\relax 
\ifx\hyper@anchor\@undefined
\global \let \oldcontentsline\contentsline
\gdef \contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global \let \oldnewlabel\newlabel
\gdef \newlabel#1#2{\newlabelxx{#1}#2}
\gdef \newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\let \contentsline\oldcontentsline
\let \newlabel\oldnewlabel}
\else
\global \let \hyper@last\relax 
\fi

\citation{ref1}
\@writefile{toc}{\contentsline {section}{\numberline {1}Definitions and main results}{1}{section.1}}
\newlabel{eq:1}{{1.1}{1}{Definitions and main results\relax }{equation.1.1}{}}
\newlabel{eq:2}{{1.2}{1}{Definitions and main results\relax }{equation.1.2}{}}
\newlabel{eq:3}{{1.3}{1}{Definitions and main results\relax }{equation.1.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {2}Proof of the Theorem }{2}{section.2}}
\newlabel{lem:1}{{2.1}{2}{Proof of the Theorem \relax }{lemma.2.1}{}}
\newlabel{eq:a}{{2.1}{2}{Proof of the Theorem \relax }{equation.2.1}{}}
\newlabel{eq:b}{{2.2}{2}{Proof of the Theorem \relax }{equation.2.1}{}}
\newlabel{lem:2}{{2.2}{3}{Proof of the Theorem \relax }{lemma.2.2}{}}
\newlabel{eq:4}{{2.3}{3}{Proof of the Theorem \relax }{equation.2.3}{}}
\newlabel{lem:3}{{2.3}{3}{Proof of the Theorem \relax }{lemma.2.3}{}}
\newlabel{eq:5}{{2.4}{4}{Proof of the Theorem \relax }{equation.2.4}{}}
\newlabel{eq:6}{{2.5}{4}{Proof of the Theorem \relax }{equation.2.5}{}}
\newlabel{lem:4}{{2.4}{4}{Proof of the Theorem \relax }{lemma.2.4}{}}
\newlabel{lem:5}{{2.5}{5}{Proof of the Theorem \relax }{lemma.2.5}{}}
\newlabel{lem:6}{{2.6}{5}{Proof of the Theorem \relax }{lemma.2.6}{}}
\bibcite{ref1}{1}
\@writefile{toc}{\contentsline {section}{\numberline {3}Acknowledgments}{6}{section.3}}