diff --git a/src/fa/convex/legendre.tex b/src/fa/convex/legendre.tex
index 1ae2fab..f10889d 100644
--- a/src/fa/convex/legendre.tex
+++ b/src/fa/convex/legendre.tex
@@ -221,14 +221,9 @@
     E \times \real \setminus \ol{\text{Conv}}(\text{epi}(f)) \subset E \times \real \setminus \text{epi}(f)
     \]
     
-    To this end, let $A = \ol{\text{Conv}}(\text{epi}(f))$ and $(x, \alpha) \in E \times \real \setminus A$. By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach-geometric-2}, there exists $\phi \in F$ and $\mu \in \real$ such that
+    To this end, let $A = \ol{\text{Conv}}(\text{epi}(f))$ and $(x, \alpha) \in E \times \real \setminus A$. By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach-geometric-2}, there exists $\phi \in F$, $\mu \in \real$, and $\alpha_0 \in (\alpha, \infty)$ such that
     \[
-    \sup_{(y, \beta) \in A}\dpn{y, \phi}{\lambda} - \mu \beta < \dpn{x, \phi}{\lambda} - \mu \alpha
-    \]
-
-    Given that the inequality is strict, there exists $\alpha_0 \in (\alpha, \infty)$ such that
-    \[
-    \sup_{(y, \beta) \in A}\dpn{y, \phi}{\lambda} - \mu \beta \le \dpn{x, \phi}{\lambda} - \mu\alpha_0
+    \sup_{(y, \beta) \in A}\dpn{y, \phi}{\lambda} - \mu \beta \le \dpn{x, \phi}{\lambda} - \mu \alpha_0 \le \dpn{x, \phi}{\lambda} - \mu \alpha
     \]
 
     For any $(y, \beta) \in A$, $\beta$ may be arbitrarily large by \autoref{lemma:closed-convex-epigraph}, so $\mu \ge 0$.