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+%%
+%% This is file `amsthm.sty',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% amsclass.dtx (with options: `amsthm')
+%% This is a generated file.
+%%
+%% Copyright 1995, 1999, 2004, 2009-2020 American Mathematical Society.
+%%
+%% American Mathematical Society
+%% Technical Support
+%% Publications Technical Group
+%% 201 Charles Street
+%% Providence, RI 02904
+%% USA
+%% tel: (401) 455-4080
+%% (800) 321-4267 (USA and Canada only)
+%% fax: (401) 331-3842
+%% email: tech-support@ams.org
+%%
+%% This work may be distributed and/or modified under the
+%% conditions of the LaTeX Project Public License, either version 1.3c
+%% of this license or (at your option) any later version.
+%% The latest version of this license is in
+%% http://www.latex-project.org/lppl.txt
+%% and version 1.3c or later is part of all distributions of LaTeX
+%% version 2005/12/01 or later.
+%%
+%% This work has the LPPL maintenance status `maintained'.
+%%
+%% The Current Maintainer of this work is the American Mathematical
+%% Society.
+%%
+%% ========================================================================
+%%
+\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
+[1995/06/01]% LaTeX date must be June 1995 or later
+\ProvidesPackage{amsthm}[2020/05/29 v2.20.6]
+\let\@xp=\expandafter
+\let\@nx=\noexpand
+\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
+\long\def\@ifempty#1{\@xifempty#1@@..\@nil}
+\long\def\@xifempty#1#2@#3#4#5\@nil{%
+ \ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
+\long\def\@ifnotempty#1{\@ifempty{#1}{}}
+\def\setboxz@h{\setbox\z@\hbox}
+\def\@addpunct#1{%
+ \relax\ifhmode
+ \ifnum\spacefactor>\@m \else#1\fi
+ \fi}
+\def\nopunct{\spacefactor 1007 }
+\def\frenchspacing{\sfcode`\.1006\sfcode`\?1005\sfcode`\!1004%
+ \sfcode`\:1003\sfcode`\;1002\sfcode`\,1001 }
+\DeclareOption*{\input{\CurrentOption .thm}}
+\ProcessOptions
+
+\newcommand{\theoremstyle}[1]{%
+ \@ifundefined{th@#1}{%
+ \PackageWarning{amsthm}{Unknown theoremstyle `#1'}%
+ \thm@style{plain}%
+ }{%
+ \thm@style{#1}%
+ }%
+}
+\newtoks\thm@style
+\thm@style{plain}
+\newtoks\thm@bodyfont \thm@bodyfont{\itshape}
+\newtoks\thm@headfont \thm@headfont{\bfseries}
+\newtoks\thm@notefont \thm@notefont{}
+\newtoks\thm@headpunct \thm@headpunct{.}
+\newskip\thm@preskip \newskip\thm@postskip
+\def\thm@space@setup{%
+ \thm@preskip=\topsep \thm@postskip=\thm@preskip
+}
+\renewcommand{\newtheorem}{\@ifstar{\@xnthm *}{\@xnthm \relax}}
+\def\@xnthm#1#2{%
+ \let\@tempa\relax
+ \@xp\@ifdefinable\csname #2\endcsname{%
+ \global\@xp\let\csname end#2\endcsname\@endtheorem
+ \ifx *#1% unnumbered, need to get one more mandatory arg
+ \edef\@tempa##1{%
+ \gdef\@xp\@nx\csname#2\endcsname{%
+ \@nx\@thm{\@xp\@nx\csname th@\the\thm@style\endcsname}%
+ {}{##1}}}%
+ \else % numbered theorem, need to check for optional arg
+ \def\@tempa{\@oparg{\@ynthm{#2}}[]}%
+ \fi
+ }%
+ \@tempa
+}
+\def\@ynthm#1[#2]#3{%
+ \ifx\relax#2\relax
+ \def\@tempa{\@oparg{\@xthm{#1}{#3}}[]}%
+ \else
+ \@ifundefined{c@#2}{%
