diff --git a/ws2019/ana/lectures/analysis.pdf b/ws2019/ana/lectures/analysis.pdf
index 460f0b0..7822bc8 100644
Binary files a/ws2019/ana/lectures/analysis.pdf and b/ws2019/ana/lectures/analysis.pdf differ
diff --git a/ws2019/ana/lectures/analysis18.tex b/ws2019/ana/lectures/analysis18.tex
index 992b955..f38d4e5 100644
--- a/ws2019/ana/lectures/analysis18.tex
+++ b/ws2019/ana/lectures/analysis18.tex
@@ -170,25 +170,20 @@
     \end{enumerate} 
 \end{satz}
 
-\begin{proof}
-    \begin{enumerate}
-        \item \begin{align*}
-                \cos (x+y) + i \sin (x+y) &= e^{i(x+y)} = e^{ix} \cdot e^{iy} \\
-                &= (\cos x + i \sin x)(\cos y + i \sin y) \\
-                &= \underbrace{\cos x \cos y - \sin x \sin y}_{\text{Re}} + i \underbrace{(\sin x \cos y + \cos x \sin y)}_{\text{Im}}
-            .\end{align*}
-        \item Setze $u := \frac{x+y}{2}, v := \frac{x - y}{2}$.
-
-            $x = u + v, y = u-v$.\\
-            \begin{align*}
-                \sin x - \sin y &= \sin (u+v) - \sin (u - v) \\
-                                &= \sin u \cdot \cos v + \cos u \cdot \sin v
-                                - (\sin u \underbrace{\cos(-v)}_{= \cos v}
-                                + \cos u \cdot \underbrace{\sin(-v)}_{- \sin v}) \\
-                                &= 2 \cos u \sin v
-                                = 2 \cos \frac{x+y}{2} \cdot \sin \frac{x - y}{2}
-            .\end{align*}
-    \end{enumerate}
+\begin{proof} 1. Mit $e^{ix} = \cos x + i \sin x$ folgt direkt
+    \begin{align*}
+            \cos (x+y) + i \sin (x+y) &= e^{i(x+y)} = e^{ix} \cdot e^{iy} \\
+            &= (\cos x + i \sin x)(\cos y + i \sin y) \\
+            &= \underbrace{\cos x \cos y - \sin x \sin y}_{\text{Re}} + i \underbrace{(\sin x \cos y + \cos x \sin y)}_{\text{Im}}
+        \intertext{2. Setze $u := \frac{x+y}{2}, v := \frac{x - y}{2}$.
+        $x = u + v, y = u-v$.}
+        \sin x - \sin y &= \sin (u+v) - \sin (u - v) \\
+                        &= \sin u \cdot \cos v + \cos u \cdot \sin v
+                        - (\sin u \underbrace{\cos(-v)}_{= \cos v}
+                        + \cos u \cdot \underbrace{\sin(-v)}_{- \sin v}) \\
+                        &= 2 \cos u \sin v
+                        = 2 \cos \frac{x+y}{2} \cdot \sin \frac{x - y}{2}
+    .\end{align*}
 \end{proof}
 
 \end{document}