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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="arytm_3.html#K7" target="_self">RAT+</a> ;<br/>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> <b> such that </b><a NAME="E1:100"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13" target="_self">{}</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:100"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a> <b> such that </b><br/><a NAME="E4:100"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T66" target="_self">Th66</a></i>;<br/>
<b>take </b>
 <a href="arytm_3.html#K13" target="_self">{}</a> 
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E5:100"/>
(  <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E4:100"><i><font color="Green">E84</font></i></a>, <a class="ref" href="arytm_3.html#D13" target="_self">Def13</a>, <a class="ref" href="arytm_3.html#T71" target="_self">Th71</a></i>;<br/>
<b>hence </b><a NAME="E6:100"/>
(  <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:100"><i><font color="Green">E82</font></i></a>, <a class="txt" href="arytm_3.html#E2:100"><i><font color="Green">E83</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:100"/>
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:100"><i><font color="Green">E82</font></i></a>, <a class="txt" href="arytm_3.html#E4:100"><i><font color="Green">E84</font></i></a>, <a class="ref" href="arytm_3.html#T56" target="_self">Th56</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:100_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:100_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E4:100"><i><font color="Green">E84</font></i></a>, <a class="ref" href="arytm_3.html#T56" target="_self">Th56</a></i>;<br/><b>hence </b><a NAME="E3:100_1"/>
contradiction
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:100"><i><font color="Green">E82</font></i></a>, <a class="ref" href="arytm_3.html#T69" target="_self">Th69</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E9:100"/>
<font color="Maroon">c<sub>4</sub></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#E1:100"><i><font color="Green">E82</font></i></a></i>;<br/>


</div>