+ \def\@tempa{\@nocounterr{#2}}%
+ }{%
+ \@xp\xdef\csname the#1\endcsname{\@xp\@nx\csname the#2\endcsname}%
+ \toks@{#3}%
+ \@xp\xdef\csname#1\endcsname{%
+ \@nx\@thm{%
+ \let\@nx\thm@swap
+ \if S\thm@swap\@nx\@firstoftwo\else\@nx\@gobble\fi
+ \@xp\@nx\csname th@\the\thm@style\endcsname}%
+ {#2}{\the\toks@}}%
+ \let\@tempa\relax
+ }%
+ \fi
+ \@tempa
+}
+\def\@xthm#1#2[#3]{%
+ \ifx\relax#3\relax
+ \newcounter{#1}%
+ \else
+ \newcounter{#1}[#3]%
+ \@xp\xdef\csname the#1\endcsname{\@xp\@nx\csname the#3\endcsname
+ \@thmcountersep\@thmcounter{#1}}%
+ \fi
+ \toks@{#2}%
+ \@xp\xdef\csname#1\endcsname{%
+ \@nx\@thm{%
+ \let\@nx\thm@swap
+ \if S\thm@swap\@nx\@firstoftwo\else\@nx\@gobble\fi
+ \@xp\@nx\csname th@\the\thm@style\endcsname}%
+ {#1}{\the\toks@}}%
+}
+\def\@thm#1#2#3{%
+ \ifhmode\unskip\unskip\par\fi
+ \normalfont
+ \trivlist
+ \let\thmheadnl\relax
+ \let\thm@swap\@gobble
+ \thm@notefont{\fontseries\mddefault\upshape}%
+ \thm@headpunct{.}% add period after heading
+ \thm@headsep 5\p@ plus\p@ minus\p@\relax
+ \thm@space@setup
+ #1% style overrides
+ \@topsep \thm@preskip % used by thm head
+ \@topsepadd \thm@postskip % used by \@endparenv
+ \def\@tempa{#2}\ifx\@empty\@tempa
+ \def\@tempa{\@oparg{\@begintheorem{#3}{}}[]}%
+ \else
+ \refstepcounter{#2}%
+ \def\@tempa{\@oparg{\@begintheorem{#3}{\csname the#2\endcsname}}[]}%
+ \fi
+ \@tempa
+}
+\def\@restorelabelsep{\relax}
+\let\@ythm\relax
+\let\thmname\@iden \let\thmnote\@iden \let\thmnumber\@iden
+\providecommand\@upn{\textup}
+\def\thmhead@plain#1#2#3{%
+ \thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2}}%
+ \thmnote{ {\the\thm@notefont(#3)}}}
+\let\thmhead\thmhead@plain
+\def\swappedhead#1#2#3{%
+ \thmnumber{#2}%
+ \thmname{\@ifnotempty{#2}{~}#1}%
+ \thmnote{ {\the\thm@notefont(#3)}}}
+\let\swappedhead@plain=\swappedhead
+\let\thmheadnl\relax
+\let\thm@indent\noindent
+\let\thm@swap\@gobble
+\def\@begintheorem#1#2[#3]{%
+ \deferred@thm@head{\the\thm@headfont \thm@indent
+ \@ifempty{#1}{\let\thmname\@gobble}{\let\thmname\@iden}%
+ \@ifempty{#2}{\let\thmnumber\@gobble}{\let\thmnumber\@iden}%
+ \@ifempty{#3}{\let\thmnote\@gobble}{\let\thmnote\@iden}%
+ \thm@swap\swappedhead\thmhead{#1}{#2}{#3}%
+ \the\thm@headpunct
+ \thmheadnl % possibly a newline.
+ \hskip\thm@headsep
+ }%
+ \ignorespaces}
+\newskip\thm@headsep
+\thm@headsep=5pt plus1pt minus1pt\relax
+\let\adjust@parskip@nobreak=\@nbitem
+\newtoks\dth@everypar
+\dth@everypar={%
+ \@minipagefalse \global\@newlistfalse
+ \@noparitemfalse
+ \if@inlabel
+ \global\@inlabelfalse
+ \begingroup \setbox\z@\lastbox
+ \ifvoid\z@ \kern-\itemindent \fi
+ \endgroup
+ \unhbox\@labels
+ \fi
+ \if@nobreak \@nobreakfalse \clubpenalty\@M
+ \else \clubpenalty\@clubpenalty \everypar{}%
+ \fi
+}%
+\def\deferred@thm@head#1{%
+ \if@inlabel \indent \par \fi % eject a section head if one is pending
+ \if@nobreak
+ \adjust@parskip@nobreak
+ \else
+ \addpenalty\@beginparpenalty
+ \addvspace\@topsep
+ \addvspace{-\parskip}%
+ \fi
+ \global\@inlabeltrue
+ \everypar\dth@everypar
+ \sbox\@labels{\normalfont#1}%
