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\author{Nobody}
\university{National Academy}

\begin{document}

\begin{problem}
Let $A,B$ be in $\mathcal{M}$. Find $A+B$.
\end{problem}

\begin{remark}
 It is an example of a problem for ISTCiM.
\end{remark}

\begin{solution}
First we need the following lemma:

\begin{lemma}\label{LEMMA} Let $X,Y\in \mathcal{M}$. 
\par Then $X+Y\in\mathcal{M}$.
\end{lemma}

\begin{proof}
It is obvious. Even when it is false.
\end{proof}
The claim is the direct application of lemma \ref{LEMMA}.
\end{solution}

\begin{solution}[ n.2]
We will prove the following claim, which is a more general fact and leads directly to the solution of the problem.

\begin{theorem}[(More general fact)] \label{THEOREM}
 Let $A_1,A_2,A_3,\dots\in \mathcal{M}$.\par 
 Then $\sum\limits_iA_i\in\mathcal{M}$.
\end{theorem}

\begin{proof} 
 It is enough to show that the thesis of the problem is trivially obvious. 
\end{proof}

\end{solution}


\end{document}