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<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16"/><i><font color="Green">E38</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 ;<br/>

<b>assume </b><a NAME="E2:16"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<b>assume </b><a NAME="E3:16"/>
not <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E4:16"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="pepin.html#T2">PEPIN:2</a></i>;<br/>
<a NAME="E5:16"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:16"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E8:16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="pepin.html#T12">PEPIN:12</a></i>;<br/>
<a NAME="E9:16"/><b>then </b>
<span class="p1">(<span class="default">0 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>;<br/>
<a NAME="E10:16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="newton.html#T16">NEWTON:16</a></i>;<br/>
<b>hence </b><a NAME="E11:16"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>;<br/>


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