lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -487,9 +487,10 @@ all \(i\) and \(j\).
   implies \(B(\lambda)\) is precisely the connected component of
   \(m(\lambda)\).
 
-  Another way of stating this is to say that \(\mExt(L(\lambda)) \cong
+  Another way of putting it is to say that \(\mExt(L(\lambda)) \cong
   \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and \(m(\mu)\) lie in the same
-  connected component. There is thus a one-to-one correspondance between
+  connected component -- which is, of course, precisely the first part of our
+  theorem! There is thus a one-to-one correspondance between
   \(\pi_0(\mathscr{B})\) and the isomorphism classes of semisimple irreducible
   coherent \(\mathfrak{sl}_n(K)\)-families. Since every connected component of
   \(\mathscr{B}\) meets \(\mathscr{B}^+\) precisely once -- again, see