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\begin{document}
\chapter{Real number}
\section{Ordered Fields}
The property of ordered fields
\subsection{Theorem}
\begin{enumerate}
\item A1. $\forall x,y\in\mathbb{R}$ and if $x=w$ and $y=z$, then $x+y=w+z$.
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\item A2. $\forall x,y\in\mathbb{R}$, $x+y=y+x$.
%-------------------
\item A3. $\forall x,y,z\in\mathbb{R}$, $x+(y+z)=(x+y)+z$.
%-------------------
\item A4. $\exists$ unique real number 0 $\ni x+0=x$ for all $x\in\mathbb{R}$.
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\subsection{Exercise}
Q3. Let $x,y,z\in\mathbb{R}$. Prove the following.\\
Q3(a) $-(-x)=x$:\\\
by M1:\\
Let $x=-1$, $y=-x$, then\\
\begin{align}
x\cdot y&=(-1)(-x)\nonumber\\
&=(-1\cdot -1)x\leftarrow\text(from,M3)\nonumber\\
&=x
\end{align}
(b) $(-x)\cdot y=-(xy)$ and $(-x)\cdot(-y)=xy$:\\\
by M3:\\\
Let $x=-1$, $y=x$,$z=y$ then\\
\begin{align}
(-x).y&=(-1.x).y\nonumber\\
&=-(x,y)\leftarrow \text{from\, M3}
\end{align}
To prove second part
by M3:\\
Let $x=-1$, $y=-x$,$z=-y$ then\\
\begin{align}
(-x)\cdot (-y)&=(-1\cdot x)(-1\cdot y)\\
&=(-1).x.(-1)y\nonumber\\
&=xy\leftarrow \text{from\, M3}
\end{align}
(e) if $x\neq 0$, then $x^2>0$\\
consider $x>0$:\\
Let $x=2$:\\
\begin{align}
x^2&=(2)^2\nonumber\\
&=4
\end{align}
consider $x<0$:\\
Let $x=-2$:\\
\begin{align}
x^2&=(-2)^2\nonumber\\
&=4
\end{align}
\end{enumerate}
\end{document}
