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>% + /X<>% + }% + /U<>% + /PC<>% + /PO<>% + }% + }% + >>% + }% + }{{\color{#5}#6}}% + \sbox\tiptext{% + \IfBooleanT{#2}{% + \ocgbase@oc@bdc{\tpTipOcg}\ocgbase@open@stack@push{\tpTipOcg}}% + %\fcolorbox{black}{#7}{#8}% + \tcbox[colframe=black,colback=#7,size=fbox,arc=1ex,sharp corners=southwest]{#8}% + \IfBooleanT{#2}{\ocgbase@oc@emc\ocgbase@open@stack@pop\tpNull}% + }% + \cListSet\tpOffsets{#9}% + \edef\twd{\the\wd\tiptext}% + \edef\tht{\the\ht\tiptext}% + \edef\tdp{\the\dp\tiptext}% + \tipshift=0pt% + \IfBooleanTF{#2}{% + %OCG-based (that is, all non-draggable) boxes should not extend beyond the + %current column as they may get overlaid by text in the neighbouring column + \setlength\whatsleft{\linegoal}% + }{% + \measureremainder{\whatsleft}% + }% + \ifdim\whatsleft<\dimexpr\twd+\cListItem\tpOffsets{1}\relax% + \setlength\tipshift{\whatsleft-\twd-\cListItem\tpOffsets{1}}\fi% + \IfBooleanF{#2}{\tpPdfXform{\tiptext}}% + \raisebox{\heightof{#6}+\tdp+\cListItem\tpOffsets{2}}[0pt][0pt]{% + \makebox[0pt][l]{\hspace{\dimexpr\tipshift+\cListItem\tpOffsets{1}\relax}% + \IfBooleanTF{#2}{\usebox{\tiptext}}{% + \tpPdfAnnot{\twd}{\tht}{\tdp}{% + /Subtype/Widget/FT/Btn/T (tip.\thetcnt)% + /AP<>% + /MK<>>>% + /Ff 65536/F 3% + /AA <<% + /U <<% + /S/JavaScript/JS(% + var fd=event.target;% + var mX=this.mouseX;var mY=this.mouseY;% + var drag=function(){% + var nX=this.mouseX;var nY=this.mouseY;% + var dX=nX-mX;var dY=nY-mY;% + var fdr=fd.rect;% + fdr[0]+=dX;fdr[1]+=dY;fdr[2]+=dX;fdr[3]+=dY;% + fd.rect=fdr;mX=nX;mY=nY;% + };% + if(!dragging\thetcnt){% + dragging\thetcnt=true;Int=app.setInterval("drag()",1);% + }% + else{app.clearInterval(Int);dragging\thetcnt=false;}% + this.dirty=false;% + )% + >>% + >>% + }% + \tpAppendToFields{\tpPdfLastAnn}% + }% + }}% + \stepcounter{tcnt}% +}} +\makeatother +\newsavebox\tiptext\newcounter{tcnt} +\newlength{\whatsleft}\newlength{\tipshift} +\newcommand{\measureremainder}[1]{% + \begin{tikzpicture}[overlay,remember picture] + \path let \p0 = (0,0), \p1 = (current page.east) in + [/utils/exec={\pgfmathsetlength#1{\x1-\x0}\global#1=#1}]; + \end{tikzpicture}% +} +\usepackage[utf8]{inputenc} +\usepackage[english]{babel} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{hyperref} +\DeclareMathOperator{\Ab}{Ab} +\DeclareMathOperator{\MuC}{Mu} + + + +\newcommand{\gdotG}[1]{\cdot_{#1}} +\newcommand{\gdot}[1]{\mbox{\tooltip{$\cdot$}{$\cdot_{#1}$}}} +\newcommand{\sGC}[1]{\vert #1 \vert} +\newcommand{\sG}[1]{\mbox{\tooltip{$#1$}{$|#1|$}}} +\newcommand{\inv}[2]{#1^{-1}} +\newcommand{\invC}[2]{#1^{-1}_{#2}} +\newcommand{\e}[1]{\mbox{\tooltip{$\mathbf{e}$}{$\eC{#1}$}}} +\newcommand{\eC}[1]{\mathbf{e}_{#1}} +%RINGS +\newcommand{\sRC}[1]{\vert #1 \vert} + +\newcommand{\tR}[1]{\mbox{\tooltip{$\cdot$}{$\cdot_{#1}$}}} +\newcommand{\pR}[1]{\mbox{\tooltip{$+$}{$+_{#1}$}}} +\newcommand{\sR}[1]{\mbox{\tooltip{$#1$}{$|#1|$}}} +\newcommand{\Mu}[1]{\mbox{\tooltip{$#1$}{$\MuC(#1)$}}} +%\newcommand{\pRC}[1]{+_{#1}} +%\newcommand{\tRC}[1]{\cdot_{#1}} +\newcommand{\mR}[1]{\mbox{\tooltip{-}{$-_{#1}$}}} +\newcommand{\zR}[1]{\mbox{\tooltip{0}{$0_{#1}$}}} +%\newcommand{\zR}[1]{0_{#1}} + +\newcommand{\naproche}{$\mathbb{N}$aproche } +\newcommand{\oR}[1]{\mbox{\tooltip{1}{$1_{#1}$}}} +%\newcommand{\oRC}[1]{1_{#1}} +%Ordnungen +\newcommand{\rest}[2]{#1 \vert_{#2}} +\newcommand{\oC}[1]{#1} +\newcommand{\oO}[1]{\mbox{\tooltip{$\le$}{$#1$}}} +\newcommand{\oBC}[1]{\vert #1 \vert} +\newcommand{\oB}[1]{\mbox{\tooltip{$#1$}{$|#1|$}}} +\newcommand{\Min}[1]{\mathcal{M}(#1)} +\newcommand{\cM}{\mathcal{M}} + + +%ganze zahlen +\newcommand{\Z}{\mathbb{Z}} +\newcommand{\lez}{\le_{\Z}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\cP}{\mathbb{P}} + +\newcommand{\NPOS}{\mathbb{N}_{>1}} + +%\newcommand{\ooDIV}{} +\newcommand{\NDIV}{\vert_{>1}} +\newcommand{\DIVI}[1]{\mbox{\tooltip{$|$}{$|_{#1}$}}} +\newcommand{\MoPos}{\mathbb{N}_{>0}} + + +\title{Euclids Theorem} +\author{Tim Lichtnau} +\date{2021} + +\usepackage{../../../../../lib/tex/naproche} +\newtheorem*{definition}{Definition} +\newtheorem{lemma}{Lemma} +\newtheorem*{theorem}{Theorem} +\newtheorem{corollary}{Corollary} +\begin{document} +\maketitle +\begin{abstract} +\noindent We present a formalization of Euclid's Theorem from an algebraic and order-theoretic approach. The \texttt{PDF}-document contains some feature which allow us to unhide suppressed information visible via a mouseover event. At this time, this extra works not for all \texttt{PDF}-viewers, but surely for \\ +\texttt{Adobe Reader} or \texttt{ Foxit Reader}. +\end{abstract} +\tableofcontents + +\section{Sets and classes} +\begin{forthel} + [synonym subset/-s] \\ + Let $X$ denote a set. + \begin{definition} + A subset of $X$ is a set $Y$ such that every element of $Y$ is an element of $X$. + \end{definition} + \begin{axiom} + Let Y be a class. Assume every element of $Y$ is an element of $X$. Then $Y$ is a set. + \end{axiom} + \begin{axiom} + Every element of every set is setsized. + \end{axiom} + +\end{forthel} +\section{Order relations} +\begin{forthel} + Let $Y \subseteq X$ stand for $Y$ is a subset of $X$. + \begin{definition} + $X$ is nonempty iff there exists an element of $X$. + \end{definition} + \begin{signature} + An order is a notion. + \end{signature} + Let $O$ denote a