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

<b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7">RAT+</a> ;<br/>

<b>given </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><a NAME="E1:22"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:22"/><i><font color="Green">E40</font></i>: 
for <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 <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="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>
 ;<br/>

<b>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="E4:22"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66">ARYTM_3:66</a></i>;<br/>
<b>consider </b><font color="Maroon">y</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="E6:22"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66">ARYTM_3:66</a></i>;<br/>
<b>consider </b><font color="Maroon">w</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="E8:22"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66">ARYTM_3:66</a></i>;<br/>
<b>take </b>
 <a href="arytm_3.html#K13">{}</a> 
;<br/>

<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>take </b>
<font color="Maroon">t</font>
;<br/>

<a NAME="E9:22"/><i><font color="Green">E44</font></i>: 
(  <a href="arytm_3.html#K13">{}</a>  <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font> &amp; <font color="Maroon">x</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font> )
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a>, <a class="ref" href="arytm_3.html#T71">ARYTM_3:71</a></i>;<br/>
<a NAME="E10:22"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">y</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">x</font> &amp; <font color="Maroon">w</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, , <a class="ref" href="arytm_3.html#D13">ARYTM_3:def 13</a></i>;<br/>
<a NAME="E11:22"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font>
 
<b>by </b><i>, , <a class="ref" href="arytm_3.html#T74">ARYTM_3:74</a></i>;<br/>
<a NAME="E12:22"/>
<font color="Maroon">w</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T74">ARYTM_3:74</a></i>;<br/>
<a NAME="E13:22"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">w</font> <a href="arytm_3.html#R3">&lt;='</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T74">ARYTM_3:74</a></i>;<br/>
<b>thus </b><a NAME="E14:22"/>
(  <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , , , </i>;<br/>

<a NAME="E15:22"/><i><font color="Green">E71</font></i>: 
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a></i>;<br/>
<a NAME="E16:22"/><i><font color="Green">E72</font></i>: 
 <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a></i>;<br/>
<a NAME="E17:22"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E2:22_1"/><b>then </b>
<font color="Maroon">t</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">t</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a></i>;<br/>
<b>hence </b><a NAME="E3:22_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T69">ARYTM_3:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:22"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:22_2"/><b>then </b>
<font color="Maroon">x</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a></i>;<br/>
<b>hence </b><a NAME="E3:22_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T69">ARYTM_3:69</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:22"/>
<font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:22_3"/><b>then </b>
<font color="Maroon">y</font> <a href="arytm_3.html#K11">+</a> <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="arytm_3.html#K11">+</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a></i>;<br/>
<b>hence </b><a NAME="E3:22_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T69">ARYTM_3:69</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:22"/>
(  <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font> &amp;  <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp;  <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp;  <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="ref" href="arytm_3.html#T56">ARYTM_3:56</a>, , , , , , , , <a class="ref" href="arytm_3.html#T73">ARYTM_3:73</a></i>;<br/>


</div>
