A numbered list with arbitrary order of items

Question

Inside a theorem environment I have a numbered list of theorem statements.

Afterward it goes a proof. In the proof I prove every of the above specified numbered statements.

But the order in the proof may be different than the order in the theorem.

Which environment best to use in the proof? I may manually write "1.", "2." or make a definition list with numeric labels, or whatever. Which way is the best?

Answer 1

As cmhughes suggested in his comment, you can use an enumerate environment with labelled \items to build the theorem, and then use \ref in the proof (this guarantees you consistency and avoids possible errors from manually numbering):

\documentclass{article}
\usepackage{amsthm}
\usepackage{amsmath}

\newtheorem{theorem}{Theorem}

\begin{document}

\begin{theorem}
For a left artin ring $\Lambda$ we have the following.
\begin{enumerate}
\item\label{th:radnil} The radical of $\Lambda$ is nilpotent.
\item\label{th:finsim} There is only a finite number of nonisomorphic simple $\Lambda$-modules.
\item\label{th:leftnoe} $\Lambda$ is left noetherian.
\end{enumerate}
\end{theorem}
\begin{proof}
\ref{th:leftnoe}. Since $\Lambda$ has a finite filtration...\par
\ref{th:radnil}. Let $I$ be an ideal in $\Lambda$...\par
\ref{th:finsim}. If for a $\Lambda$-module $A$ we have...
\end{proof}

\end{document}

Using the enumitem package one can define a customized list-like environment; in the following example the thmclaim environment works as an enumerate environment but using boldfaced labels; inside the proof environment, a description environment was used to refer to the claims (thus keeping consistent use of boldfaced type):

\documentclass{article}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{enumitem}

\newtheorem{theorem}{Theorem}

\newlist{thmclaim}{enumerate}{1}
\setlist[thmclaim,1]{label=\normalfont\textbf{\arabic*.}}

\begin{document}

\begin{theorem}
For a left artin ring $\Lambda$ we have the following.
\begin{thmclaim}
\item\label{th:radnil} The radical of $\Lambda$ is nilpotent.
\item\label{th:finsim} There is only a finite number of nonisomorphic simple $\Lambda$-modules.
\item\label{th:leftnoe} $\Lambda$ is left noetherian.
\end{thmclaim}
\end{theorem}
\begin{proof}
\begin{description}
\item[\ref{th:leftnoe}] Since $\Lambda$ has a finite filtration...\par
\item[\ref{th:radnil}] Let $I$ be an ideal in $\Lambda$...\par
\item[\ref{th:finsim}] If for a $\Lambda$-module $A$ we have...
\end{description}
\end{proof}

\end{document}

And a variation using alphabetical characters:

\documentclass{article}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{enumitem}

\newtheorem{theorem}{Theorem}

\newlist{thmclaim}{enumerate}{1}
\setlist[thmclaim,1]{label=\normalfont\textbf{(\alph*)}}

\begin{document}

\begin{theorem}
For a left artin ring $\Lambda$ we have the following.
\begin{thmclaim}
\item\label{th:radnil} The radical of $\Lambda$ is nilpotent.
\item\label{th:finsim} There is only a finite number of nonisomorphic simple $\Lambda$-modules.
\item\label{th:leftnoe} $\Lambda$ is left noetherian.
\end{thmclaim}
\end{theorem}
\begin{proof}
\begin{description}
\item[\ref{th:leftnoe}] Since $\Lambda$ has a finite filtration...\par
\item[\ref{th:radnil}] Let $I$ be an ideal in $\Lambda$...\par
\item[\ref{th:finsim}] If for a $\Lambda$-module $A$ we have...
\end{description}
\end{proof}

\end{document}

And yet another variation changing also the ref key to automatically add "Proof of claim " (this can be redundant):

\documentclass{article}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{enumitem}

\newtheorem{theorem}{Theorem}

\newlist{thmclaim}{enumerate}{1}
\setlist[thmclaim,1]{label=\normalfont\textbf{\arabic*.},ref=Proof of claim~\arabic*.}

\begin{document}

\begin{theorem}
For a left artin ring $\Lambda$ we have the following.
\begin{thmclaim}
\item\label{th:radnil} The radical of $\Lambda$ is nilpotent.
\item\label{th:finsim} There is only a finite number of nonisomorphic simple $\Lambda$-modules.
\item\label{th:leftnoe} $\Lambda$ is left noetherian.
\end{thmclaim}
\end{theorem}
\begin{proof}
\begin{description}
\item[\ref{th:leftnoe}] Since $\Lambda$ has a finite filtration...\par
\item[\ref{th:radnil}] Let $I$ be an ideal in $\Lambda$...\par
\item[\ref{th:finsim}] If for a $\Lambda$-module $A$ we have...
\end{description}
\end{proof}

\end{document}