order. + \begin{signature} + $\oBC{O}$ is a set. + \end{signature} + Let $\oB{O}$ stand for $\oBC{O}$. + + \begin{signature} + Let $x,y \in \oB{O}$. $x \oC{O} y$ is an atom. + \end{signature} + Let $x \oO{O} y$ stand for $x \oC{O} y$. +\end{forthel} +\begin{forthel} + \begin{definition} + An order on $X$ is a order $O$ such that $\oBC{O} = X$. + \end{definition} + \begin{definition} + Let $N \subseteq \oB{O}$. $O$ restricted to $N$ is an order $T$ on $N$ such that + ($x \oC{T} y$ iff $x \oC{O} y$) for all $x,y \in N$. + \end{definition} + Let $\rest{O}{N}$ stand for $O$ restricted to $N$. + \begin{definition} + A suborder of $O$ is an order $T$ such that $\oBC{T} \subseteq \oBC{O}$ and $T = \rest{O}{\oBC{T}}$. + \end{definition} + \begin{definition} + Let $N \subseteq \oB{O}$. An upper bound of $N$ by $O$ is an element $x$ of $\oB{O}$ such that $n \oO{O} x$ for all + $n \in N$. + \end{definition} +\end{forthel} +\subsection{Partial orders} +\begin{forthel} + \begin{definition} + $O$ is reflexive iff $x \oO{O} x$ for any $x \in \oB{O}$. + \end{definition} + + \begin{definition} + $O$ is antisymmetric iff \\ + $x \oO{O} y \oO{O} x$ $=>$ $x =y$ for any $x,y \in \oB{O}$. + \end{definition} + \begin{definition} + $O$ is transitive iff \\ + $x \oO{O} y \oO{O} z$ $=>$ $x \oO{O} z$ for any $x,y,z \in \oB{O}$. + \end{definition} + \begin{definition} + $O$ is partial iff $O$ is reflexive and antisymmetric and transitive. + \end{definition} + \begin{lemma} + Let $O$ be a partial order. Let $X \subseteq \oBC{O}$. $\rest{O}{X}$ is a partial order. + \end{lemma} +\end{forthel} + + +\section{Monoids} +\begin{forthel} + + %[unfoldlow on] + + \begin{signature} + A magma is a notion. + \end{signature} + Let M,G denote magma. + + \begin{signature} + $\sGC{M}$ is a set. + \end{signature} + + + \begin{signature} + Let $x,y \in \sGC{M}$. $x \gdotG{M} y$ is an element of $\sGC{M}$. + \end{signature} + Let $\sG{M}$ stand for $\sGC{M}$. + Let $x \gdot{M} y$ stand for $x \gdotG{M} y$. + + [synonym element/-s] [synonym inverse/-s] + %\begin{lemma} + % Let $x,y \in \s{M}$. Then $y \gdot{M} x \in \s{M}$. + %\end{lemma} + \begin{definition} + Let $X$ be a set. $M$ is based on $X$ iff $\sGC{M} = X$. + \end{definition} + \begin{definition} + $G$ is associative iff $(x \gdot{G} y) \gdot{G} z = x \gdot{G} (y \gdot{G} z)$ for all $x,y,z \in \sG{G}$. + \end{definition} + + \begin{definition} + $G$ is abelian iff $x\gdot{G}y = y\gdot{G}x$ for all $x,y \in \sG{G}$. + \end{definition} + + \begin{definition} + %Let $e \in \sG{G}$. $e$ is neutral in $G$ + A neutral element of $G$ is an element e of $\sG{G}$ such that $(e\gdot{G}x = x$ and $x = x \gdot{G} e)$ for all $x \in \sG{G}$. + \end{definition} + + + \begin{lemma}[uniqNeut] + % Let $e,e' \in \sG{G}$. Assume $e$ and $e'$ are neutral in $G$. + Let $e$ and $e'$ be neutral elements of $G$. + Then $e = e'$. + \end{lemma} + \begin{proof} + $e' = e \gdot{G} e' = e$. + \end{proof} + + + \begin{definition} + A Monoid is an associative Magma with a neutral element. %$e \in \sG{G}$ such that $e$ is neutral in $G$. + \end{definition} + Let $M$ denote a Monoid. + + \begin{definition} + Let $x,y \in \sG{M}$. $x$ divides $y$ in $M$ iff there exists $k \in \sG{M}$ such that $k \gdot{M} x = y$. + \end{definition} + \begin{lemma}[transitiveDiv] + Let $k,m,n \in \sG{M}$. Suppose $n$ divides $m$ in $M$ and + $m$ divides $k$ in $M$. Then $n$ divides $k$ in $M$. + \end{lemma} + \begin{proof} + Take an $l \in \sG{M}$ such that $ l \gdot{M} n = m$. \\ + Take an $p \in \sG{M}$ such that $p \gdot{M} m = k$. \\ + Then $k = p \gdot{M} m = p$ $\gdot{M} (l \gdot{M} n) = (p \gdot{M} l) \gdot{M} n $. + Thus $n$ divides $k$ in $M$. + \end{proof} +\end{forthel} + + +%\subsection{Die Teilbarkeitsrelation} +\subsection{Divisibility theory} +\begin{forthel} + Let M denote a monoid. + \begin{definition} + $\DIVI{M}$ is an order on $\sG{M}$ such that for any $x,y \in \sG{M}$ we have + $x \oC{\DIVI{M}} y$ iff $x$ divides $y$ in $M$ . + \end{definition} + \begin{lemma} + $\DIVI{M}$ is reflexive and transitive. + \end{lemma} + \begin{proof} + $\DIVI{M}$ is reflexive. Indeed $x$ divides $x$ in $M$ for all $x \in \sG{M}$. \\ + $\DIVI{M}$ is transitive (by transitiveDiv). + \end{proof} +\end{forthel} +\subsection{Finiteness} +\begin{forthel} + Let X denote a set. + + \begin{signature} + X is finite is an atom. + \end{signature} + \begin{axiom}[FiniteMultiplication] + Let M be an abelian Monoid. Let $X$ be a finite subset of $\sG{M}$. Then $X$ has an upper bound by $\DIVI{M}$. + \end{axiom} + +\end{forthel} +\subsection{Groups} +\begin{forthel} + \begin{signature} + $\eC{M}$ is a neutral element of $M$. %$$e$ of $\sG{M}$ such that $e$ is neutral in $M$. + \end{signature} + \begin{definition} + A submonoid of $M$ is a monoid $N$ such that + $\sG{N} \subseteq \sG{M}$ and $\eC{N} = \eC{M}$ and ($x \gdotG{N} y = x \gdotG{M} y$) for any $x,y \in \sG{N}$. + \end{definition} + Let $\e{M}$ stand for $\eC{M}$. + \begin{definition} + Let $x \in \sG{M}$. An inverse of $x$ in $M$ is an element $y$ of $\sG{M}$ such that $x \gdot{M} y = \eC{M}$ and $y \gdot{M} x = \eC{M}$. %$ is a neutral element of $G$ and $y\gdot{G}x$ is neutral in $G$. + \end{definition} + + + + \begin{lemma}[uniqInv] + Let $M$ be a Monoid. Let $x,y,y' \in \sG{M}$. + Assume $y$ and $y'$ are inverses of $x$ in $M$. Then $y = y'$. + \end{lemma} + \begin{proof} + $y = \e{M} \gdot{M} y = (y' \gdot{M} x) \gdot{M} y = y' \gdot{M} (x \gdot{M} y) = y' \gdot{M} \e{M} = y'$. + \end{proof} + \begin{definition} + A Group is a Monoid $G$ such that every element of $\sG{G}$ has an inverse in $G$. + \end{definition} + + Let $G$ denote a Group. + + + \begin{signature} + Let $x \in \sG{G}$. + $\invC{x}{G}$ is an inverse of $x$ in $G$. + \end{signature} +\end{forthel} + +\section{Rings} +\begin{forthel} + [synonym ring/-s] + + [synonym divisor/-s] [synonym unit/-s] + \begin{signature} + A Ring is a notion. + \end{signature} + + Let $R$ denote a ring. + \begin{signature} + $\sRC{R}$ is a set. + \end{signature} + Let $\sR{R}$ stand for $\sRC{R}$. + \begin{signature} + $\Ab(R)$ is an abelian group based on $\sR{R}$. + \end{signature} + + \begin{signature} + $\MuC(R)$ is a Monoid based on $\sR{R}$. + \end{signature} + + Let $x \pR{R} y$ stand for $x \gdotG{\Ab(R)} y$. + Let $x \tR{R} y$ stand for $x \gdotG{\MuC(R)} y$. + Let $R$ is commutative stand for $\MuC(R)$ is abelian. + + \begin{definition} + $\zR{R} = \eC{\Ab(R)}$. + \end{definition} + \begin{definition} + $\oR{R} = \eC{\MuC(R)}$. + \end{definition} + + \begin{axiom}[DistribI] + Let $x,y,z \in \sR{R}.$ $x \tR{R} (y \pR{R} z) = (x \tR{R} y) \pR{R} (x \tR{R} z)$. + \end{axiom} + \begin{axiom}[DistribII] + Let $x,y,z \in \sR{R}.$ $(x \pR{R} y) \tR{R} z = (x \tR{R} z) \pR{R} (y \tR{R} z)$. + \end{axiom} + \begin{definition} + Let $x \in \sR{R}$. $\mR{R}x = \invC{x}{\Ab(R)}$. + \end{definition} + + + Let $x \mR{R} y$ stand for $x \pR{R} (\mR{R}y)$. + + + \begin{lemma} + Let $x \in \sR{R}$. $\zR{R} \tR{R} x = \zR{R}$. + \end{lemma} + \begin{proof} + + $\zR{R} = (\zR{R} \tR{R} x) \mR{R} (\zR{R} \tR{R} x) \\ + = ((\zR{R} \pR{R} \zR{R}) \tR{R} x) \mR{R} (\zR{R} \tR{R} x) \\ + = ((\zR{R} \tR{R} x) \pR{R} (\zR{R} \tR{R} x)) \mR{R} (\zR{R} \tR{R} x) \\ + = (\zR{R} \tR{R} x) \pR{R} ((\zR{R} \tR{R} x) \mR{R} (\zR{R} \tR{R} x)) \\ + = (\zR{R} \tR{R} x) \pR{R} \zR{R}. $ + \end{proof} + + + + \begin{lemma}[Minus] + Let $k,q \in \sR{R}.$ Then $\mR{R}(k \tR{R} q)= (\mR{R}k) \tR{R} q$. + \end{lemma} + \begin{proof} + $(k \tR{R} q) \pR{R} ((\mR{R}k) \tR{R} q) = (k \mR{R}k) \tR{R} q = \zR{R} \tR{R} q = \zR{R}$. \\ + $(k \tR{R} q)$ is an inverse of $((\mR{R}k) \tR{R} q)$ in $\Ab(R)$. + \end{proof} + + + \begin{lemma}[MDistrib] + Let $x,y,z \in \sR{R}$. $(x \mR{R} y) \tR{R} z = (x \tR{R} z) \mR{R} (y \tR{R} z)$. + \end{lemma} + \begin{proof} + We have + $(x \pR{R} (\mR{R} y)) \tR{R} z = (x \tR{R} z) \pR{R} ((\mR{R} y) \tR{R} z)$ (by DistribII). \\ + $ (x \tR{R} z) \pR{R} ((\mR{R} y) \tR{R} z) = (x \tR{R} z) \mR{R} (y \tR{R} z)$ (by