Polished A-A and added new lines for broken enumerates.

src/dg/derivative/higher.tex

@@ -9,7 +9,8 @@
     \begin{enumerate}
         \item There exists $V \in \cn_E(x_0)$ such that $f$ is $(n-1)$-fold differentiable on $V$.
         \item The derivative $D_\sigma^{n-1}f: U \to B^{n-1}_\sigma(E; F)$ is derivative at $x_0$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     In which case, $D_\sigma(D_\sigma^{n-1}f)(x_0) \in L(E; B^{n-1}_\sigma(E; F))$ is the \textbf{$n$-fold $\sigma$-derivative of $f$ at $x_0$}.
 
     The mapping $f: U \to F$ is \textbf{$n$-fold $\sigma$-differentiable on $U$} if it is $n$-fold $\sigma$-differentiable at every point in $U$. Under the identification $B_\sigma(E; B^{n-1}_\sigma(E; F)) = B_\sigma^{n}(E; F)$ given by \autoref{proposition:multilinear-identify},

src/dg/derivative/mvt.tex

@@ -17,7 +17,8 @@
     \begin{enumerate}
         \item[(a)] $f, g$ are right-differentiable on $[a, b] \setminus N$.
         \item[(b)] For every $x \in [a, b] \setminus N$, $D^+f(x) \le D^+g(x)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for any $x \in [a, b]$, $f(x) - f(a) \le g(x) - g(a)$.
 \end{lemma}
 \begin{proof}
@@ -51,7 +52,8 @@
         \item $f, g$ are right-differentiable on $[a, b] \setminus N$.
         \item For each $x \in [a, b] \setminus N$, $D^+f(x) \in D^+g(x)B$.
         \item $g$ is non-decreasing.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then
     \[
     f(b) - f(a) \in [g(b) - g(a)]B

src/dg/derivative/sets.tex

@@ -8,7 +8,8 @@
         \item For each $A \in \sigma$, $r(th)/t^n \to 0$ uniformly on $A$.
         \item If $r_t(x) = r(tx)/t^n$, then $r_t \to 0$ as $t \to 0$ with respect to the $\sigma$-uniform topology on $F^E$.
         \item For each $A \in \sigma$, $\seq{a_k} \subset A$, and $\seq{t_k} \subset K \setminus \bracs{0}$ with $t_k \to 0$ as $n \to \infty$, $r(t_ka_k)/t_k^n \to 0$ as $n \to \infty$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     If the above holds, then $r$ is \textbf{$\sigma$-small of order $n$}.
 
     The set $\mathcal{R}_\sigma^n(E; F)$ is the $K$-vector space of all $\sigma$-small functions of order $n$ from $E$ to $F$. For simplicity, $\mathcal{R}_\sigma(E; F)$ denotes $\mathcal{R}_\sigma^1(E; F)$.
@@ -52,7 +53,8 @@
     \begin{enumerate}
         \item[(a)] For any $r \in \mathcal{R}_\sigma(E; F)$ and $T \in L(F; G)$, $T \circ r \in \mathcal{R}_\sigma(E; G)$.
         \item[(b)] For any $r \in \mathcal{R}_\sigma(E; F)$, $T \in L(E; F)$, and $s \in \mathcal{R}_\tau(F; G)$, $s \circ (T + r) \in \mathcal{R}_\sigma(E; G)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then for any $U \subset E$ and $V \subset F$ open, $f: U \to V$ $\sigma$-differentiable at $x_0 \in U$, $g: V \to F$ $\tau$-differentiable at $f(x_0) \in V$, $g \circ f: U \to F$ is $\sigma$-differentiable at $x_0$ with
     \[
     D_\sigma(g \circ f)(x_0) = D_\tau g(f(x_0)) \circ D_\sigma f(x_0)
@@ -84,12 +86,14 @@
     \begin{enumerate}
         \item Compact sets.
         \item Bounded sets.
-    \end{enumerate}
+    \end\{enumerate\}
+
     then
     \begin{enumerate}
         \item For any $r \in \mathcal{R}_\sigma(E; F)$ and $T \in L(F; G)$, $T \circ r \in \mathcal{R}_\sigma(E; G)$.
         \item For any $r \in \mathcal{R}_\sigma(E; F)$, $T \in L(E; F)$, and $s \in \mathcal{R}_\tau(F; G)$, $s \circ (T + r) \in \mathcal{R}_\sigma(E; G)$.
-    \end{enumerate}
+    \end\{enumerate\}
+
     and by \autoref{proposition:chain-rule-sets}, $\sigma$-derivatives and $\tau$-derivatives satisfy the Chain rule.
 \end{proposition}
 \begin{proof}