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+%First document, firstdocument.tex
+\documentclass{amsart}
+\usepackage{amssymb,latexsym}
+
+\newtheorem{theorem}{Theorem}
+
+\begin{document}
+\title{A technical result\\ for congruences of finite lattices}
+\author{G. Gr\"atzer}
+\address{Department of Mathematics\\
+ University of Manitoba\\
+ Winnipeg, MB R3T 2N2\\
+ Canada}
+\email[G. Gr\"atzer]{gratzer@me.com}
+\urladdr[G. Gr\"atzer]{http://tinyurl.com/gratzerhomepage}
+\date{March 21, 2014}
+\subjclass[2010]{Primary: 06B10.}
+\keywords{finite lattice, congruence.}
+\begin{abstract}
+We present a technical result for congruences on finite lattices.
+\end{abstract}
+\maketitle
+
+\section{Introduction}\label{S:Introduction}%Section~\label{S:Introduction}
+In some recent research, G. Cz\'edli
+and I, see \cite{gC13} and \cite{gG14}, spent quite an effort
+in proving that some equivalence relations
+on a planar semimodular lattice are congruences.
+The number of cases we had to consider
+was dramatically cut by the following result.
+
+\begin{theorem}\label{T:technical}%Theorem~\ref{T:technical}
+Let $L$ be a finite lattice.
+Let $\delta$ be an equivalence relation on $L$
+with intervals as equivalence classes.
+Then $\delta$ is a congruence relation if{}f
+the following condition and its dual hold:
+\begin{equation}\label{E:cover}%\eqref{E:cover}
+\text{If $x$ is covered by $y,z \in L$
+and $x \equiv y \pmod{\delta}$,
+then $z \equiv y + z \pmod{\delta}$.}\tag{C${}_{+}$}
+\end{equation}
+\end{theorem}
+
+\section{The proof}\label{Proof}%Section~\label{S:Proof}
+We prove the join-substitution property:
+if $x \leq y$ and $x \equiv y \pmod{\delta}$, then
+\begin{equation}\label{E:Cjoin}%\eqref{E:Cjoin}
+x + z \equiv y + z \pmod{\delta}.
+\end{equation}
+Let $U = [x, y+ z]$.
+We induct on length\,$U$, the length of $U$.
+
+Let $I=[y_1,y+ z]$ and $J=[z_1,y+ z]$.
+Then length\,$I$ and length\,$J < $ length\,$U$.
+Hence, the induction hypothesis applies to $I$
+and $\delta\rceil I$, and we obtain that
+$w \equiv y+ w \pmod{\delta}$.
+By the transitivity of $\delta$, we conclude that
+\begin{equation}\label{E:three}%\eqref{E:three}
+z_1 \equiv y+ w \pmod{\delta}.
+\end{equation}
+Therefore, applying the induction hypothesis to $J$
+and $\delta \rceil J$, we conclude \eqref{E:Cjoin}.
+
+\begin{thebibliography}{9}
+\bibitem{gC13}%G. Cz\'edli~\cite{gC13}
+G. Cz\'edli,
+\emph{Patch extensions and trajectory colorings of slim
+rectangular lattices.}
+Algebra Universalis \textbf{88} (2013), 255--280.
+
+\bibitem{gG14}%G. Gr\"atzer \cite{GS13}
+G. Gr\"atzer,
+\emph{Congruences of fork extensions of lattices.}
+Acta Sci. Math. (Szeged), \textbf{57} (2014), 417--434.
+\end{thebibliography}
+\end{document} \ No newline at end of file