+ \ignorespaces
+}
+\def\nonslanted{\relax
+ \@xp\let\@xp\@tempa\csname\f@shape shape\endcsname
+ \ifx\@tempa\itshape\upshape
+ \else\ifx\@tempa\slshape\upshape\fi\fi}
+\def\swapnumbers{\edef\thm@swap{\if S\thm@swap N\else S\fi}}
+\def\thm@swap{N}%
+\let\@opargbegintheorem\relax
+\def\th@plain{%
+%% \let\thm@indent\noindent % no indent
+%% \thm@headfont{\bfseries}% heading font is bold
+%% \thm@notefont{}% same as heading font
+%% \thm@headpunct{.}% add period after heading
+%% \let\thm@swap\@gobble
+%% \thm@preskip\topsep
+%% \thm@postskip\theorempreskipamount
+ \itshape % body font
+}
+\def\th@definition{%
+ \normalfont % body font
+}
+\def\th@remark{%
+ \thm@headfont{\itshape}%
+ \normalfont % body font
+ \thm@preskip\topsep \divide\thm@preskip\tw@
+ \thm@postskip\thm@preskip
+}
+\def\@endtheorem{\endtrivlist\@endpefalse }
+\newcommand{\newtheoremstyle}[9]{%
+ \@ifempty{#5}{\dimen@\z@skip}{\dimen@#5\relax}%
+ \ifdim\dimen@=\z@
+ \toks@{#4\let\thm@indent\noindent}%
+ \else
+ \toks@{#4\def\thm@indent{\noindent\hbox to#5{}}}%
+ \fi
+ \def\@tempa{#8}\ifx\space\@tempa
+ \toks@\@xp{\the\toks@ \thm@headsep\fontdimen\tw@\font\relax}%
+ \else
+ \def\@tempb{\newline}%
+ \ifx\@tempb\@tempa
+ \toks@\@xp{\the\toks@ \thm@headsep\z@skip
+ \def\thmheadnl{\newline}}%
+ \else
+ \toks@\@xp{\the\toks@ \thm@headsep#8\relax}%
+ \fi
+ \fi
+ \begingroup
+ \thm@space@setup
+ \@defaultunits\@tempskipa#2\thm@preskip\relax\@nnil
+ \@defaultunits\@tempskipb#3\thm@postskip\relax\@nnil
+ \xdef\@gtempa{\thm@preskip\the\@tempskipa
+ \thm@postskip\the\@tempskipb\relax}%
+ \endgroup
+ \@temptokena\@xp{\@gtempa
+ \thm@headfont{#6}\thm@headpunct{#7}%
+ }%
+ \@ifempty{#9}{%
+ \let\thmhead\thmhead@plain
+ }{%
+ \@namedef{thmhead@#1}##1##2##3{#9}%
+ \@temptokena\@xp{\the\@temptokena
+ \@xp\let\@xp\thmhead\csname thmhead@#1\endcsname}%
+ }%
+ \@xp\xdef\csname th@#1\endcsname{\the\toks@ \the\@temptokena}%
+}
+\DeclareRobustCommand{\qed}{%
+ \ifmmode \mathqed
+ \else
+ \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill
+ \quad\hbox{\qedsymbol}%
+ \fi
+}
+\let\QED@stack\@empty
+\let\qed@elt\relax
+\newcommand{\pushQED}[1]{%
+ \toks@{\qed@elt{#1}}\@temptokena\expandafter{\QED@stack}%
+ \xdef\QED@stack{\the\toks@\the\@temptokena}%
+}
+\newcommand{\popQED}{%
+ \begingroup\let\qed@elt\popQED@elt \QED@stack\relax\relax\endgroup
+}
+\def\popQED@elt#1#2\relax{#1\gdef\QED@stack{#2}}
+\newcommand{\qedhere}{%
+ \begingroup \let\mathqed\math@qedhere
+ \let\qed@elt\setQED@elt \QED@stack\relax\relax \endgroup
+}
+\newif\ifmeasuring@
+\newif\iffirstchoice@ \firstchoice@true
+\def\setQED@elt#1#2\relax{%
+ \ifmeasuring@
+ \else \iffirstchoice@ \gdef\QED@stack{\qed@elt{}#2}\fi
+ \fi
+ #1%
+}
+\def\qed@warning{%
+ \PackageWarning{amsthm}{The \@nx\qedhere command may not work
+ correctly here}%
+}
+\newcommand{\mathqed}{\quad\hbox{\qedsymbol}}
+\def\linebox@qed{\hfil\hbox{\qedsymbol}\hfilneg}
+\@ifpackageloaded{amsmath}{%
+ \def\math@qedhere{%
+ \@ifundefined{\@currenvir @qed}{%
+ \qed@warning\quad\hbox{\qedsymbol}%
+ }{%
