%% This is file `amsart.sty' generated from `amsart.doc'
%% on 31-JUL-1991 by the docstrip.ams utility (1.0). 
%% 
%%@texfile{%
%% filename="amsart.sty"
%% version="1.1b",
%% date="31-JUL-1991",
%% filetype="AMS-LaTeX: documentstyle",
%% copyright="Copyright (C) American Mathematical Society",
%% author="American Mathematical Society",
%% address="American Mathematical Society,
%%     Technical Support Department,
%%     P. O. Box 6248,
%%     Providence, RI 02940,
%%     USA",
%% telephone="401-455-4080 or (in the USA) 800-321-4AMS",
%% email="Internet: Tech-Support@Math.AMS.org",
%% codetable="ISO/ASCII",
%% checksumtype="line count",
%% checksum="650",
%% keywords="latex, amslatex, ams-latex",
%% abstract="This file is part of the AMS-\LaTeX{} package, version 1.1.
%%   It is a LaTeX documentstyle that produces output in the style of
%%   the principal American Mathematical Society journals: Journal of
%%   the A.M.S, Transactions of the A.M.S, Proceedings of the A.M.S.,
%%   etc."
%%}
\expandafter\ifx\csname amsart.sty\endcsname\relax
  \expandafter\gdef\csname amsart.sty\endcsname{}
\else \message{already loaded}\endinput\fi
\def\filename{amsart.sty}
\def\fileversion{1.1b}       \def\filedate{31-JUL-1991}
%%% end of file header
\ifx\undefined\if@article % If \if@article is undefined,
\expandafter\let          % we will end up with \let\if@article\iftrue
\else \expandafter\@gobbletwo\fi % otherwise \@gobbletwo\if@article\iftrue
\if@article\iftrue       % which removes this line.
\if@article
\immediate\write16{%
AMS-LaTeX document style `AMSART' (\fileversion, \filedate)}
\fi
\input amstex.sty
\def\@ptsize{0}
\@namedef{ds@11pt}{\def\@ptsize{1}}
\@namedef{ds@12pt}{\def\@ptsize{2}}
\@namedef{ds@twoside}{\@empty}
\def\ds@draft{\overfullrule 5\p@}
\@namedef{ds@twocolumn}{\typeout{Sorry---the `twocolumn' option
      is not available in the AMSART and AMSBOOK documentstyles.}}
\@namedef{ds@openbib}{\typeout{Sorry---the `openbib' option
      is not available in the AMSART and AMSBOOK documentstyles.}}
\@namedef{ds@fleqn}{\typeout{Sorry---the `fleqn' option
      is not available in the AMSART and AMSBOOK documentstyles.}}
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\@namedef{ds@titlepage}{\typeout{Sorry---the `titlepage' option
      is not available in the AMSART documentstyle.}}
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\@options
\let\ds@twocolumn\relax       \let\ds@openbib\relax
\let\ds@fleqn\relax
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\let\ds@titlepage\relax
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\input{margid.sty}
\def\@plus{plus}
\def\@string{\expandafter\@gobble\string}
\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
\long\def\@ifempty#1{%
 \expandafter\ifx\@car#1@\@nil @\@empty
  \expandafter\@leftmark\else\expandafter\@rightmark\fi}
\long\def\@ifnotempty#1{\@ifempty{#1}{}}
\long\def\@leftmark#1#2{#1}
\long\def\@rightmark#1#2{#2}
\def\@andify#1#2{%
  \expandafter\in@\expandafter\and\expandafter{#2}%
  \ifin@
  \begingroup\global\toks\@ne{}% initialize to empty
  \def\and##1\and{\@ifnotempty{##1}{\advance\@tempcnta\@ne\and}}%
  \@tempcnta\z@\expandafter\and#2\and%
  \ifnum\@tempcnta>\tw@
    \def\and##1\and{\advance\@tempcnta\m@ne
