How to create a proof environment with a different typeset name in addition to the original proof environment, in a non-English language?

Question

This post is a direct extension of this one. How can I define a proof environment with a different typeset name than "Proof" in a non-English document in addition to the original proof environment, rather than in lieu of it?

The language I am interested in is Hebrew, and the LaTeX language support I use is provided by the polyglossia package in tandem with the xelatex engine, but I'm also interested in the combination babel + lualatex.

Following is a LaTeX document, saved as ~/test.tex, as a starting point.

\documentclass{article}
\usepackage{amsthm}
\newtheorem{theorem}{משפט}

\usepackage{polyglossia}
\setmainlanguage{hebrew}
\setmainfont{Arial}
\begin{document}

Following is an interesting theorem.
\begin{theorem}
A house is not a home.
\end{theorem}
\begin{proof}
   A chair is still a chair, even when there's no one sitting there.
   But a chair is not a house, and a house is not a home.
\end{proof}

\end{document}

When the following instructions are executed in the Terminal:

> cd ~
> xelatex test

the file ~/test.pdf is created. When opened in a PDF viewer, the file displays as follows.

Note that Hebrew is a right-to-left language.

The desired solution would enable me to add the following to ~\test.tex immediately below the \end{proof} line:

\begin{explanation}
   A chair is still a chair, even when there's no one sitting there.
   But a chair is not a house, and a house is not a home.
\end{explanation}

with the resulting PDF file displaying like this:

Note that the desired output can be accomplished on an ad-hoc basis by adding the option [הסבר] to the proof environment, thus: \begin{proof}[הסבר] ... \end{proof}. However, I'm looking for a solution that would enable me to declare a new environment with a different name than proof.

Answer 1

You want to add to \captions<language>.

\documentclass{article}
\usepackage{polyglossia}
\usepackage{amsthm}

\setmainlanguage{hebrew}
\setotherlanguage{english}
\setmainfont{Arial}

\newtheorem{theorem}{\theoremname}
\newenvironment{explanation}
  {\let\proofname\explanationname\begin{proof}}%
  {\end{proof}}

\newcommand{\theoremname}{Theorem}% default
\newcommand{\explanationname}{Explanation}% default
\appto{\captionshebrew}{%
  \renewcommand\theoremname{משפט}%
  \renewcommand\explanationname{הסבר}%
}
\appto{\captionsenglish}{%
  \renewcommand\theoremname{Theorem}%
  \renewcommand\explanationname{Explanation}%
}

\begin{document}

Following is an interesting theorem.

\begin{theorem}
A house is not a home.
\end{theorem}
\begin{proof}
   A chair is still a chair, even when there's no one sitting there.
   But a chair is not a house, and a house is not a home.
\end{proof}
\begin{explanation}
   When there's no one there to hold you tight,
   and no one there you can kiss goodnight,
   a house is not a home.
\end{explanation}

\selectlanguage{english}

\begin{theorem}
A house is not a home.
\end{theorem}
\begin{proof}
   A chair is still a chair, even when there's no one sitting there.
   But a chair is not a house, and a house is not a home.
\end{proof}
\begin{explanation}
   When there's no one there to hold you tight,
   and no one there you can kiss goodnight,
   a house is not a home.
\end{explanation}

\end{document}

A slightly different approach, perhaps simpler.

\documentclass{article}
\usepackage{polyglossia}
\usepackage{amsthm}

\setmainlanguage{hebrew}
\setotherlanguage{english}
\setmainfont{Arial}

\newcommand{\theoremname}{Theorem}% default
\newcommand{\explanationname}{Explanation}% default
\appto{\captionshebrew}{%
  \renewcommand\theoremname{משפט}%
  \renewcommand\explanationname{הסבר}%
}
\appto{\captionsenglish}{%
  \renewcommand\theoremname{Theorem}%
  \renewcommand\explanationname{Explanation}%
}

