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

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





<b>assume </b><a NAME="E1:65"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">i</font>,<font color="Maroon">p</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp;  <a href="pepin.html#K4" target="_self">order</a> <font color="Maroon">i</font>,<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:65"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" target="_self">order</a> <font color="Maroon">i</font>,<font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><a href="pepin.html#K4" target="_self">order</a> <font color="Maroon">i</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="newton.html#T14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="pepin.html#K4" target="_self">order</a> <font color="Maroon">i</font>,<font color="Maroon">p</font></span>)</span></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">t</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
<br/>.= 
1 <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="newton.html#T15">NEWTON:15</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<b>hence </b><a NAME="E6:65"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


</div>
