documentclass[12pt,a4paper]{article}
     %------------------------------------------------------------
      usepackage{amsmath,amssymb,amsthm}
        %------------------------------------------------------------
         usepackage[utf8]{inputenc}
         usepackage[T1]{fontenc}
    % ------------------------------------------------------------
    newtheorem{theorem}{Theorem}[section]
     newtheorem{definition}{Definition}[section]
     newtheorem{definitions}{Definitions}[section]
     newtheorem{notation}{Notation}[section]
    newtheorem{corollary}{Corollary}[section]
   newtheorem{proposition}{Proposition}[section]
  newtheorem{lemma}{Lemma}[section]
  newtheorem{remark}{Remark}[section]
  newtheorem{example}{Example}[section]
  numberwithin{equation}{section}
    begin{document}
    section{section1}
    begin{theorem}
    end{theorem}
    begin{proof}
    end{proof}
     end{document}