Theorem and Proof In Latex 2

Unnumbered Theorem-Like Environments

To add remarks, comments, note environment or examples to a mathematical document. The amsthm package provides this functionality. It's defined using the starred version of \newtheorem. 

Example:

Suppose I want to have an unnumbered remark environment, I can define the environment like this:

\documentclass[12pt,letterpaper]{article}

% in the preamble:

\newtheorem*{note}{Note}

\begin{document}

% later in the body document:

\begin{note}

This is a note about something.

\end{note}

\end{document}

Proof Environments

Proofs are the core of mathematical papers and books and it is customary to keep them visually apart from the normal text in the document. The amsthm package provides the environment proof for this.

The word Proof is italicized and there is some extra spacing, also a special symbol is used to mark the end of the proof. This symbol ( QED ) can be easily changed, to learn how see the next section.

\documentclass{article}

\usepackage[english]{babel}

\usepackage{amsthm}

\begin{document}

\section{Introduction}

\begin{proof}

Here is some text for the proof.

\end{proof}

\end{document}

Proof Environment With Math Equation

Occasionally you will want to end a proof with an equation; it’s generally a recommended style to the symbol (QED) will appear on a same line by itself, like:

\documentclass{article}

\usepackage[english]{babel}

\usepackage{amsmath}

\usepackage{amsthm}

% use math equation inside proof theorem environment

\begin{proof}

\begin{equation}       % instead of $

f(x)=y\tag*{\qedhere}  % the \tag*{} is new 

\end{equation}         % instead of $

\end{proof}

\end{document}

Changing the QED symbol

The symbol printed at the end of a proof is called the “QED symbol / halmos ”, named after the Hungarian mathematician Paul Halmos, who was renowned for his expository skills. To quote the meaning of QED from Wikipedia:

QED is an initialism of the Latin phrase quod erat demonstrandum, meaning “thus it has been demonstrated”.

It is straightforward to use a symbol, or wording, of your choice to represent the QED symbol. The LaTeX command:

\renewcommand\qedsymbol{$\blacksquare$}

can be used to replace the default white square for a black square printed by $\blacksquare$, the parameter inside the braces. 

Theorem styles

The package amsthm provide special commands to accomplish this.

Here is a list of the possible pre-defined Theorem styles:

• definition boldface title, Roman body. Commonly used in definitions, conditions, problems, and examples.
  Sample appearance: Definition 2. Definition text.
• plain boldface title, italicized body. Commonly used in theorems, lemmas, corollaries, propositions and conjectures. (default).
  Sample appearance: Theorem 1. Theorem text.
• remark italicized title, Roman body. Commonly used in remarks, notes, annotations, claims, cases, acknowledgments and conclusions.
  Sample appearance: Remark 3. Remark text.
  

Notes: 

• If you are using either packages with amsmath, you must load amsmath first, example:
  \usepackage{amsmath}
  \usepackage{amsthm}
• General LaTeX guidelines, Avoid manual formatting, use instead appropriate LaTeX constructs.

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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 )

https://uwaterloo.ca/ ( university )

latex document typeset, typesetting