diff --git a/main.tex b/main.tex
index b316fa80e615a19d4765c69cbf456815cf9f8f0e..ac2e57e6e333187ab336f077d83c4bf0a78b2cd3 100644
--- a/main.tex
+++ b/main.tex
@@ -229,6 +229,7 @@ This can be rearranged to express a bound on $d$ as follows:
 \end{sagesilent}
 
 \begin{dmath}
+	\label{eqn-bgmlv1_d_lowerbound}
 	\sage{bgmlv1_d_ineq}
 \end{dmath}
 
@@ -242,33 +243,40 @@ This can be rearranged to express a bound on $d$ as follows:
 	assert bgmlv1_d_lowerbound_const_term == v.twist(beta_min).ch[2]/2 + beta_min*q, "fail"
 	assert bgmlv1_d_lowerbound_linear_term == (v.twist(beta_min).ch[2]/2 + beta_min^2/2 + beta_min*q)*r, "fail"
 
-	foo = (
+	assert bgmlv1_d_lowerbound_exp_term == (
 		- 2*R*v.twist(beta_min).ch[2]
 		- 3*R^2*beta_min^2
 		- 4*R*beta_min*q
 		+ C*q
 		- q^2
-	).expand()/(R-2*r)
-	foo = (
+	).expand()/(R-2*r), "fail"
+
+	assert bgmlv1_d_lowerbound_exp_term == (
 		- 2*D*R + (C^2)/4
 		- ((C - 4*R*beta_min)/2 - q)^2
-	).expand()/(R-2*r)
-	foo = (
+	).expand()/(R-2*r), "fail"
+
+	assert bgmlv1_d_lowerbound_exp_term == (
 		(2*R*beta_min + q)
 		*(2*R*beta_min + q - C)
 		+ 2*D*R
-	).expand()/(2*r - R)
+	).expand()/(2*r - R), "fail"
 \end{sagesilent}
 
-\begin{equation}
-	\sage{bgmlv1_d_lowerbound_exp_term}
-\end{equation}
-\begin{equation}
-	\sage{foo}
-\end{equation}
-
 \noindent
-In the case $\beta = \beta_{-}$ (or $\beta_{+}$) this can be simplified.
+Viewing equation \ref{eqn-bgmlv1_d_lowerbound} as a lower bound for $d$ given
+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
+$(\chern^{\beta}_2(F)/2 + \beta^2/2 + \beta q)r$.
+Finally, there's an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
+and can be written:
+$\frac{-2R\chern^{\beta}_2(F) - 3R^2\beta^2 - 4Rq\beta + Cq - q^2}{R-2r}$ or
+$\frac{(2R\beta + q)(2R\beta + q - C) + 2DR}{2r-R}$.
+In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
+$\chern^{\beta}_2(F) = 0$,
+so some of these expressions simplify.
 
 \subsection{$\Delta(E) \geq 0$}