%%% This is a file to test some environments like theorem+proof, lemma+proof and claims

\documentclass[thesis]{subfiles}

\begin{document}
\newpage % Use newpage to start on a new page
\section{Theorems and proofs}
\label{sec:theorems} %Use this label to refer to this section, this will appear in another subfile as ?? but if you compile the main file (thesis.tex) it will be a correct reference to this section.

\subsection{Not my theorem}
\begin{theorem}[Mine] % The brackets are usefull for adding stuff like (Pythagoras) for his theorem).
The unit sphere is not compact in $l^2$.
\begin{proof} 
The sequence $(1,0,0,0,...),(0,1,0,0,...),(0,0,1,0,...),...$ does not have a converging subsequence so it is not sequentially compact hence not compact.
\end{proof} 
\end{theorem}

\thm{This is a shorter theorem.}{Look at the source code.} % Using \thm instead of environments

\begin{theorem}
Adding a [title] is not necessary.
\begin{proof}
Proof by lack of contradicition.
\end{proof}
\end{theorem}

% here we use one of our custom commands:
\lm{This lemma keeps the same numbering as the theorems.}{Proof by picture: look at the number.}

%This is an example of a lemma with a proof, further in the text.
\begin{lemma}
The proof of this lemma will come later, refer to it using the label given in the source code.
\label{lm.laterproof} % Note the \label that is used.
\end{lemma}


\subsection{More numbering}

\begin{lemma}
We now see that the $2^{nd}$ number increased but the last number went back to 1.
\begin{proof}
\clm{The $2^{nd}$ number increased.}
We use a claim here, using the `\textbackslash clm' command.
\clm{The last number went back to one.}
\end{proof}
\end{lemma}

\vspace{0.5cm}
\fbox{\begin{minipage}{0.99\textwidth}
\thm{This Theorem is very important!}{Although the equation
$$\boxed{E=mc^2}$$ is very important, this Theorem is even more important because it has a bigger box and more space around it!}
\end{minipage}}
\vspace{0.5cm}

\begin{remark}
Whenever we make a claim in a proof, the claim counter starts back at one.
\begin{proof}
Proof by picture:
\clm{Here we see a one.}
\end{proof}
\end{remark}

\clm{This is a maintext claim. Now the next claim after this will start with a two.}

\begin{remark}
As a test, we do a proof with claim:
\begin{proof}
Proof by picture:
\clm{Here we see a two.}
\end{proof}
\end{remark}

\begin{remark}
The effect only works for one proof:
\begin{proof}
Proof by picture:
\clm{Here we see a one.}
\end{proof}
\end{remark}

We did not prove the earlier lemma yet, so let's do that now.
% Here we prove the lemma we made earlier but didn't prove. Using a label and reference command is VERY much advised.
\begin{proof}[Proof of lemma \ref{lm.laterproof}]
Here be the proof of thy lemma.
\end{proof}

\end{document}