\newtheorem{theorem}{\theoremname}

\NewDocumentEnvironment{explanation}{o}
 {\IfNoValueTF{#1}{\proof[\explanationname]}{\proof[#1]}}
 {\endproof}

\begin{document}

Following is an interesting theorem.

\begin{theorem}
A house is not a home.
\end{theorem}
\begin{proof}
   A chair is still a chair, even when there's no one sitting there.
   But a chair is not a house, and a house is not a home.
\end{proof}
\begin{explanation}
   When there's no one there to hold you tight,
   and no one there you can kiss goodnight,
   a house is not a home.
\end{explanation}
\begin{explanation}[הסבר נוסף]
   When there's no one there to hold you tight,
   and no one there you can kiss goodnight,
   a house is not a home.
\end{explanation}

\selectlanguage{english}

\begin{theorem}
A house is not a home.
\end{theorem}
\begin{proof}
   A chair is still a chair, even when there's no one sitting there.
   But a chair is not a house, and a house is not a home.
\end{proof}
\begin{explanation}
   When there's no one there to hold you tight,
   and no one there you can kiss goodnight,
   a house is not a home.
\end{explanation}
\begin{explanation}[Further explanation]
   When there's no one there to hold you tight,
   and no one there you can kiss goodnight,
   a house is not a home.
\end{explanation}

\end{document}

Answer 2

And for babel with either luatex or (with a minimal change) xetex:

\documentclass{article}

\usepackage{amsthm}
\newtheorem{theorem}{\theoremname}

\newenvironment{explanation}%
   {\renewcommand\proofname{\explanationname}%
    \begin{proof}}
   {\end{proof}}

% For LuaTeX. For XeTeX use bidi=bidi
\usepackage[bidi=basic, english, hebrew, provide=*]{babel}
\babelfont{rm}{FreeSans}

\setlocalecaption{hebrew}{theorem}{משפט}
\setlocalecaption{hebrew}{explanation}{הסבר}
\setlocalecaption{english}{theorem}{Theorem}
\setlocalecaption{english}{explanation}{Explanation}

\begin{document}

Following is an interesting theorem.
\begin{theorem}
  A house is not a home.
\end{theorem}
\begin{proof}
  A chair is still a chair, even when there's no one sitting there.
  But a chair is not a house, and a house is not a home.
\end{proof}
\begin{explanation}
  A chair is still a chair, even when there's no one sitting there.
  But a chair is not a house, and a house is not a home.
\end{explanation}

\selectlanguage{english}

Following is an interesting theorem.
\begin{theorem}
  A house is not a home.
\end{theorem}
\begin{proof}
  A chair is still a chair, even when there's no one sitting there.
  But a chair is not a house, and a house is not a home.
\end{proof}
\begin{explanation}
  A chair is still a chair, even when there's no one sitting there.
  But a chair is not a house, and a house is not a home.
\end{explanation}

\end{document}

Answer 3

The following solution is based on Mico's answer to my other question. However, this solution falls short for a bilingual document in which one wants the explanation environment to be language-sensitive the way the proof environment is language-sensitive.

\documentclass{article}
\usepackage{amsthm}
\newtheorem{theorem}{משפט}

% The following is based on Mico's answer here:
% https://tex.stackexchange.com/a/664972/21685
% BEGIN
\newcommand\explanationname{Explanation} % could be made language-dependent
\newenvironment{explanation}%
   {\begingroup
    \renewcommand\proofname\explanationname%
    \begin{proof}[הסבר]}%
   {\end{proof}%
    \endgroup}
% END

\usepackage{polyglossia}
\setmainlanguage{hebrew}
\setmainfont{Arial}
\begin{document}

Following is an interesting theorem.

\begin{theorem}
A house is not a home.
\end{theorem}
\begin{proof}
   A chair is still a chair, even when there's no one sitting there.
   But a chair is not a house, and a house is not a home.
\end{proof}
\begin{explanation}
   When there's no one there to hold you tight,
   and no one there you can kiss goodnight,
   a house is not a home.
\end{explanation}

\end{document}