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

<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_2.html#K2" target="_self">REAL+</a> , <font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/>

<b>given </b><font color="Maroon">x0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><a NAME="E1:45"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>given </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><a NAME="E2:45"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>assume </b><a NAME="E3:45"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">x</font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Olive">y</font>
 ;<br/>

<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1"/>
(  ex <font color="Olive">z'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  iff <font color="Olive">x'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">z'</font> ) or for <font color="Olive">z'</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">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  iff <font color="Olive">x'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">z'</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>suppose </b></a><a NAME="E1:45_1_1"/>
 ex <font color="Olive">z'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  iff <font color="Olive">x'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">z'</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z'</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="E3:45_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  iff <font color="Olive">x'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">z'</font> )
 ;<br/><a NAME="E4:45_1_1"/>
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z'</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">z</font>
;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E6:45_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E7:45_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>thus </b><a NAME="E8:45_1_1"/>
<font color="Maroon">x</font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1_1_1"/>
<font color="Maroon">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E3:45_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E4:45_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x0</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:45_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma47</font></i></a></i>;<br/>
<a NAME="E7:45_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">z'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, <a class="txt" href="arytm_2.html#E2:45"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E8:45_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="arytm_2.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E9:45_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_2.html#D3" target="_self">Def3</a>, </i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E9:45_1_1"/>
<font color="Maroon">y</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><b>then consider </b><font color="Maroon">y0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E11:45_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">y0</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E12:45_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">y0</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E13:45_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">y</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">y0</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma47</font></i></a></i>;<br/><a NAME="E15:45_1_1"/>
<font color="Maroon">y'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a>, <a class="txt" href="arytm_2.html#E2:45"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E16:45_1_1"/><b>then </b>
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="arytm_2.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E17:45_1_1"/><b>then </b>
 ex <font color="Olive">y''</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">y''</font> &amp;  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">y''</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) )
 ;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> , <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E19:45_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E20:45_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E21:45_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">y1</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x1</font>
 ;<br/><a NAME="E22:45_1_1"/>
<font color="Maroon">y</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x1</font>
 <b>by </b><i>, , , <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>hence </b><a NAME="E23:45_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_2.html#T2" target="_self">Th2</a>, , <a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma51</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><b>suppose </b></a><a NAME="E1:45_1_2"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">z'</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">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  iff <font color="Olive">x'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">z'</font> )
 ;<br/><div class="add"><div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  <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="E3:45_1_2_1"/><i><font color="Green">E80</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  <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="E4:45_1_2_1"/>
<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="E5:45_1_2_1"/>
for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  holds <br/> ( <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  iff <font color="Olive">x'</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="45_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x'</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="E1:45_1_2_1_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span> 
 ;<br/><a NAME="E2:45_1_2_1_1_1"/><b>then </b>
 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#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Olive">s</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="E3:45_1_2_1_1_1"/>
<font color="Maroon">x'</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="E2:45_1_2_1_1"/>
( <font color="Maroon">x'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">t</font> implies <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:45_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_2.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><a NAME="E2:45_1_2_2"/><b>then </b><i><font color="Green">E82</font></i>: 
for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  holds <br/><font color="Olive">x'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K13">{}</a> 
 ;<br/><a NAME="E3:45_1_2_2"/>
for <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">x'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a>  <a href="arytm_3.html#K13">{}</a>  holds <br/><font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/><b>hence </b><a NAME="E4:45_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_2.html#D3" target="_self">Def3</a>, <a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma47</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:45_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</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="45_1_2_3"><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="E1:45_1_2_3"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:45_1_2_3"/><b>then </b>
 ex <font color="Olive">z'</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">z'</font> &amp;  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:45_1_2_3"/>
<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">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">z'</font> where <font color="Olive">z'</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  :  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i/>;<br/><a NAME="E6:45_1_2"/>
for <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</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">t</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</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">Z</font> &amp; <font color="Olive">t</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="45_1_2_4"><b>proof </b></a><div class="add">

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

<a NAME="E1:45_1_2_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">y0</font> = <font color="Maroon">t</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>assume </b><a NAME="E3:45_1_2_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E4:45_1_2_4"/><b>then </b>
 ex <font color="Olive">z'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Olive">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp;  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) )
 
;<br/>
<b>then consider </b><font color="Maroon">x0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E6:45_1_2_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E7:45_1_2_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">y0</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x0</font>
 ;<br/>
<b>thus </b><a NAME="E8:45_1_2_4"/>
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">t</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_4_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/>

