lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -95,14 +95,14 @@ combinatorial counterpart.
 % TODO: Remark that the probability of σ_β ∙ λ ∈ P+ is slight: there precisely
 % one element in the orbit of λ which is dominant integral, so the odds are
 % 1/|W ∙ λ|
-\begin{proposition}\label{thm:lemma6.1}
+\begin{lemma}\label{thm:lemma6.1}
   Let \(\beta \in \Sigma\) and \(\lambda \notin P^+\) be such that.
   \(L(\lambda)\) is bounded and \(\lambda(H_\beta) \notin \mathbb{N}\). Then
   \(L(\sigma_\beta \bullet \lambda) \subset \mExt(L(\lambda))\). In particular,
   if \(\sigma_\beta \bullet \lambda \notin P^+\) then \(L(\sigma_\beta)\) is a
   bounded infinite-dimensional \(\mathfrak{g}\)-module and
   \(\mExt(L(\sigma_\beta \bullet \lambda)) \cong \mExt(L(\lambda))\).
-\end{proposition}
+\end{lemma}
 
 % TODOO: Treat the case of sl(2) here
 
@@ -220,7 +220,7 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   = \sigma_n \bullet \lambda\). Since \(m(\lambda) \in \mathscr{B}\),
   \(\lambda(H_{\beta_n}) \in \sfrac{1}{2} + \mathbb{Z}\) and thus
   \(\lambda(H_{\beta_n}) \notin \mathbb{N}\). Hence by
-  Proposition~\ref{thm:lemma6.1} \(L(\mu) \subset \mExt(L(\lambda))\) and
+  Lemma~\ref{thm:lemma6.1} \(L(\mu) \subset \mExt(L(\lambda))\) and
   \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\).
 
   For each semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-family
@@ -482,7 +482,7 @@ all \(i\) and \(j\).
   % Great migué
   In this situation, \(m(\mu) \in \mathscr{B}^+\) implies \(\mu(H_{\beta_1}) =
   m(\mu)_1 - m(\mu)_2 \in \mathbb{Z}\) is negative. But it follows from
-  Proposition~\ref{thm:lemma6.1} that for each \(\beta \in \Sigma\) there is at
+  Lemma~\ref{thm:lemma6.1} that for each \(\beta \in \Sigma\) there is at
   most one \(\mu \notin P^+\) with \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\)
   such that \(\mu(H_\beta)\) is a negative integer -- see Lemma~6.5 of
   \cite{mathieu}. Hence there is at most one \(m' \in \mathscr{B}^+ \cap W