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<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:63"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:63"/><i><font color="Green">E45</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 1,<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E3:63"/>
<font color="Maroon">p</font> <a href="nat_1.html#K10">hcf</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64">NEWTON:64</a></i>;<br/>
<a NAME="E4:63"/><b>then </b><i><font color="Green">E49</font></i>: 
1,<font color="Maroon">p</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<span class="p1">(<span class="default">1 <a href="wsierp_1.html#K2">|^</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> = 
1 <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="pepin.html#E4:63"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<a NAME="E6:63"/><b>then </b>
 <a href="pepin.html#K4" target="_self">order</a> 1,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E4:63"><i><font color="Green">E49</font></i></a>, , </i>;<br/>
<a NAME="E7:63"/><b>then </b>
(  <a href="pepin.html#K4" target="_self">order</a> 1,<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 or  <a href="pepin.html#K4" target="_self">order</a> 1,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E8:63"/><b>then </b>
(  <a href="pepin.html#K4" target="_self">order</a> 1,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 or  <a href="pepin.html#K4" target="_self">order</a> 1,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>hence </b><a NAME="E9:63"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E4:63"><i><font color="Green">E49</font></i></a>, , , </i>;<br/>


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