diff --git a/inputs/lecture_12.tex b/inputs/lecture_12.tex
index 6f88217..2eaf19e 100644
--- a/inputs/lecture_12.tex
+++ b/inputs/lecture_12.tex
@@ -20,7 +20,7 @@
         T \in \IF &\iff& \exists \beta \in \cN .~\forall n \in \N.~T(\beta\defon{n}) = 1.
     \end{IEEEeqnarray*}
 
-    Consider $\{(T, \beta) \in \Tr \times  \cN : \forall n.~ T(\beta\defon{n}) = 1\}$.
+    Consider $D \coloneqq \{(T, \beta) \in \Tr \times  \cN : \forall n.~ T(\beta\defon{n}) = 1\}$.
     Note that this set is closed in $\Tr \times \cN$,
     since it is a countable intersection of clopen sets.
     % TODO Why clopen?
@@ -123,9 +123,9 @@ For the proof we need some prerequisites:
     $\IF$ is $\Sigma^1_1$-complete.
 \end{corollary}
 \begin{proof}
-    Let $A \subseteq X$ is analytic
-    and $X$ Polish and uncountable,
-    then
+    Let $X$ be Polish.
+    Suppose that $A \subseteq X$ is analytic and uncountable.
+    Then
     % https://q.uiver.app/#q=WzAsNSxbMCwwLCJYIl0sWzEsMCwiXFxjTiJdLFsyLDAsIlxcVHIiXSxbMCwxLCJBIl0sWzEsMSwiYihBKSJdLFsxLDIsImYiXSxbMCwxLCJiIl0sWzMsMCwiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbNCwxLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dXQ==
 \[\begin{tikzcd}
 	X & \cN & \Tr \\
@@ -135,6 +135,7 @@ For the proof we need some prerequisites:
 	\arrow[hook, from=2-1, to=1-1]
 	\arrow[hook, from=2-2, to=1-2]
 \end{tikzcd}\]
+    where $f$ is chosen as in \yaref{thm:lec12:1}.
 
     If $X$ is Polish and countable and $A \subseteq  X$ analytic,
     just consider