From 92b52ef1b856188bbee00b211cca874a08a05181 Mon Sep 17 00:00:00 2001
From: Joshua Potter <jrpotter2112@gmail.com>
Date: Thu, 29 Jun 2023 14:05:08 -0600
Subject: [PATCH] Enderton. Infinite cartesian products.

---
 Bookshelf/Enderton/Set.tex | 21 +++++++++++++++++++++
 preamble.tex               |  2 +-
 2 files changed, 22 insertions(+), 1 deletion(-)

diff --git a/Bookshelf/Enderton/Set.tex b/Bookshelf/Enderton/Set.tex
index d3965c6..3fc6896 100644
--- a/Bookshelf/Enderton/Set.tex
+++ b/Bookshelf/Enderton/Set.tex
@@ -31,6 +31,12 @@
 For any relation $R$ there is a function $H \subseteq R$ with
   $\dom{H} = \dom{R}$.
 
+\section{\partial{Axiom of Choice, Second Form}}%
+\label{ref:axiom-of-choice-2}
+
+For any set $I$ and any function $H$ with domain $I$, if $H(i) \neq \emptyset$
+  for all $i \in I$, then $$\bigtimes_{i \in I} H(i) \neq \emptyset.$$
+
 \section{\defined{Composition}}%
 \label{ref:composition}
 
@@ -3929,4 +3935,19 @@ Define
 
 \end{proof}
 
+\section{Exercise 9}%
+\label{sec:exercise-9}
+
+\subsection{\unverified{Exercise 9.31}}%
+\label{sub:exercise-9.31}
+
+Show that from the first form of the axiom of choice we can prove the second
+  form, and conversely.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
 \end{document}
diff --git a/preamble.tex b/preamble.tex
index 27ca766..4a6203e 100644
--- a/preamble.tex
+++ b/preamble.tex
@@ -6,7 +6,7 @@
 \usepackage{environ}
 \usepackage{fancybox}
 \usepackage{fontawesome5}
-\usepackage{mathrsfs}
+\usepackage{mathabx, mathrsfs}
 \usepackage{soul}
 \usepackage{stmaryrd}
 \usepackage[usenames,dvipsnames]{xcolor}