\documentclass{amsart}

\usepackage{amsthm}

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

\begin{document}

\section{A section}
\subsection{A subsection}
\begin{theorem}\end{theorem}
\begin{proposition}\end{proposition}
\begin{lemma}\end{lemma}
\subsection{A subsection}
\begin{theorem}\end{theorem}
\begin{proposition}\end{proposition}
\begin{lemma}\end{lemma}

\section{A section}
\subsection{A subsection}
\begin{theorem}\end{theorem}
\begin{proposition}\end{proposition}
\begin{lemma}\end{lemma}
\subsection{A subsection}
\begin{theorem}\end{theorem}
\begin{proposition}\end{proposition}
\begin{lemma}\end{lemma}

\end{document}