Theorem and Proof In Latex 1

Latex documents include elements that require special formatting and numbering such as ( theorems, definitions, propositions, remarks, corollaries, lemmas, problems, examples, conjecture and so on ) in environment.

Minimal Working Example

\documentclass[12pt,letterpaper]{article}

\newtheorem{theorem}{Theorem}

\begin{document}

\section{Introduction}

Theorems can easily be defined:

\begin{theorem}

A definition is an exact and concise explanation of a single concept or word.

\end{theorem}

\end{document}

Numbered Theorems, Definitions, Corollaries and Lemmas

Numbered environments in LaTeX can be defined by means of the command \newtheorem :

\newtheorem{environment name}{Text}[parent counter]

\newtheorem{environment name}[shared counter]{Text}

1. Put command in the preamble.
2. The name/label {environment name} of the environment that is defined.
3. The name {environment name} you will use to reference inside a theorem-like environment \begin{environment name}[environment caption]
   \end{environment name}.
4. The word {Text/Printed Output} that will be printed, in boldface font, at the beginning of the environment.
5. The additional argument / optional argument [parent counter/sequential level/numberby/hierarchy name/outer counter] is the name of the sequential level (often by section/subsection/etc.) at which the numbering is to take place / the environment counter is reset every time parent counter is incremented.
6. The additional argument / optional argument [shared counter/sequential level/numberby/hierarchy name/inner counter] is the name of the sequential level at which the numbering is to take place / the environment counter is reset every time shared counter is incremented.

The statement of any theorem can then be enclosed between \begin{theorem} and \end{theorem}:

\begin{environment name}[environment caption]

\end{environment name}

1. Put command into the body of your document.
2. You can put a environment caption / theorem’s traditional name, date, discoverer/scientist, or the similar in square brackets. The environment caption prints at the beginning of the paragraph and inside Parentheses / round bracket.
3. The theorem will be labelled/titled and numbered by LaTeX, and the statement of the theorem with be automatically italicised.

Referencing Numbers

• To introduce cross-references, you can give the command \label{name} within the statement of a theorem, lemma, proposition or corollary in order assign the given name to that result. 
• To reference the number assigned to that result in some other theorem, one gives the command \ref{name}; this will be replaced by the number automatically assigned by LaTeX to that result when the LaTeX program is executed. 
• We recommend using consistent label names like eq:whatever and sec:whatever, but it is not required.
• The necessary space above and below the statement of the theorem will automatically be generated by LaTeX.

Example

\documentclass[12pt,letterpaper]{article}

\newtheorem{theorem}{Theorem}[section]

\newtheorem{corollary}{Corollary}[theorem]

\newtheorem{lemma}[theorem]{Lemma}

\begin{document}

\section{Introduction}

Theorems can easily be defined:

\begin{theorem}

Let \(f\) be a function whose derivative exists in every point, then \(f\) is 

a continuous function.

\end{theorem}

\begin{theorem}[Pythagorean theorem]

\label{pythagorean} % To introduce cross-references number

This is a theorem about right triangles and can be summarised in the next 

equation 

\[ x^2 + y^2 = z^2 \]

\end{theorem}

% to assign a referencing numbers

And a consequence of theorem \ref{pythagorean} is the statement in the next 

corollary.

\begin{corollary}

There's no right rectangle whose sides measure 3cm, 4cm, and 6cm.

\end{corollary}

You can reference theorems such as \ref{pythagorean} when a label is assigned.

\begin{lemma}

Given two line segments whose lengths are \(a\) and \(b\) respectively there is a 

real number \(r\) such that \(b=ra\).

\end{lemma}

Notes

1. By default, each theorem-like uses its own counter. However it is common for similar types of theorems (e.g. Theorems, Lemmas and Corollaries)  to share a counter.
2. for above example "Theorem 1.2" \label{pythagorean}refers to the 2nd theorem in the 1st section of a document. In this case, specify the theorem in the preamble as follows:

There are three new environments defined in the preamble.

1. \newtheorem{theorem}{Theorem}[section]
   This is the example presented in the introduction but it has the additional parameter [section] that restarts the theorem counter at every new section.
2. \newtheorem{corollary}{Corollary}[theorem]
   An environment called corollary is created, the counter of this new environment will be reset every time a new theorem environment is used.
3. \newtheorem{lemma}[theorem]{Lemma}
   In this case, the even though a new environment called lemma is created, it will use the same counter as the theorem environment. In this case, define subsequent theorems from {theorem}.

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Bibliography

http://www.mnstate.edu/ ( university )

http://www.ams.org/

https://www.tcd.ie/ ( university )

https://www.overleaf.com/

http://www.ntnu.no/ ( university )

https://en.wikibooks.org/

http://web.mit.edu/ ( university )

https://www.brown.edu/ ( university )

https://illinois.edu/ ( university )

https://www.impan.pl/ ( university )

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