+ \@xp\aftergroup\csname\@currenvir @qed\endcsname
+ }%
+ }
+ \def\displaymath@qed{%
+ \relax
+ \ifmmode
+ \ifinner \aftergroup\linebox@qed
+ \else
+ \eqno
+ \let\eqno\relax \let\leqno\relax \let\veqno\relax
+ \hbox{\qedsymbol}%
+ \fi
+ \else
+ \aftergroup\linebox@qed
+ \fi
+ }
+ \@xp\let\csname equation*@qed\endcsname\displaymath@qed
+ \def\equation@qed{%
+ \iftagsleft@
+ \hbox{\phantom{\quad\qedsymbol}}%
+ \gdef\alt@tag{%
+ \rlap{\hbox to\displaywidth{\hfil\qedsymbol}}%
+ \global\let\alt@tag\@empty
+ }%
+ \else
+ \gdef\alt@tag{%
+ \global\let\alt@tag\@empty
+ \vtop{\ialign{\hfil####\cr
+ \tagform@\theequation\cr
+ \qedsymbol\cr}}%
+ \setbox\z@
+ }%
+ \fi
+ }
+ \def\qed@tag{%
+ \global\tag@true \nonumber
+ &\omit\setboxz@h {\strut@ \qedsymbol}\tagsleft@false
+ \place@tag@gather
+ \kern-\tabskip
+ \ifst@rred \else \global\@eqnswtrue \fi \global\advance\row@\@ne \cr
+ }
+ \def\split@qed{%
+ \def\endsplit{\crcr\egroup \egroup \ctagsplit@false \rendsplit@
+ \aftergroup\align@qed
+ }%
+ }
+ \def\align@qed{%
+ \ifmeasuring@ \tag*{\qedsymbol}%
+ \else \let\math@cr@@@\qed@tag
+ \fi
+ }
+ \@xp\let\csname align*@qed\endcsname\align@qed
+ \@xp\let\csname gather*@qed\endcsname\align@qed
+%% Needs some patching up for amsmath 1.2
+}{% end of amsmath branch, start plain LaTeX branch
+ \def\math@qedhere{%
+ \@ifundefined{\@currenvir @qed}{%
+ \qed@warning \aftergroup\displaymath@qed
+ }{%
+ \@xp\aftergroup\csname\@currenvir @qed\endcsname
+ }%
+ }
+ \def\displaymath@qed{%
+ \relax
+ \ifmmode
+ \ifinner \aftergroup\aftergroup\aftergroup\linebox@qed
+ \else
+ \eqno \def\@badmath{$$}%
+ \let\eqno\relax \let\leqno\relax \let\veqno\relax
+ \hbox{\qedsymbol}%
+ \fi
+ \else
+ \aftergroup\linebox@qed
+ \fi
+ }
+ \@ifundefined{ver@leqno.clo}{%
+ \def\equation@qed{\displaymath@qed \quad}%
+ }{%
+ \def\equation@qed{\displaymath@qed}%
+ }
+ \def\@tempa#1$#2#3\@nil#4{%
+ \def#4{#1$#2\def\@currenvir{displaymath}#3}%
+ }%
+ \expandafter\ifx\csname[ \endcsname\relax
+ \expandafter\@tempa\[\@nil\[%
+ \else
+ \expandafter\expandafter\expandafter\@tempa\csname[
+ \expandafter\endcsname\expandafter\@nil
+ \csname[ \endcsname
+ \fi
+}
+\@ifpackageloaded{amstex}{%
+ \def\@tempa{TT}%
+}{%
+ \@ifpackageloaded{amsmath}{%
+ \def\@tempb#1 v#2.#3\@nil{#2}%
+ \ifnum\@xp\@xp\@xp\@tempb\csname ver@amsmath.sty\endcsname v0.0\@nil
+ <\tw@
+ \def\@tempa{TT}%
+ \else
+ \def\@tempa{TF}%
+ \fi
+ }{%
+ \def\@tempa{TF}
+ }%
+}
+\if\@tempa
+ \renewcommand{\math@qedhere}{\quad\hbox{\qedsymbol}}%
+\fi
+\newcommand{\rgddopenbox}{\leavevmode
+ \hbox to.77778em{%
+ \hfil\vrule
+ \vbox to.675em{\hrule width.6em\vfil\hrule}%
+ \vrule\hfil}}
+\DeclareRobustCommand{\textsquare}{%
+ \begingroup \usefont{U}{msa}{m}{n}\thr@@\endgroup
+}
+\providecommand{\qedsymbol}{\rgddopenbox}
+\newenvironment{proof}[1][\proofname]{\par
+ \pushQED{\qed}%
+ \normalfont \topsep6\p@\@plus6\p@\relax
+ \trivlist
+ \item[\hskip\labelsep
+ \itshape
+ #1\@addpunct{.}]\ignorespaces
+}{%
+ \popQED\endtrivlist\@endpefalse
+}
+\providecommand{\proofname}{Proof}
+\endinput
+%%
+%% End of file `amsthm.sty'.