}
-
-\begin{equation}
-	\sage{0 <= bgmlv3_with_q}
-\end{equation}
-
-
-\noindent
-This can be rearranged to express a bound on $d$ as follows:
-
-\begin{sagesilent}
 var("r_alt",domain="real") # r_alt = r - R temporary substitution
 
 bgmlv3_with_q_reparam = (
@@ -1191,9 +1170,7 @@ bgmlv3_d_ineq = (
 assert bgmlv3_d_ineq.lhs() == d
 
 bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
-\end{sagesilent}
 
-\begin{sagesilent}
 # Seperate out the terms of the lower bound for d
 
 bgmlv3_d_upperbound_without_hyp = (
@@ -1263,8 +1240,8 @@ assert bgmlv3_d_upperbound_exp_term == (
 \end{sagesilent}
 
 \bgroup
-\def\psi{\chern_1^{\beta}(F)}
-\def\phi{\chern_2^{\beta}(F)}
+\def\psi{\chern_1^{\beta}(v)}
+\def\phi{\chern_2^{\beta}(v)}
 \begin{dmath}
 	\label{eqn-bgmlv3_d_upperbound}
 	d \leq
@@ -1276,22 +1253,16 @@ assert bgmlv3_d_upperbound_exp_term == (
 
 
 \noindent
-Viewing equation \ref{eqn-bgmlv3_d_upperbound} as an upper bound for $d$ give:
-as a function of $r$, the terms can be rewritten as follows.
-The constant term in $r$ is
-$\chern^{\beta}_2(F) + \beta q$.
-The linear term in $r$ is
-$\sage{bgmlv3_d_upperbound_linear_term}$.
-Finally, there is an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
-and can be written:
-\bgroup
-\def\psi{\chern_1^{\beta}(F)}
-$\sage{bgmlv3_d_upperbound_exp_term_alt2}$.
-\egroup
-In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
-$\chern^{\beta}_2(F) = 0$,
-so some of these expressions simplify, and in particular, the constant and
-linear terms match those of the other bounds in the previous subsections.
+Viewing the right hand side of equation \ref{eqn-bgmlv3_d_upperbound}
+as a function of $r$, the linear and constant terms almost match up with the
+ones in the previous section, up to
+$\chern_2^{\beta}(v)$.
+
+
+However, when specializing to problem \ref{problem:problem-statement-2} again
+(with $\beta = \beta_{-}$), then we have $\chern^{\beta}_2(v) = 0$.
+And so in this context, the linear and constant terms do match up with the
+previous subsubsections.
 
 \subsubsection{All Bounds on $d$ together}
 %% RECAP ON INEQUALITIES TOGETHER