\documentclass[12pt]{amsart}
\usepackage{amssymb, amsmath}

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\title{A Short Paper}
\author{Al G. Ebra}
\date{\today}

\newtheorem{thm}{Theorem}[section]
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}

\newcommand{\Z}{{\mathbb Z}}
\newcommand{\E}{{\mathcal E}}

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%    Proof:}\hspace{0.5em}}{\hfill\rule{1ex}{1ex}\vspace{1ex}}

\numberwithin{equation}{section}        % This includes the section number
                                % in the equation numbering, and starts
                                % each section with equation 1.

\numberwithin{table}{section}   % do the same for tables

\begin{document}
\maketitle

\section{Introduction}

We will exemplify here how to number, label and refer to sections, theorems
etc. in a \LaTeX{} paper. Along the way, we will prove a remarkable result
--- see Section~\ref{sec.main-results}.

% Use the tilde in ``Section~\ref{sec.main-results}'' above to make sure
% that ``Section'' and the number \ref{sec.main-results} are not broken on
% two lines (in case this would occur at the end of a line).

\section{Preliminaries}

We want to prove that the even numbers form a ring with the operations of
addition and multiplication. We begin with two propositions.

\begin{prop}
  \label{th.sum}      % It helps to specify what the label refers
                      % to; here it is a Proposition, which falls
                      % under ``Theorems'', thus the prefix th.
                      % 
                      % For labels of equations one can use the prefix eq., etc.
                      % 
                      % Pick a suggestive name for the label.
  The sum and difference of two even numbers is even.
\end{prop}

\begin{proof}
  Let
  \begin{equation}
    \label{eq.n}
    n=2k
  \end{equation}
  and
  \begin{equation}
    \label{eq.m}
    n=2\ell
  \end{equation}
  be two even numbers. Hence $k, \ell \in \Z$.

  Then
  \begin{eqnarray}
    % Since we do not want equation numbers, there is no reason to define
    % labels.
    %
    % If needed, can use & to allign the formulas.
    % 
    % Instead, we can use the eqnarray* environment, which does not produce
    % numbers.
    n+m  = 2k+2\ell = 2(k+\ell)\nonumber\\
    n-m  = 2k-2\ell = 2(k-\ell)\nonumber
  \end{eqnarray}
  therefore $n\pm m$ is a multiple of two as well.
\end{proof}

\begin{prop}
  \label{th.prod}
  The product of two even numbers is even.
\end{prop}

\begin{proof}
  Let $n$ and $m$ be as in \eqref{eq.n}, \eqref{eq.m}, with $k,\ell \in
  \Z$. Then
  \[
  n\cdot m= (2k)(2\ell)=2(2k\ell),
  \]
  hence $n\cdot m$ is even (actually, even a multiple of 4).
\end{proof}

\section{The Main Theorem}\label{sec.main-results}

\begin{thm}[Our Main Result]
  \label{th.main}
  The set $\E:=\{2k \mid k\in \Z\}$ forms a ring.
\end{thm}

\begin{proof}
  Since $\Z$ is a ring and $\E\subset \Z$, we only have to check that
  \begin{eqnarray}
    x,y\in\E \Rightarrow x-y\in \E, \label{eq.ring_sum}\\
    x,y\in\E \Rightarrow x y\in \E. \label{eq.ring_prod}
  \end{eqnarray}
  
  But \eqref{eq.ring_sum} was proven in Proposition~\ref{th.sum}, and
  \eqref{eq.ring_prod} in Proposition~\ref{th.prod}.
\end{proof}

\section{Remarks}

Following a similar approach, one can show that the following rules apply
for addition (see Table~\ref{tab:add-table}) and multiplication:

\begin{table}[h]
  \centering
  \begin{tabular}{c||c|c|}
    +& even & odd\\
    \hline
    \hline
    even& even & odd\\
    \hline
    odd& odd & even\\
    \hline
  \end{tabular}
  \caption{mod 2 addition table}
  \label{tab:add-table}
\end{table}


\begin{center}
  \begin{tabular}{c|cc}
    $\cdot$& even & odd\\
    \hline
    even& even & even\\
    odd& even & odd\\
  \end{tabular}
\end{center}

\section{Bibliography, BibTeX}

You can also add references to your paper, e.g. Strassen~\cite{Strassen67}
and Takahashi~\cite{Takahashi72} (of course, do not mix numerical and
alphabetical labels).

One very convenient way to do this is BibTeX. One creates a database of
bibliographical references, and then formats it easily in the desired
style. See more under the BibTeX entry.


\begin{thebibliography}{9999}
  % the parameter 9999 indicates the widest label (four digits here);
  % normally two digits suffice
 
\bibitem{Strassen67}
  V.~Strassen. {Almost sure behavior of sums of independent random
    variables and martingales}. \emph{Proc. 5th Berkeley Symp. Math.
    Statist. Probab.}, \textbf{2} (1967), 315--343.

\bibitem[Ta72]{Takahashi72}
  S.~Takahashi. {Notes on the law of the iterated logarithm}. \emph{Studia
    Sci.  Math. Hung.} \textbf{7} (1972), 21--24.
  
\end{thebibliography}
\end{document}