\newtheorem{theorem}{Theorem}[section]
\newtheorem{theoreme}{Th\'eor\`eme}[section]
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{lemme}[theoreme]{Lemme}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{corollaire}[theoreme]{Corollaire}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}{Example}[section]

\newenvironment{remark}{\textbf{Remark: }}{}
\newcommand{\note}{{\it Note.}\ \ }

\newcommand{\proof}{\noindent {\bf Proof.}\ \ }
\newcommand{\proofsketch}{\noindent {\bf Sketch of proof.}\ \ }
\newcommand{\proofidea}{\noindent {\bf Idea of proof.}\ \ }

\newcommand{\Wolog}{Without loss of generality}
\newcommand{\wolog}{without loss of generality}
\newcommand{\st}{such that }