<a NAME="E1:45_1_2_4_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">s</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>assume </b><a NAME="E3:45_1_2_4_1"/>
<font color="Maroon">s</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E4:45_1_2_4_1"/><b>then </b>
<font color="Maroon">z</font> <a href="arytm_2.html#R1" target="_self">&lt;='</a> <font color="Maroon">y0</font>
 
<b>by </b><i><a class="txt" href="arytm_2.html#E2:45"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E5:45_1_2_4_1"/><b>then </b>
<font color="Maroon">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:45_1_2_4_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma47</font></i></a></i>;<br/>


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

<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E10:45_1_2_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E11:45_1_2_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x0</font>
 <b>and </b><br/><a NAME="E12:45_1_2_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">y0</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a>, <a class="ref" href="arytm_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z'</font>
;<br/>

<b>thus </b><a NAME="E14:45_1_2_4"/>
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma47</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E15:45_1_2_4"/>
<font color="Maroon">t</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="arytm_2.html#E2:45"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:45_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">Z</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_5"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_5_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  or  not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_5_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  ;<br/><a NAME="E3:45_1_2_5_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E4:45_1_2_5_1_1"/><b>then </b>
 ex <font color="Olive">z'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp;  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) )
 ;<br/><b>hence </b><a NAME="E5:45_1_2_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E10"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="arytm_2.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_5_1_2"/><i><font color="Green">E90</font></i>: 
not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E2:45_1_2_5_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E3:45_1_2_5_1_2"/><b>then </b>
<font color="Maroon">x1</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> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> <b> such that </b><br/><a NAME="E5:45_1_2_5_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:45_1_2_5_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">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="Maroon">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="Maroon">A</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) )
 ;<br/><a NAME="E7:45_1_2_5_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x'</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="E9:45_1_2_5_1_2"/><i><font color="Green">E92</font></i>: 
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T49">SUBSET_1:49</a></i>;<br/><a NAME="E10:45_1_2_5_1_2"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">x2</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/><a NAME="E12:45_1_2_5_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">x1</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , <a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/><a NAME="E13:45_1_2_5_1_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E14:45_1_2_5_1_2"/><b>then </b>
 ex <font color="Olive">z'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp;  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) )
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E16:45_1_2_5_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E17:45_1_2_5_1_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">x2</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E18:45_1_2_5_1_2"/>
<font color="Maroon">x1</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E18"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>hence </b><a NAME="E19:45_1_2_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_2.html#E10"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="arytm_2.html#T2" target="_self">Th2</a>, <a class="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma53</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E9:45_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">Z</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>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K7">RAT+</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><a NAME="E10:45_1_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">{<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>  <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K7">RAT+</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <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">z</font> = <font color="Maroon">Z</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/><div><i><font color="Green">E121</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_6"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><a NAME="E2:45_1_2_6"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="arytm_3.html#K13">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:45_1_2_6"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>take </b>
<font color="Maroon">z</font>
;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E13:45_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E14:45_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:45_1_2_7"/>
<font color="Maroon">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">x0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E3:45_1_2_7"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E4:45_1_2_7"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">x0</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:45_1_2_7"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i/>;<br/><a NAME="E7:45_1_2_7"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma53</font></i></a></i>;<br/><b>hence </b><a NAME="E8:45_1_2_7"/>
contradiction
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E18"><i><font color="Green">Lemma54</font></i></a>, </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E16:45_1_2"/>
<font color="Maroon">y</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><b>then consider </b><font color="Maroon">y0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E18:45_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">y0</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E19:45_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">y0</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E20:45_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">y</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">y0</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#T1" target="_self">Th1</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  <b>by </b><i/>;<br/><a NAME="E22:45_1_2"/>
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="arytm_2.html#E17"><i><font color="Green">Lemma53</font></i></a>, </i>;<br/><a NAME="E23:45_1_2"/><b>then </b>
 ex <font color="Olive">z'</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7">RAT+</a>  st <br/>( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp;  ex <font color="Olive">x</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z'</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Olive">x</font> ) )
 ;<br/><b>then consider </b><font color="Maroon">x0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_2.html#K2" target="_self">REAL+</a> <b> such that </b><br/><a NAME="E25:45_1_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E26:45_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">y0</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x0</font>
 ;<br/><a NAME="E27:45_1_2"/>
<font color="Maroon">y</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E18"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="arytm_2.html#E20"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a></i>;<br/><b>hence </b><a NAME="E28:45_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_2.html#T2" target="_self">Th2</a>, <a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="arytm_2.html#E19"><i><font color="Green">Lemma55</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
