Add proof for 3.5.7

LinAlg2.tex

@@ -3649,15 +3649,36 @@ $ \implies q(\tilde x_1, \tilde x_2) = \lambda_1 \tilde x_1^2 + \lambda_2 \tilde
 \end{defin}
 
 \begin{lemma}
-	Hermitesche Formen und hermitesche Sesquilinearformen entsprechen einander eineindeutig
-	\begin{proof}
-		\leavevmode
-		\begin{itemize}
-			\item $\rho$ hermitesche Form, $\sigma$ wie oben in c) $\implies \sigma$ hermitesche Sesquilinearform.
-			\item $\sigma$ hermitesche Sesquilinearform, $\rho(v) := \frac 12 \sigma(v, v) \overset{\text{\tl UE\br}}
-				      \implies \rho$ ist hermitesche Form.
-		\end{itemize}
-	\end{proof}
+Hermitesche Formen und hermitesche Sesquilinearformen entsprechen einander eineindeutig
+\begin{proof}
+Für hermitesche Form ist durch Definition \ref{theo:3.5.6} c) eine hermitesche Sesquilinearform definiert. \\
+Sei umgekehrt $\sigma$ hermitesche Sesquilinearform. Dann ist $\rho(v) := \frac12 \sigma(v, v)$ hermitesche
+Form:
+\begin{enumerate}[label=\alph*)]
+\item \checkmark
+\item \begin{align*}
+\rho(u+v) + \rho(u - v) &= \sigma(u+v, u+v) + \sigma(u-v, u-v) \\
+&\begin{multlined}= \sigma(u, u) + \sigma(v, v) + \sigma(u, v) + \sigma(v, u)
+	+ \sigma(u, u)\\ + \sigma(v, v) - \sigma(u, v) - \sigma(v, u)
+\end{multlined} \\
+&= 2\sigma(u, u) + 2\sigma(v, v) \\
+&= 2(\rho(u) + \rho(v))
+\end {align*}
+\item \begin{align*}
+	\frac12 (\rho(u+v) + i\rho(u+iv)
+	 & - (1+i)(\rho(u)+\rho(v))) =                                                               \\
+	 & \begin{multlined}
+		   = \sigma(u+v, u+v) + i \sigma(u+iv,u+iv) \\- \sigma(u, u) - \sigma(v, v) - i\sigma(u, u) -
+		   i \sigma(v, v) \end{multlined} \\
+	 & = \sigma(u, v) + \sigma(v, u) + i \sigma(iv, u) + i \sigma(u, iv)                         \\
+	 & = \sigma(u, v) + \overline{\sigma(u, v)} + i \overline{\sigma(u, iv)} + \sigma(u, v)      \\
+	 & = \sigma(u, v) + \overline{\sigma(u, v)} + i \cdot \overline{\overline{i}} \cdot
+	\overline{\sigma(u, v)}
+	+ \sigma(u, v)                                                                               \\
+	 & = 2 \sigma(u, v)
+\end{align*}
+\end{enumerate}
+\end{proof}
 \end{lemma}
 
 \subsubsection{Bemerkung}