      \global\toks\@ne\expandafter{\the\toks\@ne ##1}%
      \edef\@tempa{\the\toks\@ne\ifnum\@tempcnta=\@ne\unskip,#1\else
          \ifnum\@tempcnta>\@ne\unskip,\ \fi\fi}%
      \global\toks\@ne\expandafter{\@tempa \ignorespaces}%
      \ifnum\@tempcnta>\z@\expandafter\and\fi}%
  \else % if NOT > 2
    \def\and##1\and{\advance\@tempcnta\m@ne
       \global\toks\@ne\expandafter{\the\toks\@ne ##1}%
       \edef\@tempa{\the\toks\@ne
         \ifnum\@tempcnta=\@ne\unskip#1\ignorespaces\fi}%
       \global\toks\@ne\expandafter{\@tempa}%
       \ifnum\@tempcnta>\z@\expandafter\and\fi}%
  \fi % ifnum
 \expandafter\and#2\relax\endgroup
 \edef#2{\the\toks\@ne}%
\fi % ifin@
}
\def\uppercasetext@#1{%
   {\spaceskip1.3\fontdimen2\the\font plus1.3\fontdimen3\the\font
    \upchars@\skipmath@#1$\skipmath@$}}
\def\upchars@{\def\ss{SS}\let\i=I\let\j=J\let\ae\AE\let\oe\OE
  \let\o\O\let\aa\AA\let\l\L}
\def\skipmath@#1$#2${\skipmath@b#1\(\skipmath@b\)%
  \ifx\skipmath@#2\else$#2$\expandafter\skipmath@\fi}
\def\skipmath@b#1\(#2\){\uppercase{#1}%
  \ifx\skipmath@b#2\else\(#2\)\expandafter\skipmath@b\fi}
\def\today{\ifcase\month\or
 January\or February\or March\or April\or May\or June\or
 July\or August\or September\or October\or November\or December\fi
 \space\number\day, \number\year}
\def\LaTeX{L\raise.42ex\hbox{\kern-.33em\protect\scriptsize A\kern-.15em}\TeX}
\def\@ifundefined#1{\expandafter\ifx\csname#1\endcsname\relax
        \expandafter\@leftmark
        \else\expandafter\@rightmark\fi}
\def\ref#1{\@ifundefined{r@#1}{{\normalshape\bf ??}\iffirstchoice@
  \@warning{Reference `#1' on page \thepage \space undefined}\fi}%
  {\rom{\expandafter\expandafter\expandafter
    \@car\csname r@#1\endcsname\@nil\null}}}
\def\defaultfont{\family\default@family \series\default@series
  \shape\default@shape \selectfont}
\def\title{\@dblarg\@xtitle}
\def\@title{}
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\@xtitle[#1]#2{\def\sh@rttitle{\def\\{\unskip\space\ignorespaces}%
   \ignorespaces#1\unskip}%
   \def\@title{\ignorespaces#2\unskip}}
\def\sh@rttitle{}
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\author{\@dblarg{\@xauthor}}
\def\@authors{}
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\@xauthor[#1]#2{\expandafter\def\expandafter
  \@authors\expandafter{\@authors#2\and}%
  \@ifnotempty{#1}{\expandafter\def\expandafter\sh@rtauthor
       \expandafter{\sh@rtauthor#1\and}}%
}
\def\sh@rtauthor{}
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\newtoks\@addresstoks
\def\address{\@oparg\@xaddress[]}
\def\@xaddress[#1]#2{%
  \@addresstoks\expandafter{\the\@addresstoks\address{#1}{#2}{}{}{}{}}}
\def\@setaddress{%
  \par\expandafter\@ifnotempty\expandafter{\the\@addresstoks}{%
  \nobreak \begingroup \small
  \def\\{\unskip, \ignorespaces}%
  \interlinepenalty\@M
  \def\address##1##2##3##4##5##6{%
    \nobreak\bigskip\indent
    \@ifnotempty{##1}{(\ignorespaces##1\unskip) }%
    {\sc\ignorespaces##2}\par
    \@ifnotempty{##4}{\nobreak\indent{\it Current address}%
      \@ifnotempty{##3}{, \ignorespaces##3\unskip}\/:\space
      ##4\par}%
    \@ifnotempty{##6}{\nobreak\indent{\it E-mail address}%
      \@ifnotempty{##5}{, \ignorespaces##5\unskip}\/:\space
      ##6\par}%
  }% End definition of \address.