Minus). + \end{proof} + +\end{forthel} +%############ Teilbarkeitstheorie ################################ + +\begin{forthel} + + Let $R$ denote a commutative ring. + Let $\Mu{R}$ stand for $\MuC(R)$. + %Let $x$ divides $y$ in $R$ stand for $x$ divides $y$ in $\Mu{R}$. + %\begin{definition} + % Let $x \in \sR{R}$. A divisor of $x$ in $R$ is an element $y$ of $\sR{R}$ such that $y$ divides $x$ in $\Mu{R}$. + %\end{definition} + + \begin{lemma}[divDif] + Let $k,m,n \in \sR{R}$. Assume $k$ divides $m$ in $\Mu{R}$ and $k$ divides $n$ in $\Mu{R}$. Then $k$ divides $m \mR{R} n$ in $\Mu{R}$. + \end{lemma} + \begin{proof} + Take $x \in \sR{R}$ such that $x \tR{R} k = m$. \\ + Take $y \in \sR{R}$ such that $y \tR{R} k = n$. + $(x \mR{R} y) \tR{R} k = m \mR{R} n$ (by MDistrib). + \end{proof} +\end{forthel} + + +\section{Wellfounded orders} +\begin{forthel} + \begin{definition} + Let $N \subseteq \oBC{O}$. + A minimum of $N$ with $O$ is an element $x$ of $N$ such that + $y \oO{O} x$ $=>$ $x = y$ for all $y \in N$ . + \end{definition} + + \begin{definition} + O is wellfounded iff O is a partial order and for all nonempty subsets $S$ of $\oBC{O}$ + there exists a minimum of $S$ with $O$. + \end{definition} + + [synonym predecessor/-s] + \begin{definition} + A minimum of $O$ is a minimum of $\oBC{O}$ with $O$. + \end{definition} + \begin{definition} + $\Min{O}$ = \{$m \in \oB{O}$ $|$ $m$ is a minimum of $O$ \}. + \end{definition} + + \begin{definition} + Let $x \in \oB{O}$. A predecessor of $x$ by $O$ is an element $y$ of $\oB{O}$ such that $y \oO{O} x$. + \end{definition} + + \begin{lemma} + Let $O$ be a wellfounded order. Let $x \in \oB{O}$. + Then there exists a predecessor $y$ of $x$ by $O$ such that $y$ is a minimum of $O$. + \end{lemma} + \begin{proof} + Define $X$ = \{$z \in \oB{O}$ $|$ $z \oO{O} x$\}. $X \subseteq \oB{O}$. + Take a minimum $z$ of $X$ with $O$. $z$ is a minimum of $O$. + \end{proof} + \begin{definition} + Let $x \in \oB{O}$. A stranger of $x$ in $O$ is an element $y$ of $\oB{O}$ such that + $x$ and $y$ have no common predecessors by $O$. + \end{definition} + \begin{theorem}[StrangerTheorem] + Let $O$ be a wellfounded order. Assume every element of $\oB{O}$ has a stranger in $O$. + Then $\Min{O}$ has no upper bound by $O$. + \end{theorem} +\end{forthel} +\section{The ring of integers} +\begin{forthel} + [synonym number/-s] + \begin{signature} + $\Z$ is a commutative ring. + \end{signature} + + \begin{definition} + $\MoPos$ is a submonoid of $\MuC(\Z)$. + \end{definition} + \begin{definition} + A positive number is an element of $\sG{\MoPos}$. + \end{definition} + Let $n,m$ denote positive numbers. + + \begin{lemma} + $\oR{\Z}$ is a positive number. + \end{lemma} + + \begin{axiom}[MultEquiv] + Assume $n$ divides $m$ in $\Mu{\Z}$ and $m$ divides $n$ in $\Mu{\Z}$. Then $n$ = $m$. + \end{axiom} + + \begin{lemma} + $\DIVI{\MoPos}$ is a partial order. + \end{lemma} + \begin{proof} + $\DIVI{\MoPos}$ is antisymmetric (by MultEquiv). + Indeed (if $x \oC{\DIVI{\MoPos}} y$ then $x$ divides $y$ in $\Mu{\Z}$) for all $x,y \in \sG{\MoPos}$. \\ + \end{proof} + Let $n$ is nontrivial stand for $n \neq \oR{\Z}$. + \begin{definition} + $\NPOS$ = \{$x \in \sG{\MoPos}$ $|$ $x$ is nontrivial \}. + \end{definition} + \begin{axiom} + $n \pR{\Z} m$ is a nontrivial positive number. + \end{axiom} + \begin{axiom} + Let $S$ be a nonempty subset of $\NPOS$. Then there exists a minimum of $S$ with $\DIVI{\MoPos}$. + \end{axiom} + \begin{definition} + $\NDIV$ is $\DIVI{\MoPos}$ restricted to $\NPOS$. + \end{definition} + \begin{lemma}[Wellfounding] + $\NDIV$ is a wellfounded order. + \end{lemma} + \begin{proof} + $\NDIV$ is a partial order. Every nonempty subset of $\NPOS$ has a minimum with $\DIVI{\MoPos}$. + \end{proof} +\end{forthel} +\section{Euclids Theorem} +\begin{forthel} + \begin{lemma}[ExistenceOfStrangers] + Every element of $\NPOS$ has a stranger in $\NDIV$. + \end{lemma} + + \begin{proof} + Let $x \in \NPOS$. Consider $y = \oR{\Z} \pR{\Z} x$. $y \in \NPOS$. + Let us show that $x$ and $y$ have no common predecessor by $\NDIV$. \\ + Proof by contradiction. + Assume the contrary. + Take a common predecessor $k$ of $x$ and $y$ by $\NDIV$. + $k$ divides $(\oR{\Z} \pR{\Z} x)$ in $\Mu{\Z}$ and $k$ divides $x$ in $\Mu{\Z}$. + $(\oR{\Z} \pR{\Z} x) \mR{\Z} x = \oR{\Z}$. + Then $k$ divides $\oR{\Z}$ in $\Mu{\Z}$(by divDif). + $\oR{\Z}$ divides $k$ in $\Mu{\Z}$. $\oR{\Z}$ and $k$ are positive numbers. + Thus $k = \oR{\Z}$(by MultEquiv) . contradiction. + qed. + \end{proof} + + \begin{definition} + Let $p \in \NPOS$. $p$ is prime iff for all $d \in \NPOS$ we have \\ + ($d \oC{\DIVI{\MoPos}} p$) $=>$ $d = p$. + \end{definition} + \begin{definition} + $\cP$ = \{ $p \in \NPOS$ $|$ $p$ is prime \}. + \end{definition} + \begin{theorem} [Euclid] + $\cP$ is not finite. + \end{theorem} + \begin{proof} + Proof by contradiction. + Assume $\cP$ is finite. \\ + $\cP = \Min{\NDIV}$. \\ + $\Min{\NDIV}$ has no upper bound by $\NDIV$ (by Wellfounding , ExistenceOfStrangers , StrangerTheorem). \\ + $\Min{\NDIV}$ is a finite subset of $\sG{\MoPos}$. \\ + $\MoPos$ is an abelian monoid. \\ + Take an upper bound b of $\Min{\NDIV}$ by $\DIVI{\MoPos}$ (by FiniteMultiplication). \\ + b = $\oR{\Z}$. $\Min{\NDIV}$ is nonempty. \\ + contradiction. + \end{proof} +\end{forthel} + +\end{document}