\documentclass{report}
\usepackage{amssymb, amsmath, hyperref}

\theoremstyle{plain}
\newtheorem{theorem}[subsection]{Theorem}
\newtheorem{proposition}[subsection]{Proposition}
\newtheorem{lemma}[subsection]{Lemma}

\theoremstyle{definition}
\newtheorem{definition}[subsection]{Definition}
\newtheorem{example}[subsection]{Example}
\newtheorem{exercise}[subsection]{Exercise}
\newtheorem{situation}[subsection]{Situation}

\theoremstyle{remark}
\newtheorem{remark}[subsection]{Remark}
\newtheorem{remarks}[subsection]{Remarks}

\numberwithin{equation}{subsection}

\newcommand{\hi}{gamer}

\begin{document}

\chapter{A first chapter}
\label{chapter:first}

\section{A first section}
\label{section:first}

\begin{lemma}
  \label{lemma:pythagoras}
  $a^2=b^2+c^2 \hello$
\hi
\end{lemma}
\begin{proof}
hihihihihihihi
\[\xymatrix{ &  Y_1 &  \\  X \ar[ru]^{f_1} \ar[r]^{f_3} \ar[rd]_{f_2} &  Y_3 \ar[u]_{a_{31}} \ar[d]^{a_{32}} &  Y \ar[lu]_{s_1} \ar[l]_{s_3} \ar[ld]^{s_2} \\  &  Y_2 &  }\]
\end{proof}

\section{A second section}
\label{section:second}


\chapter{A second chapter}
\label{chapter:second}

\section{A third section}
\label{section:third}

\section{A fourth section}
\label{section:fourth}

\end{document}
