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

<b>let </b><font color="Maroon">r1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> , <font color="Maroon">r2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> , <font color="Maroon">s1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> , <font color="Maroon">s2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/>

<b>assume </b><a NAME="E1:108"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">r1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">s2</font>
 ;<br/>

<b>assume </b><a NAME="E2:108"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">s1</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">s2</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r2</font> )
 ;<br/>

<a NAME="E3:108"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">s1</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">s2</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r2</font> )
 
;<br/>
<a NAME="E4:108"/>
(  <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">s1</font> &amp;  <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">s2</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E5:108"/><b>then </b>
<font color="Maroon">r1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a> 
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E6:108"/><b>then </b>
( <font color="Maroon">s1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  &amp; <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E45"><i><font color="Green">Lemma86</font></i></a>, </i>;<br/>
<a NAME="E7:108"/><b>then </b>
( <font color="Maroon">s1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">s2</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">s1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">s2</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:108"/><b>then </b>
( <font color="Maroon">s1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">s2</font> <a href="arytm_3.html#NR4" target="_self">&lt;</a> <font color="Maroon">r1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">r1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">s2</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">r1</font> <a href="arytm_3.html#K12" target="_self">*'</a> <font color="Maroon">r2</font> )
 
<b>by </b><i>, <a class="ref" href="arytm_3.html#T26" target="_self">Th26</a>, </i>;<br/>
<b>hence </b><a NAME="E9:108"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E45"><i><font color="Green">Lemma86</font></i></a></i>;<br/>


</div>