  \the\@addresstoks
  \endgroup
}% End of \@ifnotempty{\the\@addresstoks}
}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of \@setaddress
\def\curraddr{\@oparg\@xcurraddr[]}
\def\@xcurraddr[#1]#2{\@address@concat\curraddr{#1}{#2}{##5}{##6}}
\def\email{\@oparg\@xemail[]}
\def\@xemail[#1]#2{\@address@concat\email{##3}{##4}{#1}{#2}}
\def\@address@concat#1#2#3#4#5{%
  \expandafter\@ifempty\expandafter{\the\@addresstoks}%
  {\errmessage{\string#1 can't precede \string\address.}}%
  {\begingroup \global\toks\@ne{}\global\toks\thr@@{}%
    \def\address##1##2##3##4##5##6{\global\toks\thr@@\expandafter{%
      \the\toks\expandafter\thr@@\the\toks\@ne}%
      \global\toks\@ne{\address{##1}{##2}{##3}{##4}{##5}{##6}}}%
    \the\@addresstoks
    \def\address##1##2##3##4##5##6{\global\toks\thr@@\expandafter{%
      \the\toks\thr@@\address{##1}{##2}{#2}{#3}{#4}{#5}}}%
    \the\toks\@ne
    \endgroup \@addresstoks\toks\thr@@
}}
\def\date#1{\def\@date{#1}}
\def\@date{}
\def\thanks#1{\ifx\@empty\@thanks
   \def\@thanks{#1}%
 \else \expandafter\def\expandafter\@thanks\expandafter
   {\@thanks\@@par#1}\fi
}
\def\@thanks{}
\def\dedicatory#1{\def\@dedicatory{#1}}
\def\@dedicatory{}
\def\keywords#1{\def\@keywords{#1}}
\def\@keywords{}
\def\subjclass#1{\def\@subjclass{#1}}
\def\@subjclass{}
\def\translator#1{\def\@translator{#1}}
\def\@translator{}
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\maketitle{\par
  \@topnum\z@ % this prevents figures from falling at the top of page 1
  \ifx\@empty\sh@rtauthor \let\sh@rtauthor\sh@rttitle\fi
  \begingroup
  \@maketitle
  \endgroup
  \@andify{ AND }\sh@rtauthor
  \thispagestyle{plain}%
  \c@footnote\z@
  \def\do##1{\let##1\relax}%
  \do\maketitle \do\@maketitle
  \do\title \do\@xtitle \do\@title
  \do\author \do\@xauthor \do\@authors
  \do\address \do\@xaddress
  \do\email \do\@xemail \do\curraddr \do\@xcurraddr
  \do\dedicatory \do\@dedicatory
  \do\thanks \do\@thanks
  \do\keywords \do\@keywords
  \do\subjclass \do\@subjclass
  \do\@andify
}
\def\@maketitle{%
  \defaultfont\normalsize
  \let\@makefnmark\relax  \let\@thefnmark\relax
  \ifx\@empty\@subjclass\else
   \@footnotetext{1991 {\it Mathematics Subject
     Classification}.\enspace
        \@subjclass.}\fi
  \ifx\@empty\@keywords\else
   \@footnotetext{{\it Key words and phrases.}\enspace \@keywords.}\fi
\ifx\@empty\@thanks\else
   \@footnotetext{\@thanks}\fi
\topskip66\p@ % 6.5 picas to the base of the first title line
  \vtop{\centering{\baselineskip14\p@\bf
    \expandafter\uppercasetext@\expandafter{\@title}\@@par}%
   \global\dimen@i\prevdepth}%
  \prevdepth\dimen@i
  \ifx\@empty\@authors
  \else
    \baselineskip32\p@
    \vtop{\@andify{ AND }\@authors
      \centering{\small
  \expandafter\uppercasetext@\expandafter{\@authors}\@@par}%
