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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_1"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <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">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:32_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <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">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:32_1_1_1_1_1"/>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:32_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:32_1_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <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">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span>  is    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <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">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> ;<br/><a NAME="E2:32_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <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">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_1_1_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <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">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:32_1_1_1_2_1"/><b>then </b>
 ex <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
;<br/>
<b>hence </b><a NAME="E3:32_1_1_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <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">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  ;<br/><a NAME="E4:32_1_1_1_2"/>
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_1_1_1_2_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">s0</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="E3:32_1_1_1_2_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font>
 <b>and </b><br/><a NAME="E4:32_1_1_1_2_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:32_1_1_1_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>
<b>thus </b><a NAME="E6:32_1_1_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>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_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/>

<b>assume </b><a NAME="E1:32_1_1_1_2_2_1"/>
<font color="Maroon">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r</font>
 ;<br/>

<b>then consider </b><font color="Maroon">t0</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="E3:32_1_1_1_2_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>and </b><br/><a NAME="E4:32_1_1_1_2_2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">t0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s0</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T87">ARYTM_3:87</a></i>;<br/>
<a NAME="E5:32_1_1_1_2_2_1"/>
<font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E4"><i><font color="Green">Lemma34</font></i></a>, , <a class="txt" href="arytm_2.html#E1:32_1_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:32_1_1_1_2_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:32_1_1_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="arytm_2.html#E5"><i><font color="Green">Lemma35</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>consider </b><font color="Maroon">t0</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="E8:32_1_1_1_2_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E9:32_1_1_1_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">s0</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E4"><i><font color="Green">Lemma34</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1_2_2_2"/>
( <font color="Maroon">r0</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a>  or <font color="Maroon">r0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_2_2_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_2_2_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">r1</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="E3:32_1_1_1_2_2_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:32_1_1_1_2_2_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i>, </i>;<br/><b>take </b>
<font color="Maroon">r1</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
;<br/><b>thus </b><a NAME="E5:32_1_1_1_2_2_2_1"/>
<font color="Maroon">r1</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5"><i><font color="Green">Lemma35</font></i></a>, </i>;<br/><a NAME="E6:32_1_1_1_2_2_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T54">ARYTM_3:54</a></i>;<br/><a NAME="E7:32_1_1_1_2_2_2_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">r</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r1</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/><a NAME="E8:32_1_1_1_2_2_2_1"/>
<font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/><a NAME="E9:32_1_1_1_2_2_2_1"/><b>then </b>
<font color="Maroon">r1</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 <b>by </b><i><a class="txt" href="arytm_2.html#E8:32_1_1_1_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="arytm_3.html#T86">ARYTM_3:86</a></i>;<br/><b>hence </b><a NAME="E10:32_1_1_1_2_2_2_1"/>
<font color="Maroon">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self">Th1</a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_2_2_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2_2_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">r0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
;<br/><b>thus </b><a NAME="E2:32_1_1_1_2_2_2_2"/>
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E5:32_1_1_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="arytm_2.html#E5"><i><font color="Green">Lemma35</font></i></a></i>;<br/><a NAME="E3:32_1_1_1_2_2_2_2"/>
<font color="Maroon">s0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t0</font>
 <b>by </b><i/>;<br/><a NAME="E4:32_1_1_1_2_2_2_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T62">ARYTM_3:62</a></i>;<br/><a NAME="E5:32_1_1_1_2_2_2_2"/>
<font color="Maroon">r</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T90">ARYTM_3:90</a></i>;<br/><b>hence </b><a NAME="E6:32_1_1_1_2_2_2_2"/>
<font color="Maroon">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">t0</font>
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T75">ARYTM_3:75</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:32_1_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  : for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Olive">r</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) &amp;  ex <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) ) </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">r0</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="E7:32_1_1_1_2"/><i><font color="Green">E81</font></i>: 
not <font color="Maroon">r0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">s0</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="E9:32_1_1_1_2"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1_2_3"/>
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/><b>then consider </b><font color="Maroon">r1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <font color="Maroon">s1</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="E3:32_1_1_1_2_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">r0</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="arytm_3.html#K12">*'</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E4:32_1_1_1_2_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:32_1_1_1_2_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E6:32_1_1_1_2_3"/>
( <font color="Maroon">r0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">r1</font> or <font color="Maroon">s0</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">s1</font> )
 <b>by </b><i><a class="txt" href="arytm_2.html#E5"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="arytm_3.html#T91">ARYTM_3:91</a></i>;<br/><b>hence </b><a NAME="E7:32_1_1_1_2_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E4"><i><font color="Green">Lemma34</font></i></a>, , , <a class="txt" href="arytm_2.html#E1:32_1_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E11:32_1_1_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>hence </b><a NAME="E12:32_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">r</font> <a href="arytm_3.html#K12">*'</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a> , <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">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span>  is    <a href="subset_1.html#M2">Element</a> of  <a href="arytm_2.html#K1" target="_self">DEDEKIND_CUTS</a> 
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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