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<b>let </b><font color="Maroon">A</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">t</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">E13</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" target="_self">{}</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:100"/><i><font color="Green">E14</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" target="_self">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" target="_self">&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" target="_self">RAT+</a> <b> such that </b><br/><a NAME="E4:100"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T22" target="_self">Th22</a></i>;<br/>
<b>take </b>
 <a href="arytm_3.html#K13" target="_self">{}</a> 
;<br/>

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

<b>take </b>
<font color="Maroon">t</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">t</font> &amp; <font color="Maroon">x</font> <a href="arytm_3.html#R3" target="_self">&lt;='</a> <font color="Maroon">t</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T33" target="_self">Th33</a>, <a class="ref" href="arytm_3.html#T23" target="_self">Th23</a>, <a class="ref" href="arytm_3.html#T25" target="_self">Th25</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">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><a class="ref" href="arytm_3.html#T31" target="_self">Th31</a>, <a class="ref" href="arytm_3.html#T32" target="_self">Th32</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">x</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T31" target="_self">Th31</a>, <a class="ref" href="arytm_3.html#T33" target="_self">Th33</a>, </i>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:100_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E2:100_1"/><b>then </b>
<font color="Maroon">t</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">t</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T33" target="_self">Th33</a>, </i>;<br/><b>hence </b><a NAME="E3:100_1"/>
contradiction
 <b>by </b><i><a class="ref" href="arytm_3.html#T31" target="_self">Th31</a>, <a class="ref" href="arytm_3.html#T24" target="_self">Th24</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E9:100"/>
<font color="Maroon">t</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#T31" target="_self">Th31</a></i>;<br/>


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