         \global\dimen@i\prevdepth}\relax
    \prevdepth\dimen@i
  \fi
  \ifx\@empty\@dedicatory
  \else
    \baselineskip18\p@
  \vtop{\centering{\small\it\@dedicatory\@@par}%
      \global\dimen@i\prevdepth}\prevdepth\dimen@i
  \fi
  \ifx\@empty\@date\else
  \baselineskip24\p@
    \vtop{\centering\@date\@@par
      \global\dimen@i\prevdepth}\prevdepth\dimen@i
  \fi
  \normalsize
  \dimen@32\p@ \advance\dimen@-\baselineskip
  \vskip\dimen@\@plus14\p@
  } % end \@maketitle
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\abstractname{Abstract}
\def\abstract{%
  \global\let\abstract\relax
  \defaultfont\small  \skip@28\p@ \advance\skip@-\lastskip
  \advance\skip@-\baselineskip \vskip\skip@
  \moveright 3pc\vtop \bgroup
  \advance \hsize -6pc
  \trivlist \labelsep.5em\item[\hskip\labelsep
    {\sc\abstractname}.]\ignorespaces
}
\def\endabstract{\endtrivlist
  \global\let\endabstract\relax
  \global\dimen@i\prevdepth \egroup \prevdepth\dimen@i
  \skip@32\p@\@plus 14\p@ \advance\skip@-\baselineskip
  \vskip\skip@
}
\def\titlepage{\thispagestyle{empty}\c@page\z@}
\def\endtitlepage{\newpage}
\def\labelenumi{(\@arabic\c@enumi)}
\def\theenumi{\@arabic\c@enumi}
\def\labelenumii{(\@alph\c@enumii)}
\def\theenumii{\@alph\c@enumii}
\def\p@enumii{\theenumi}
\def\labelenumiii{(\@roman\c@enumiii)}
\def\theenumiii{\@roman\c@enumiii}
\def\p@enumiii{\theenumi(\theenumii)}
\def\labelenumiv{(\@Alph\c@enumiv)}
\def\theenumiv{\@Alph\c@enumiv}
\def\p@enumiv{\p@enumiii\theenumiii}
\def\labelitemi{$\m@th\bullet$}
\def\labelitemii{\bf --}% \normalshape already done by \itemize
\def\labelitemiii{$\m@th\ast$}
\def\labelitemiv{$\m@th\cdot$}
\def\verse{\let\\\@centercr
 \list{}{\itemsep\z@ \itemindent -1.5em\listparindent\itemindent
 \rightmargin\leftmargin \advance\leftmargin 1.5em}\item[]}
\let\endverse=\endlist
\def\quotation{\list{}{\listparindent 1.5em\itemindent\listparindent
 \rightmargin\leftmargin \parsep\z@ \@plus\p@}\item[]}
\let\endquotation=\endlist
\def\quote{\list{}{\rightmargin\leftmargin}\item[]}
\let\endquote=\endlist
\def\trivlist{\parsep\parskip
  \@trivlist \labelwidth\z@ \leftmargin\z@
  \itemindent\z@ \def\makelabel##1{\normalshape##1}}
\def\enumerate{\ifnum \@enumdepth >3 \@toodeep\else
      \advance\@enumdepth \@ne
      \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list
      {\csname label\@enumctr\endcsname}{\usecounter
        {\@enumctr}\def\makelabel##1{\hss\llap{\normalshape##1}}}\fi}
\def\itemize{\ifnum\@itemdepth>3 \@toodeep
  \else \advance\@itemdepth\@ne
  \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
  \list{\csname\@itemitem\endcsname}%
  {\def\makelabel##1{\hss\llap{\normalshape##1}}}\fi}
\def\descriptionlabel#1{\hspace\labelsep \normalshape\bf #1:}
\def\description{\list{}{%
\advance\leftmargin6\p@ \itemindent-6\p@
