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<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>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>assume </b><a NAME="E1:43"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">y</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; not <font color="Maroon">y</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="43_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:43_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:43_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</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#E10"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="arytm_2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:43_1_1"/>
<font color="Maroon">y'</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#E8"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="arytm_2.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><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="E6:43_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">y'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">z'</font> &amp; <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="ref" href="arytm_3.html#T101">ARYTM_3:101</a></i>;<br/><a NAME="E7:43_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>thus </b><a NAME="E9:43_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a>  &amp; <font color="Maroon">z</font> <a href="arytm_2.html#NR2" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</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#D3" target="_self">Def3</a>, <a class="txt" href="arytm_2.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:43_1_2"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:43_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E3:43_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E4:43_1_2"/><b>then </b>
<font color="Maroon">x</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="txt" href="arytm_2.html#E10"><i><font color="Green">Lemma42</font></i></a>, , <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="E6:43_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E7:43_1_2"/><i><font color="Green">E38</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>   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/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</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="ref" href="arytm_2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:43_1_2"/>
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="arytm_2.html#E10"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">s</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="E11:43_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E12:43_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">y'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D3" target="_self">Def3</a>, </i>;<br/><a NAME="E13:43_1_2"/>
<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>take </b>
<font color="Maroon">z</font>
;<br/><b>thus </b><a NAME="E15:43_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>thus </b><a NAME="E16:43_1_2"/>
<font color="Maroon">z</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#E10"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="arytm_2.html#D3" target="_self">Def3</a>, , </i>;<br/><b>thus </b><a NAME="E17:43_1_2"/>
<font color="Maroon">y</font> <a href="arytm_2.html#NR2" target="_self">&lt;</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>, <a class="txt" href="arytm_2.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:43_1_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:43_1_3"/><i><font color="Green">E74</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E3:43_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E4:43_1_3"/><b>then </b>
<font color="Maroon">y</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="arytm_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">B</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="E6:43_1_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E7:43_1_3"/><i><font color="Green">E76</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>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</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">B</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">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) )
 ;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</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#E10"><i><font color="Green">Lemma42</font></i></a></i>;<br/><a NAME="E9:43_1_3"/><i><font color="Green">E77</font></i>: 
not <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="arytm_2.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E10:43_1_3"/>
<font color="Maroon">B</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_2.html#T2" target="_self">Th2</a>, <a class="ref" href="arytm_2.html#D3" target="_self">Def3</a></i>;<br/><b>then consider </b><font color="Maroon">y1</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="E12:43_1_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E13:43_1_3"/>
 <a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/><a NAME="E14:43_1_3"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">x'</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_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/><a NAME="E15:43_1_3"/>
 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="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">x'</font> &amp; not <font color="Olive">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_3_1"/><i><font color="Green">E80</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 <font color="Olive">z'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">x'</font> or <font color="Olive">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 ;<br/>

<b>set </b><font color="Maroon">C</font> = <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">x'</font> </span>}</span> ;<br/>
<a NAME="E2:43_1_3_1"/>
<font color="Maroon">y</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">x'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1_3_1_1"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</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">x'</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1_1_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="E1:43_1_3_1_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">e</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="ref" href="arytm_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:43_1_3_1_1_1"/>
not <font color="Maroon">x'</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">z'</font>
 
<b>by </b><i><a class="ref" href="arytm_2.html#D3" target="_self">Def3</a>, , , </i>;<br/>
<b>hence </b><a NAME="E4:43_1_3_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</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">x'</font> </span>}</span> 
 ;<br/>


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

<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="E2:43_1_3_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</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">x'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">s</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="E4:43_1_3_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:43_1_3_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">s</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">x'</font>
 ;<br/>
<b>thus </b><a NAME="E6:43_1_3_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="txt" href="arytm_2.html#E16"><i><font color="Green">Lemma51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:43_1_3_1"/><b>then </b>
<font color="Maroon">y</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> 
 
<b>by </b><i><a class="ref" href="arytm_2.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:43_1_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><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="E17:43_1_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">z'</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E18:43_1_3"/><i><font color="Green">E85</font></i>: 
not <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E19:43_1_3"/>
<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>thus </b><a NAME="E21:43_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>thus </b><a NAME="E22:43_1_3"/>
<font color="Maroon">z</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#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/><b>thus </b><a NAME="E23:43_1_3"/>
<font color="Maroon">y</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#T2" target="_self">Th2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:43_1_4"/><i><font color="Green">E86</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 <b>and </b><br/><a NAME="E2:43_1_4"/><i><font color="Green">E87</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><div class="add"><a NAME="E3:43_1_4"/><i><font color="Green">E88</font></i>: 
not <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="arytm_2.html#E8"><i><font color="Green">Lemma41</font></i></a>, <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/><a NAME="E4:43_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E5:43_1_4"/><b>then </b>
<font color="Maroon">x</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="txt" href="arytm_2.html#E10"><i><font color="Green">Lemma42</font></i></a>, , <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="E7:43_1_4"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E8:43_1_4"/>
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="E9:43_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="arytm_2.html#K2" target="_self">REAL+</a> 
 ;<br/><a NAME="E10:43_1_4"/><b>then </b>
<font color="Maroon">y</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="arytm_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">B</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="E12:43_1_4"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E13:43_1_4"/>
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">B</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">B</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">B</font> &amp; <font color="Olive">r</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Olive">s</font> ) )
 ;<br/><b>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="E15:43_1_4"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E16:43_1_4"/><i><font color="Green">E92</font></i>: 
not <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="arytm_2.html#D3" target="_self">Def3</a>, , , <a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/><a NAME="E17:43_1_4"/>
<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>thus </b><a NAME="E19:43_1_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="arytm_3.html#K7">RAT+</a> 
 ;<br/><b>thus </b><a NAME="E20:43_1_4"/>
<font color="Maroon">z</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#E10"><i><font color="Green">Lemma42</font></i></a>, , <a class="txt" href="arytm_2.html#E14"><i><font color="Green">Lemma47</font></i></a>, </i>;<br/><b>thus </b><a NAME="E21:43_1_4"/>
<font color="Maroon">y</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#T2" target="_self">Th2</a>, , <a class="ref" href="arytm_2.html#D4" target="_self">Def4</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
