diff --git a/main.tex b/main.tex
index ea2c4504fedbbf3e245c8ef5e4caeb8b7a165bfc..3003b47d6cfc0bf506d3ed2ca8600bb671e8741e 100644
--- a/main.tex
+++ b/main.tex
@@ -377,7 +377,7 @@ $\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on
 $\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0)$
 (as per subsection \ref{subsect:bertrams-nested-walls}).
 
-\subsection{Bertram's nested wall theorem}
+\subsection{Bertram's Nested Wall Theorem}
 \label{subsect:bertrams-nested-walls}
 
 Although Bertram's nested wall theorem can be proved more directly, it's also
@@ -390,13 +390,16 @@ This fact, along with the hindsight knowledge that non-vertical walls are
 circles with centers on the $\beta$-axis, gives an alternative view to see that
 the circular walls must be nested and non-intersecting.
 
-\subsection{Characteristic curves for pseudo-semistabilizers}
+\subsection{Characteristic Curves for Pseudo-semistabilizers}
 
 These characteristic curves introduced are convenient tools to think about the
 numerical conditions that can be used to test for pseudo-semistabilizers, and
-for solutions to the problem (\ref{problem:problem-statement-1}).
+for solutions to the problems
+(\ref{problem:problem-statement-1},\ref{problem:problem-statement-2})
+tackled in this article (to be introduced later).
 In particular, problem (\ref{problem:problem-statement-1}) will be translated to
 a list of numerical inequalities on it's solutions $u$.
+% ref to appropriate lemma when it's written
 
 The next lemma is a key to making this translation and revolves around the
 geometry and configuration of the characteristic curves involved in a