  \labelwidth\z@ \let\makelabel\descriptionlabel}}
\let\enddescription=\endlist
\input amsart1\@ptsize.sty\relax
\def\@startsection#1#2#3#4#5#6{%
\if@noskipsec \leavevmode \fi
 \par \@tempskipa #4\relax
 \@afterindenttrue
 \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse\fi
 \if@nobreak \everypar{}\else
     \addpenalty\@secpenalty\addvspace\@tempskipa\fi
 \@ifstar{\@dblarg{\@sect{#1}{\@m}{#3}{#4}{#5}{#6}}}%
         {\@dblarg{\@sect{#1}{#2}{#3}{#4}{#5}{#6}}}%
}
\def\@sect#1#2#3#4#5#6[#7]#8{%
\ifnum #2>\c@secnumdepth
   \let\@svsec\@empty
 \else
   \refstepcounter{#1}%
\edef\@svsec{\ifnum#2<\@m
             \@ifundefined{#1name}{}{\csname #1name\endcsname\ }\fi
\noexpand\rom{\csname the#1\endcsname.}\enspace}\fi
 \@tempskipa #5\relax
 \ifdim \@tempskipa>\z@ % then this is not a run-in section heading
   \begingroup #6\relax
   \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty\@M #8\par}%
   \endgroup
   \if@article\else\csname #1mark\endcsname{%
        \ifnum \c@secnumdepth >#2\relax\csname the#1\endcsname. \fi#7}\fi
\ifnum#2>\@m \else
       \addcontentsline{toc}{#1}%
{\ifnum #2>\c@secnumdepth \else
             \protect\numberline{%
               \ifnum#2<\@m
               \@ifundefined{#1name}{}{\csname #1name\endcsname\ }\fi
               \csname the#1\endcsname.}\fi
           #8}%
     \fi
 \else
  \def\@svsechd{#6\hskip #3\@svsec
    \@ifnotempty{#8}{\ignorespaces#8\unskip
       \ifnum\spacefactor<1001.\fi}%
        \ifnum#2>\@m \else
          \addcontentsline{toc}{#1}%
            {\ifnum #2>\c@secnumdepth \else
              \protect\numberline{%
                \ifnum#2<\@m
                \@ifundefined{#1name}{}{\csname #1name\endcsname\ }\fi
                \csname the#1\endcsname.}\fi
             #8}\fi}%
 \fi
\@xsect{#5}}
\def\frenchspacing{\count@ 1001 \sfcode`\.\count@ \sfcode`\?\count@
  \sfcode`\!\count@ \sfcode`\:\count@ \sfcode`\;\count@
  \sfcode`\,\count@}
\def\@xsect#1{\@tempskipa #1\relax
 \ifdim \@tempskipa>\z@ % then this is not a run-in heading
   \par \nobreak \vskip\@tempskipa
   \@afterheading
 \else % otherwise it IS a run-in heading
   \global\@nobreakfalse \global\@noskipsectrue
   \everypar{\if@noskipsec \global\@noskipsecfalse
               \clubpenalty\@M \kern-\parindent
               \begingroup \@svsechd \endgroup \hskip -#1\relax
             \else \clubpenalty \@clubpenalty\everypar{}\fi
            }%
 \fi\ignorespaces}
\def\@afterheading{\global\@nobreaktrue
   \everypar{\if@nobreak \global\@nobreakfalse \clubpenalty\@M
               \if@afterindent \else {\setbox\z@\lastbox}\fi
             \else \clubpenalty\@clubpenalty \everypar{}\fi}}
\def\@hangfrom#1{\setboxz@h{#1}\hangindent\wdz@ \noindent\box\z@}
\let\@ssect\relax % to save memory.  \@ssect not needed in amsart.sty.
