diff --git a/main.tex b/main.tex
index ba80f4110c85fec348a1c5ba1a9af422b145504e..8367c5b2b179f8c4b5ec5e869c4eaaf5af372e36 100644
--- a/main.tex
+++ b/main.tex
@@ -705,6 +705,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 	Are precisely given by integers $r,c,d$ satisfying the following conditions:
 	\begin{enumerate}
 		\item $r > 0$
+			\label{item:rankpos:lem:num_test_prob1}
 		\item $\Delta(u) \geq 0$
 		\item $\Delta(v-u) \geq 0$
 		\item $\mu(u)=\frac{c}{r}<\mu(v)$
@@ -1043,27 +1044,21 @@ amounts to:
 \end{align}
 
 \subsubsection{
+	Semistability of the Semistabilizer:
 	\texorpdfstring{
-		$\Delta(E) \geq 0$
+		$\Delta(u) \geq 0$
 	}{
-		Δ(E) ≥ 0
+		Δ(u) ≥ 0
 	}
 }
 
-This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
-
 \begin{sagesilent}
 # First Bogomolov-Gieseker form expression that must be non-negative:
 bgmlv2 = Δ(u)
 \end{sagesilent}
 
-\begin{equation}
-	\sage{0 <= bgmlv2.expand() }
-\end{equation}
-
-
 \noindent
-Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
+Expressing $\Delta(u)\geq 0$ in terms of $q$ as defined in eqn \ref{eqn-cintermsofm}
 we get the following:
 
 \begin{sagesilent}
@@ -1080,7 +1075,10 @@ bgmlv2_with_q = (
 
 
 \noindent
-This can be rearranged to express a bound on $d$ as follows:
+This can be rearranged to express a bound on $d$ as follows
+(recall from condition \ref{item:rankpos:lem:num_test_prob1}
+in lemma \ref{lem:num_test_prob1} or corrolary
+\ref{cor:num_test_prob2} that $r>0$):
 
 \begin{sagesilent}
 bgmlv2_d_ineq = (
@@ -1128,9 +1126,10 @@ a linear term
 $\sage{bgmlv2_d_upperbound_linear_term}$,
 and a hyperbolic term
 $\sage{bgmlv2_d_upperbound_exp_term}$.
-Notice that for $\beta = \beta_{-}$ (or $\beta_{+}$), that is when
-$\chern^{\beta}_2(F)=0$, the constant and linear terms match up with the ones
-for the bound found for $d$ in subsection \ref{subsect-d-bound-bgmlv1}.
+Notice that in the context of problem \ref{problem:problem-statement-2}
+($\beta = \beta_{-}$),
+the constant and linear terms match up with the ones
+for the bound found for $d$ in subsubsection \ref{subsect-d-bound-radiuscond}.
 
 \subsubsection{
 	\texorpdfstring{