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I have two questions. 1. Can I delete the red-colored space? 2. If I use enumerate in proof, QED symbol appears the next line of the end, as above. If I do not want to type manually \qedhere, should I use ntheorem?

\documentclass[a4paper]{article}
\usepackage[scale=0.75]{geometry}

\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{textcomp}
\usepackage{enumitem}
\setlist[enumerate]{nosep, label=(\arabic*)}

\theoremstyle{definition}
\newtheorem{lem}{Lemma}

\begin{document}

\begin{lem}
\begin{enumerate}
\item xxxx
\item yyyy
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}
\item xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. 
\item yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. \end{enumerate}
\end{proof}

\end{document}
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A low-tech solution to both problems is to not use the enumerate environment, but just type (1) and (2) manually and use paragraphs. –  Jeroen Zuiddam Mar 5 at 18:05

3 Answers 3

up vote 3 down vote accepted

amsthm already has the mechanism, called \qedhere:

\documentclass[a4paper]{article}

\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{textcomp}
\usepackage{enumitem}
\setlist[enumerate]{nosep, label=(\arabic*)}

\theoremstyle{definition}
\newtheorem{lem}{Lemma}

\begin{document}

\begin{lem}
\begin{enumerate}
\item xxxx
\item yyyy
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}

\item xxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx 
xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx 
xxxxxx.

\item yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy 
yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. 
yyyyy yyy yyyy. yyyyy yyy yyyy.\qedhere
\end{enumerate}

\end{proof}

\end{document}

You can use it also at the end of displayed equations.

enter image description here

If you want to remove the indent, just don't use enumerate.

\documentclass[a4paper]{article}

\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{textcomp}
\usepackage{enumitem}
\setlist[enumerate]{nosep, label=(\arabic*)}

\theoremstyle{definition}
\newtheorem{lem}{Lemma}

\begin{document}

\begin{lem}
\begin{enumerate}
\item xxxx
\item yyyy
\end{enumerate}
\end{lem}
\begin{proof}
(1) xxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx 
xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx xxxxxx. xxxxx xxx xxxxxx xxxxx 
xxxxxx.

(2) yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy 
yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. yyyyy yyy yyyy. 
yyyyy yyy yyyy. yyyyy yyy yyyy.
\end{proof}

\end{document}

enter image description here

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Here are the right enumitem parameters for the enumerate environments:

        \documentclass[a4paper]{article}
        \usepackage[scale=0.75]{geometry}

        \usepackage[english]{babel}
        \usepackage[utf8x]{inputenc}
        \usepackage{amsmath}
        \usepackage{amsthm}
        \usepackage{amssymb}
        \usepackage{textcomp}
        \usepackage{enumitem}
        \setlist[enumerate]{nosep, label=(\arabic*)}

        \theoremstyle{definition}
        \newtheorem{lem}{Lemma}

        \begin{document}

        \begin{lem}
        \begin{enumerate}[wide,labelindent = 0pt, labelwidth = ! ]
        \item xxxx
        \item yyyy
        \end{enumerate}
        \end{lem}
        \begin{proof}
        \begin{enumerate}[wide,labelindent = 0pt, labelwidth = ! ]
        \item xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxxxxxxx.
        \item yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. \end{enumerate}
        \end{proof}

        \end{document} 

enter image description here

If you want the labels to be indented, don't write anything about labelindent: with the wide parameter, it is set by default to labelindent=parindent`, and the parameter given here simply overwrites this value.

As for the theorem symbol, ntheorem handles it automatically(I used the standard set of theorems, but it can be entirely customised):

        \documentclass[a4paper]{article}
        \usepackage[showframe, nomarginpar,  noheadfoot]{geometry}

        \usepackage[english]{babel}
        \usepackage[utf8x]{inputenc}
        \usepackage{amsmath}
        %\usepackage{amsthm}
        \usepackage[standard, thmmarks, amsmath]{ntheorem}
        \usepackage{enumitem}
        \setlist[enumerate]{nosep, label=(\arabic*)}

        \theoremstyle{definition}
        \newtheorem{lem}{Lemma}

        \begin{document}

        \begin{Lemma}
        \begin{enumerate}[wide,labelindent = 0pt, labelwidth = ! ]
        \item xxxx
        \item yyyy
        \end{enumerate}
        \end{Lemma}

        \begin{Proof}
        \begin{enumerate}[wide,labelindent = 0pt, labelwidth = ! ]
        \item xxxxxxxxxxxxxxxxxxxxxxxxx. xxxxxxxxxxxxxxxxxxxx xxxxx. xxxxx xxxxxx xxxx xx xxxxx xxx. xxxxx xxxxxxx xxx xxx xxx xxxx. xxxxx xxxxx xxxx xxxx xxxxxxx. xxxx xxxx xxxxxxx xxxxx xxxxx.
        \item yyyyy yyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyy yyyyyyy. yyyyyyyy yyyy. yyyyyy yyyyyy. yyyyy yyyyyyy. yyyyy yyy yyyy. yyyyyy yyyyyy. yyyyyyyyyyyy. yyyyyyyyyyyy. yyyyy yyyyyyy. \end{enumerate}

        \end{Proof}

        \end{document} 

enter image description here

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In such cases I use \qedhere at the end of the environment. And to avoid the awkward Proof: (1) Bla bla bla line, I'd write a short introduction as to why the proof is split into cases. Your gentle reader will thank you.

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