\newcounter{part}
\newcounter{section}
\newcounter{subsection}[section]
\newcounter{subsubsection}[subsection]
\def\thepart{\@Roman\c@part}
\def\thesection{\@arabic\c@section}
\def\thesubsection{\thesection.\@arabic\c@subsection}
\def\thesubsubsection{\thesubsection.\@arabic\c@subsubsection}
\setcounter{secnumdepth}{3}
\def\@pnumwidth{1.6em}
\def\@tocrmarg {2.6em}
\setcounter{tocdepth}{2}
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\tableofcontents{%
\removelastskip\vskip\z@skip
  \@startsection{}\@M\z@{12\p@\@plus12\p@}{6\p@}%
        {\centering\sc}\contentsname
 \@starttoc{toc}%
 \addvspace{32\p@\@plus14\p@}%
\let\tableofcontents\relax
\let\l@part\relax \let\l@section\relax
\let\l@subsection\relax \let\l@subsubsection\relax
}
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\contentsname{Contents}
\def\@tocline#1#2#3#4#5{\par
  \addpenalty\@secpenalty\addvspace{#1}%
  \begingroup   \@tempdima#2\relax % used by \numberline
  \parindent\z@ \rightskip\@pnumwidth \parfillskip-\@pnumwidth
  {#3\leavevmode #4\hfil \hbox to\@pnumwidth{\hss #5}}\par\nobreak
  \endgroup}
\def\l@part{\@tocline\bigskipamount{2em}{\normalshape\bf}}
\def\l@section{\@tocline\medskipamount{2em}{\normalshape\bf}}
\def\@dottedtocline#1#2#3#4#5{\ifnum #1>\c@tocdepth \else
  \vskip \z@\@plus\p@
  {\leftskip #2\relax \rightskip\@tocrmarg \parfillskip-\rightskip
   \parindent #2\relax \@afterindenttrue
   \interlinepenalty\@M \leavevmode
   \@tempdima #3\relax \advance\leftskip\@tempdima
    \null \hskip-\leftskip #4\nobreak
        \hfill\nobreak
  \ifnum#1<\c@tocdepth\hbox to\@pnumwidth{\hfil\normalshape#5}\fi
  \null\par}\fi}
\def\numberline#1{\setboxz@h{#1\enspace}\ifdim\wdz@>\@tempdima
  \box\z@\else \hbox to\@tempdima{#1\hfil}\fi}
\def\l@subsection{\@dottedtocline\tw@{1.5em}{2.3em}}
\def\l@subsubsection{\@dottedtocline\thr@@{3.8em}{3.2em}}
\def\l@paragraph{\@dottedtocline4{7em}{4.1em}}
\def\refname{References}
\def\bibname{Bibliography}
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\thebibliography#1{\section*\refname
  \defaultfont\small\labelsep .5em\relax
  \list{\@arabic\c@enumi.}{\settowidth\labelwidth{#1.}%
  \leftmargin\labelwidth \advance\leftmargin\labelsep
  \usecounter{enumi}}%
  \sloppy \clubpenalty4000\relax \widowpenalty\clubpenalty
  \sfcode`\.\@m}
\let\endthebibliography=\endlist
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\bysame{\leavevmode\hbox to3em{\hrulefill}\,}
\def\newblock{}
\def\footnoterule{\kern-.4\p@
        \hrule\@width 5pc\kern11\p@\kern-\footnotesep}
\def\@makefnmark{\hbox{$\m@th^{\@thefnmark}$}}
\def\@makefntext{\indent\@makefnmark}
\long\def\@footnotetext#1{\insert\footins{%
    \defaultfont\footnotesize
    \interlinepenalty\interfootnotelinepenalty
    \splittopskip\footnotesep \splitmaxdepth \dp\strutbox
    \floatingpenalty\@MM \hsize\columnwidth
\@parboxrestore \parindent\normalparindent \sloppy
  \edef\@currentlabel{\p@footnote\@thefnmark}\@makefntext
 {\rule\z@\footnotesep\ignorespaces#1\unskip\strut\par}}}
\def\sloppy{\tolerance9999\relax}
\hfuzz=1pt \vfuzz=\hfuzz
\setcounter{topnumber}{2}
\def\topfraction{.7}
\setcounter{bottomnumber}{1}
\def\bottomfraction{.3}
\setcounter{totalnumber}{3}
\def\textfraction{.2}
\def\floatpagefraction{.5}
\setcounter{dbltopnumber}{2}
\def\dbltopfraction{.7}
\def\dblfloatpagefraction{.5}
\long\def\@makecaption#1#2{%
\setbox\@tempboxa\vbox{\advance\hsize-6pc\noindent
    \sc#1\expandafter\@ifnotempty\expandafter
        {\@cdr#2\@nil}{.\normalshape\enspace#2}%
    \unskip\kern-6pc\par
    \global\setbox\@ne\lastbox}%
\ifhbox\@ne % the normal case
  \setbox\@ne\hbox{\unhbox\@ne\unskip\unskip\unpenalty\unkern}%
\fi
  \ifdim\wd\@tempboxa=\z@ % this means caption will fit on one line
    \setbox\@ne\hbox to\columnwidth{\hss\kern-6pc\box\@ne\hss}%
  \else % tempboxa contained more than one line (or other vertical mode stuff)
    \setbox\@ne\vbox{\unvbox\@tempboxa\parskip\z@skip
        \noindent\unhbox\@ne\advance\hsize-6pc\par}%
\fi
  \ifnum\@tempcnta<64 % if the float IS a figure...
