Adjusted citation formats. Moved citation off of named theorems if possible.

src/dg/derivative/higher.tex

@@ -20,11 +20,11 @@
     is the \textbf{$n$-fold $\sigma$-derivative of $f$ at $x_0$}.
 \end{definition}
 
-\begin{theorem}[{{\cite[Theorem 5.1.1]{Cartan}}}]
+\begin{theorem}[Symmetry of Higher Derivatives]
 \label{theorem:derivative-symmetric-frechet}
     Let $E, F$ be Banach spaces, $U \subset E$ be open, $n \in \natp$, and $f: U \to F$ be a function $n$-times Fréchet-differentiable at $x \in U$, then $D^nf(x) \in L^n(E; F)$ is symmetric.
 \end{theorem}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Theorem 5.1.1]{Cartan}}}. ]
     First suppose that $n = 2$. Let $r > 0$ such that $B(x, 2r) \subset U$, and define $A: B_E(0, r) \times B_E(0, r) \to F$ by
     \[
     A(h, k) = f(x + h + k) - f(x + h) - f(x + k) + f(x)
@@ -85,11 +85,11 @@
     Since any element $\sigma \in S_n$ that does not fix $x_1$ is the composition of the transposition $(12)$ and an element that fixes $x_1$, $Df(x)$ is symmetric.
 \end{proof}
 
-\begin{theorem}[{{\cite[Proposition 4.5.14]{Bogachev}}}]
+\begin{theorem}[Symmetry of Higher Derivatives]
 \label{theorem:derivative-symmetric}
     Let $E$ be a topological vector space over $K \in \RC$, $\sigma \subset B(E)$ be an upward-directed system that includes all bounded sets contained in finite-dimensional spaces, $F$ be a separated locally convex space over $K$, $U \subset E$ be open, and $f: E \to F$ be $n$-fold $\sigma$-differentiable at $x_0 \in U$, then $D_\sigma^nf(x_0) \in B_\sigma^n(E; F)$ is symmetric.
 \end{theorem}
-\begin{proof}
+\begin{proof}[Proof {{\cite[Proposition 4.5.14]{Bogachev}}}. ]
     Let $\seqf{h_j} \subset E$, $E_0$ be the subspace generated by $\seqf{h_j}$, and $g = f|_{E_0 \cap U}: E_0 \cap U \to F$. Since $\sigma$ includes all bounded sets contained in finite-dimensional spaces, for any $\phi \in F^*$, the mapping $\phi \circ g: E_0 \cap U \to K$ is $n$-times Fréchet-differentiable, with
     \[
     D_{B(E_0)}^n(\phi \circ g)(x_0) = \phi \circ D_\sigma^n g(x_0)