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

<b>let </b><font color="Maroon">a</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">b</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:84"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:84"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">b</font>;<br/>
<b>assume </b><a NAME="E3:84"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">b</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:84"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">a</font></span>)</span>
 <b>and </b><br/><a NAME="E6:84"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="txt" href="nat_3.html#E6:84"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<a NAME="E8:84"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_3.html#E6:84"><i><font color="Green">E47</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>
<a NAME="E9:84"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:84"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E11:84"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:84"/>
<font color="Maroon">x</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">x</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_3.html#E8:84"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nat_3.html#E9:84"><i><font color="Green">E50</font></i></a>, </i>;<br/>
<a NAME="E13:84"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E11:84"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E14:84"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>
<a NAME="E15:84"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:84"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E10:84"><i><font color="Green">E51</font></i></a>, </i>;<br/>
<a NAME="E17:84"/>
<font color="Maroon">x</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">x</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#T3" target="_self">Th3</a>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E18:84"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E19:84"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#R1">divides</a> 1
 
<b>by </b><i>, <a class="txt" href="nat_3.html#E13:84"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E20:84"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_3.html#E10:84"><i><font color="Green">E51</font></i></a>, <a class="ref" href="wsierp_1.html#T20">WSIERP_1:20</a></i>;<br/>


</div>