    \vskip 1pc%
    \moveright 3pc\box\@ne
  \else % if the float IS NOT a figure...
    \moveright 3pc\box\@ne
    \vskip 1pc%
  \fi
\relax
}
\newcounter{figure}
\def\thefigure{\@arabic\c@figure}
\def\fps@figure{tbp}
\def\ftype@figure{1}
\def\ext@figure{lof}
\def\fnum@figure{\figurename\ \thefigure}
\def\figurename{Figure}
\def\figure{\@float{\@string\figure}}
\let\endfigure=\end@float
\newcounter{table}
\def\thetable{\@arabic\c@table}
\def\fps@table{tbp}
\def\ftype@table{2}
\def\ext@table{lot}
\def\fnum@table{\tablename\ \thetable}
\def\tablename{Table}
\def\table{\@float{\@string\table}}
\let\endtable=\end@float
\input theorem.sty
\theorembodyfont{}
\def\@begintheorem#1#2[#3]{\item[\hskip\labelsep \theorem@headerfont
       \theorem@indent#1\ \rom{#2}\unskip
      \@ifnotempty{#3}{\ \rom(#3\rom)}.]\ignorespaces}
\def\@thm#1#2{\refstepcounter{#1}%
   \trivlist
   \@topsep \theorempreskipamount               % used by first \item
   \@topsepadd \theorempostskipamount           % used by \@endparenv
   \@oparg{\@begintheorem{#2}{\csname the#1\endcsname}}[]}
\let\@opargbegintheorem\relax \let\@ythm\relax
\def\th@plain{\it}
\def\th@definition{\theorempreskipamount6\p@\@plus6\p@
    \theorempostskipamount\theorempreskipamount
    \normalshape}
\def\th@remark{\theorempreskipamount6\p@\@plus6\p@
    \theorempostskipamount\z@\@plus6\p@
    \def\theorem@headerfont{\it}\normalshape}
\def\theorem@headerfont{\normalshape\bf}
\let\theorem@indent\noindent
\theorempreskipamount=6pt plus9pt
\theorempostskipamount=\theorempreskipamount
\def\@endtheorem{\endtrivlist\@endpefalse }
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\pf{%
  \par\topsep6\p@\@plus6\p@
  \trivlist
  \item[\hskip\labelsep\it\proofname.]\ignorespaces}
\def\endpf{\qed\endtrivlist}
\@namedef{pf*}#1{\par\toks@\expandafter{\proofname}%
  \edef\restoreproofname{\def\noexpand\proofname{\the\toks@}}%
  \def\proofname{#1}\pf\restoreproofname\ignorespaces}
\expandafter\let\csname endpf*\endcsname=\endpf
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\qedsymbol{\RIfM@\bgroup\else$\bgroup\aftergroup$\fi
  \vcenter{\hrule\hbox{\vrule\@height.6em\kern.6em\vrule}\hrule}\egroup}
\def\qed{\RIfM@\else\unskip\nobreak\fi\quad\qedsymbol}
\def\proofname{Proof}
\def\enddocument{\@checkend{document}\par
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
  \ifx\@empty\@translator \else
    \addvspace{6\p@\@plus9\p@}%
    \hbox to\columnwidth{\hss\defaultfont\normalsize
     Translated by \expandafter\uppercasetext@\expandafter{\@translator}}%
  \fi
 \removelastskip\@setaddress
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
 \clearpage\begingroup
 \if@filesw \immediate\closeout\@mainaux
 \def\global\@namedef##1##2{}\def\newlabel{\@testdef r}%
 \def\bibcite{\@testdef b}\@tempswafalse \makeatletter\input \jobname.aux
 \if@tempswa \@warning{Label(s) may have changed.  Rerun to get
 cross-references right}\fi\fi\endgroup\deadcycles\z@\@@end}
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\ps@headings{\let\@mkboth\@gobbletwo
  \let\partmark\@gobble\let\sectionmark\@gobble
