diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index b967fa2a0a9e5da659b6c3f52ee0c58f992560bd..811e3c96fae61859ad2c2d188be67ff18cb486ad 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -297,9 +297,10 @@ condition $\chern_2^P(u) = \sage{problem1.radius_condition_before_sub}$ yields:
 Expanding $\chern^{\beta_0}_2(u)$ in terms of $r$, $c$, $d$, and rearranging for
 $d$ then yields:
 
-\begin{equation*}
+\begin{equation}
+	\label{eqn:radius_condition_d_bound}
 	\sage{problem1.radius_condition_d_bound}
-\end{equation*}
+\end{equation}
 
 
 \subsubsection{Semistability of the Semistabilizer:
@@ -360,14 +361,78 @@ from plots_and_expressions import bgmlv3_d_upperbound_terms
 \end{equation}
 
 \noindent
-If $r=R$, then $\Delta(v-u)=(C-c)^2 \geq 0$ is always true, and for $r<R$ it gives a lower
-bound on $d$, but it is weaker than the one given by the lower bound
+If $r=R$, then $\Delta(v-u)=(C-c)^2 \geq 0$ is always true, and for $r<R$
+the expression on the right hand side of Equation \ref{eqn-bgmlv3_d_upperbound}
+gives a lower bound for $d$ instead.
+However it is weaker than lower bound
 given by $\chern^P_2(u)>0$ if $u$ already satisfies Equations
-\ref{lem:eqn:cond-for-fixed-q}.
-We see this by comparing the unique terms from either bound:
+\ref{lem:eqn:cond-for-fixed-q} as will be shown now:
 
-
-{\color{red} THIS IS BECAUSE TODO}
+Since $r, R-r>0$, we have:
+\begin{equation}
+	\label{lem:proof:slope-order-rltR}
+	\beta_0, \mu(u) < \mu(v) < \mu(v-u)
+\end{equation}
+\noindent
+The first inequality coming from $P \in \Theta_v^{-}$ and Equation
+\ref{lem:eqn:cond-for-fixed-q}, and the second inequality following by the
+see-saw principle.
+% TODO maybe cover the see-saw principle
+\begin{align*}
+	\left(
+		\frac{\chern^{\beta_0}_1(v-u)}{\chern_0(v-u)}
+	\right)^2
+	&=
+	\left(
+		\mu(v-u) - \beta_0
+	\right)^2
+\\
+	&>
+	\left(
+		\mu(v) - \beta_0
+	\right)^2
+	&\text{by Equation \ref{lem:proof:slope-order-rltR}}
+\\
+	&=
+	\left(
+		\frac{\chern^{\beta_0}_1(v)}{\chern_0(v)}
+	\right)^2
+\\
+	&\geq
+	2 \frac{\chern^{\beta_0}_2(v)}{\chern_0(v)}
+	&\text{since }\Delta(v) \geq 0
+	\:\text{and }\chern_0(v) > 0
+\\
+	\frac{
+		\left(
+		q-\chern^{\beta_0}_1(v)
+		\right)^2
+	}{
+		\left(
+		R-r
+		\right)^2
+	}
+	&>
+	2 \frac{\chern^{\beta_0}_2(v)}{R}
+\\
+	\chern_2^{\beta_0}(v)	
+	- \frac{
+		\left(
+		q-\chern^{\beta_0}_1(v)
+		\right)^2
+	}{
+		2\left(
+		R-r
+		\right)
+	}
+	&<
+	\frac{r\chern^{\beta_0}_2(v)}{R}
+\end{align*}
+\noindent
+Showing that the unique terms of Equation
+\ref{eqn:radius_condition_d_bound}
+are greater than those of Equation
+\ref{eqn-bgmlv3_d_upperbound}.
 
 
 \subsubsection{All Bounds on \texorpdfstring{$d$}{d} Together for Problem