Centralized label in multline mode

Question

I would like to know how I can copy the following lemma.

I tried the following code

   \begin{lemma}
   On any $t-$manifold there exist $\rho \in \mathcal{C}^{\infty}(X)$ such 
   that
   \begin{multline}\label{1.6.5}
   \rho>0 \hbox{ \  on \ } X^\circ, \rho=0  \hbox{ \  on \ } \partial X 
   \hbox{ \  and \ } \\
   \hbox{ \  in local coordinates at \ } p \in \partial X, \rho=a(x)x_1\dots 
   x_k, a(p)>0
   \end{multline}
   where $a$ is smooth.
   \end{lemma}

But the label stays above and the sentences are not centralized.

Answer 1

Labelling an equation in the middle across a number of lines can be achieved using an array/tabular structure, or gathered:

\documentclass[leqno]{report}

\usepackage{amsthm,amsmath}
\newtheorem{lemma}{Lemma}
\renewcommand{\thelemma}{\thesection.\arabic{lemma}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}

\begin{document}

\chapter{A chapter}
\section{A section}
\begin{lemma}
On any $t-$manifold there exist $\rho \in \mathcal{C}^{\infty}(X)$ such that
\begin{equation}
  \begin{gathered}
    \text{$\rho > 0$ on $X^\circ$, $\rho = 0$ on $\partial X$ and} \\
    \text{in local coordinates at $p \in \partial X$, $\rho = a(x) x_1 \dots x_k$, $a(p) > 0$}
  \end{gathered}
\end{equation}
where $a$ is smooth.
\end{lemma}

\begin{lemma}
On any $t-$manifold there exist $\rho \in \mathcal{C}^{\infty}(X)$ such that
\begin{equation}
  \begin{tabular}{c}
    $\rho > 0$ on $X^\circ$, $\rho = 0$ on $\partial X$ and \\
    in local coordinates at $p \in \partial X$, $\rho = a(x) x_1 \dots x_k$, $a(p) > 0$
  \end{tabular}
\end{equation}
where $a$ is smooth.
\end{lemma}

\begin{lemma}
On any $t-$manifold there exist $\rho \in \mathcal{C}^{\infty}(X)$ such that
\setlength{\abovedisplayskip}{0pt}
\setlength{\belowdisplayskip}{0pt}
\begin{equation}
  \begin{tabular}{c}
    $\rho > 0$ on $X^\circ$, $\rho = 0$ on $\partial X$ and \\
    in local coordinates at $p \in \partial X$, $\rho = a(x) x_1 \dots x_k$, $a(p) > 0$
  \end{tabular}
\end{equation}
where $a$ is smooth.
\end{lemma}

\end{document}

The final option (Lemma 1.1.3) provides some way of removing the introduced vertical spacing above/below an equation. This may be needed since the Lemma is written as a single paragraph/sentence (with some numbering).

Answer 2

Just for fun. BTW, centertags was not used here. I just threw it in as a reminder.

\documentclass{article}
\usepackage[leqno,centertags]{amsmath}
\newtheorem{lemma}{Lemma}

\begin{document}
\begin{lemma}
   On any $t-$manifold there exist $\rho \in \mathcal{C}^{\infty}(X)$ such 
   that
   \begin{equation}
   \begin{cases}
   \rho>0 & \text{on $X^\circ$} \\
   \rho=0 & \text{on  $\partial X$} \\
   \rho=a(x)x_1\dots &\text{in local coordinates at $p \in \partial X$, $x_k$, $a(p)>0$}
   \end{cases}
   \end{equation}
   where $a$ is smooth.
   \end{lemma}
\end{document}

Answer 3

I propose this layout with alignat:

\documentclass{book}
\usepackage[leqno]{mathtools}
\usepackage[amsmath, thmmarks]{ntheorem}
\theoremheaderfont{\scshape}
\theoremseparator{. }
\theorembodyfont{\itshape}
\newtheorem{lemma}{Lemma}[section]

\begin{document}
\setcounter{chapter}{1}
\setcounter{section}{6}
\setcounter{lemma}{1}

\begin{lemma}
  On any $t$-manifold there exist $ρ ∈ \mathcal{C}^{∞}(X)$ such
  that
  \begin{alignat}{3}
    \label{1.6.5}
     & & & \rho>0 \text{ on } X^\circ, & & \rho=0 \text{ on \ } ∂ X \hspace{6em}\notag \\
     & \text{and in }\rlap{local coordinates at $ p ∈ ∂ X $,}\hspace{5em} & & & \\
     & & & \rho=a(x)x₁ ... x_k, & \qquad & a(p)>0 \notag
  \end{alignat}
  where $a$ is smooth.
\end{lemma}

\end{document} 

Answer 4

A simple way of achieve a Lemma as I understood you want, could be this one:

\documentclass[leqno]{article} 
\usepackage[utf8]{inputenc} 
\usepackage{amsmath} 
\usepackage{ntheorem}

\newtheorem{lemma}{Lemma}

\begin{document}

\begin{lemma}    
On any $t-$manifold there exist $\rho \in \mathcal{C}^{\infty}(X)$ 
such that    
\begin{equation}    
\rho>0 \text{ on} X^\circ, \rho=0  \text{  on } \partial X \text{ and}       
\end{equation}    
in local coordinates at %
$p \in \partial X$, $\rho=a(x)x_1\dots x_k, a(p)>0$ %
where $a$ is smooth.    
\end{lemma}    
\end{document}

This is quite simpler than the Werner's answer. I understood that you wanted to replicate the aspect of the quoted page, in that sense I believe that the label in the middle of the text is not a problem per se, but depends on the page layout and other design decisions that are irrelevant as to how to raise a lemma. Of course you can do it in tabular form as Werner has shown you, my doubt is is it really necessary? I don't know, in my experience it's not, but there may be some context in which yes and then the Werner solution can be very useful to you. However, for the simplest thing, this may be enough.

I think the trick for this to work is to learn how to use the amsmath package that is very helpful in cases like this one, and either theorem or ntheorem packages, which are the ones that make it simpler to work with phrases like lemmas. In the case of John Kormylo's answer, amsmath is also key. Whatever the solution you prefer, I suggest you learn how to use that package.

Answer 5

Use the split environment inside the equation environ, as suggested by https://www.latex-project.org/help/documentation/amsldoc.pdf

\begin{equation}\begin{split}
   \rho>0 \hbox{ \  on \ } X^\circ, \rho=0  \hbox{ \  on \ } \partial X 
   \hbox{ \  and \ } \\
   \hbox{ \  in local coordinates at \ } p \in \partial X, \rho=a(x)x_1\dots 
   x_k, a(p)>0
\end{split}\end{equation}