  \let\subsectionmark\@gobble
  \let\@oddfoot\@empty\let\@evenfoot\@empty%
  \def\@evenhead{\defaultfont\small
      \rlap{\thepage}\hfil
      \expandafter\uppercasetext@\expandafter{\sh@rtauthor}\hfil}%
  \def\@oddhead{\defaultfont\small \hfil
      \expandafter\uppercasetext@\expandafter{\sh@rttitle}\hfil
      \llap{\thepage}}%
}
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\def\ps@myheadings{\let\@mkboth\@gobbletwo
  \let\@oddfoot\@empty\let\@evenfoot\@empty
  \def\@oddhead{\null\defaultfont\small\rightmark \hfil\thepage}%
  \def\@evenhead{\defaultfont\small \thepage\hfil\leftmark\null}%
}
\def\ps@plain{\let\@mkboth\@gobbletwo
     \let\@oddhead\@empty \let\@evenhead\@empty
     \def\@oddfoot{\defaultfont\small \hfil\thepage\hfil}%
     \let\@evenfoot\@oddfoot}
\def\@cite#1#2{\rom{\mediumseries[{#1\if@tempswa , #2\fi}]}}
\def\rom#1{\leavevmode\skip@\lastskip\unskip\/%
        \ifdim\skip@=\z@\else\hskip\skip@\fi
   {\normalshape#1}}
\atdef@-{\let\next-%%
  \DN@{\ifx\next-%%
    \def\next-{\setboxz@h{-}\unhbox\z@\FN@\next@}%%
    \expandafter\next \else\nobreak\fi}%%
  \next@-}%%
\def\eqref#1{\rom{\tagform@{\ref{#1}}}}
\def\theequation{\@arabic\c@equation}
\ctagsplit@true
\if@article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
\advance\textheight by -\headheight
\advance\textheight by -\headsep
\advance\textheight by -\normalbaselineskip
\advance\textheight by \topskip
\oddsidemargin=8.5 true in % standard U.S. paper width
\advance\oddsidemargin by -\textwidth
\oddsidemargin=.5\oddsidemargin
\ifdim\oddsidemargin < .5 true in
  \oddsidemargin = .5 true in \fi
\advance\oddsidemargin by -1 true in
\evensidemargin=\oddsidemargin
\topmargin=11 true in % standard U.S. paper width
\advance\topmargin by -\textheight
\advance\topmargin by -\headheight
\advance\topmargin by -\headsep
\advance\topmargin by -\footheight
\advance\topmargin by -\footskip
\topmargin=.5\topmargin
\ifdim\topmargin<.5 true in
  \topmargin=.5 true in \fi
\advance\topmargin by -1 true in
\pagenumbering{arabic}
\onecolumn
\@twosidetrue \@mparswitchtrue   % this = `twoside' option
\mark{{}{}}
\ps@headings
\fi %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARTICLE ONLY
%% \CharacterTable
%%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
%%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
%%   Digits        \0\1\2\3\4\5\6\7\8\9
%%   Exclamation   \!     Double quote  \"     Hash (number) \#
%%   Dollar        \$     Percent       \%     Ampersand     \&
%%   Acute accent  \'     Left paren    \(     Right paren   \)
%%   Asterisk      \*     Plus          \+     Comma         \,
%%   Minus         \-     Point         \.     Solidus       \/
%%   Colon         \:     Semicolon     \;     Less than     \<
%%   Equals        \=     Greater than  \>     Question mark \?
%%   Commercial at \@     Left bracket  \[     Backslash     \\
%%   Right bracket \]     Circumflex    \^     Underscore    \_
%%   Grave accent  \`     Left brace    \{     Vertical bar  \|
%%   Right brace   \}     Tilde         \~}
\endinput
%% 
%% End of file `amsart.sty'.