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\documentclass[a4paper,12pt,twoside]{book}
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\begin{document}
\author{Michael Dykes and Croix Snapp}
\title{Real and Complex Analysis}
\date{\today}
\chapter{Logic, Set Theory, Functions, and Relations.}
\section{Basic Set Theory.}
\subsection{Introduction.}
\begin{them}
Let $X$ be any set. Then:
\begin{enumerate}
\item $\emptyset \subseteq X.$
\item $X \subseteq X.$
\end{enumerate}
\end{them}
\begin{proof}\mbox{}
\begin{enumerate}
\item Let $X$ be any set, and let $x$ be any object. Then, let $x \in \emptyset.$ Hence $(\forall x) (X \in \emptyset \Rightarrow x \in X). \,\therefore \emptyset \subseteq X$ [since the hypothesis is false].
\item Let $X$ be any set, and let $x$ be any object. Then $x \in X \Rightarrow x \in X.$ Hence $(\forall x)(x \in X \Rightarrow x \in X). \, \therefore X \subseteq X.$
\end{enumerate}
\end{proof}
\end{document}