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

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">l</font> be  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/>





<b>assume </b><a NAME="E1:26"/>
<font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:26"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:26"/>
<font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">m</font> <a href="arytm_3.html#K2" target="_self">+^</a> <font color="Maroon">l</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being  <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:26"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">m</font> <a href="arytm_3.html#K2" target="_self">+^</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E7:26"/><i><font color="Green">E15</font></i>: 
not <font color="Maroon">n</font> <a href="arytm_3.html#R2" target="_self">divides</a> <font color="Maroon">l</font>
 ;<br/>

<font color="Maroon">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">b</font></span>)</span> <a href="ordinal3.html#K5">-^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="ordinal3.html#T65">ORDINAL3:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="ordinal3.html#K5">-^</a> <font color="Maroon">a</font></span>)</span> <a href="arytm_3.html#K3" target="_self">*^</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="ordinal3.html#T76">ORDINAL3:76</a></i>
;<br/>
<b>hence </b><a NAME="E9:26"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div>
