<?xml version="1.0"?>
<div class="add">

<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><a NAME="E1:10_1_2_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> iff <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> )
 ;<br/><a NAME="E2:10_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  <b>by </b><i/>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E4:10_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><b>thus </b><a NAME="E5:10_1_2_1"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> iff <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:10_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   holds <br/>not for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> iff <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">r</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">t</font> </span>}</span>  where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E3:10_1_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font> </span>}</span> 
 <b>and </b><br/><a NAME="E4:10_1_2_2"/>
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><a NAME="E5:10_1_2_2"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> iff <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:10_1_2_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:10_1_2_2_1_1"/><b>then </b>
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:10_1_2_2_1_1"/>
<font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font>
 ;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:10_1_2_2_1"/>
( <font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font> implies <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:10_1_2_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:10_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a>  <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">t</font> </span>}</span>  where <font color="Olive">t</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  : <font color="Olive">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E6:10_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>


</div>
