A custom theorem-like environment without \qed at the end (with the journal class file)

Inspired by the style of old French papers, see for instance Borel et Tits, Groupes réductifs, I use in my papers my custom theorem-like environment subsec:

\newtheorem{subsec}[equation]{} 

It works fine with the amsart class: 

\documentclass[a4paper,11pt]{amsart}    \numberwithin{equation}{section}    \theoremstyle{plain}       \newtheorem{thm}[equation]{Theorem}    \theoremstyle{definition}       \newtheorem{rem}[equation]{Remark}       \newtheorem{subsec}[equation]{} \begin{document} \section{Motivation} \label{s:motivation}    \begin{subsec}\label{ss:mod-Y}        Throughout the article, $k_0$ is a field of characteristic $0$,        and  $k$ is  a fixed algebraic closure of $k_0$.    \end{subsec}    \begin{rem}       We use the term ``$k_0$-model'' rather than ``$k_0$-form''       because the term ``form'' is overloaded.    \end{rem}    \begin{thm}       For any two  numbers $x,y\in{\mathbb R}$, we have       \begin{equation} x+y=y+x.\end{equation}    \end{thm}    \begin{proof} This is well known. \end{proof}    \begin{subsec}      A $k_0$-torus $T_0$ is called {\em split} if it is isomorphic to $({\mathbb G}_{m,k_0})^n$       for some natural number $n$, where ${\mathbb G}_{m,k_0}$       is the multiplicative group over $k_0$.    \end{subsec} \end{document} 

Now my paper has been accepted for publication in the International Mathematics Research Notices (IMRN), and I must prepare it with the journal class file oupau.cls. This class file can be downloaded from the journal webpage (click Tex template).See below the corresponding file, for which the only essential difference with the previous file is the first line \documentclass{oupau} instead of  \documentclass[a4paper,11pt]{amsart} :

\numberwithin{equation}{section}    \theoremstyle{definition}       \newtheorem{thm}[equation]{Theorem}       \newtheorem{rem}[equation]{Remark}       \newtheorem{subsec}[equation]{} \begin{document} \section{Motivation} \label{s:motivation}    \begin{subsec}\label{ss:mod-Y}       Throughout the article, $k_0$ is a field of characteristic $0$,       and  $k$ is  a fixed algebraic closure of $k_0$.    \end{subsec}    \begin{rem}       We use the term ``$k_0$-model'' rather than ``$k_0$-form''       because the term ``form'' is overloaded.    \end{rem}    \begin{thm}       For any two  numbers $x,y\in{\mathbb R}$, we have       \begin{equation} x+y=y+x.\end{equation}    \end{thm}    \begin{proof} This is well known. \end{proof}    \begin{subsec}       A $k_0$-torus $T_0$ is called {\em split} if it is isomorphic to $({\mathbb G}_{m,k_0})^n$      for some natural number $n$, where ${\mathbb G}_{m,k_0}$       is the multiplicative group over $k_0$.   \end{subsec} \end{document}``` See the result at the end of my question (I could not put it here...) As you can see, the ```oupau```` class puts a *solid square* at the end of the proof, and it puts an *empty square* ```\qed``` at the end of each theorem-like environment. In particular, each time when I write ```\end{subsec}``` I get a ```\qed``` symbol, which I don't need! > **Request.**  Please write for me a theorem-like environment  ```subsec``` that will not put a ```\qed```-symbol when I write ```\end{subsec}```. See below the relevant part (?) of the ```oupau``` class file. 

\DeclareRobustCommand{\qedthm}{%   \ifmmode \mathqed   \else     \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill     \quad\hbox{\qedsymbolthm}%   \fi } \def\@begintheorem#1#2[#3]{%   \pushQED{\qedthm}\deferred@thm@head{\the\thm@headfont \thm@indent     \@ifempty{#1}{\let\thmname\@gobble}{\let\thmname\@iden}%     \@ifempty{#2}{\let\thmnumber\@gobble}{\let\thmnumber\@iden}%     \@ifempty{#3}{\let\thmnote\@gobble}{\let\thmnote\@iden}%     \thm@swap\swappedhead\thmhead{#1}{#2}{#3}%     \the\thm@headpunct     \thmheadnl % possibly a newline.     \hskip\thm@headsep   }%   \ignorespaces} \def\@endtheorem{\popQED\endtrivlist\@endpefalse } % \theoremstyle{definition} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} % \newcommand{\filledbox}{\leavevmode   \hbox to.77778em{%   \hfil\vbox to.675em{\hrule width.6em height.6em}\hfil}} \renewcommand{\qedsymbol}{\filledbox} \renewenvironment{proof}[1][\proofname]{\par\noindent   \pushQED{\qed}%   \normalfont \topsep6\p@\@plus6\p@\relax   \trivlist   \item[\hskip\labelsep         \textbf     #1\@addpunct{.}]\ignorespaces }{%   \popQED\endtrivlist\@endpefalse 

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