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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>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>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>assume </b><a NAME="E1:64"/><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">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <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 &amp; <font color="Maroon">i</font>,<font color="Maroon">p</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:64"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<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> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E3:64"/><i><font color="Green">E49</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> <font color="Maroon">i</font>,<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:64"/>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</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="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</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="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">I</font> = <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</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>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<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> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pepin.html#K4" target="_self">order</a> <font color="Maroon">i</font>,<font color="Maroon">p</font> )
 <b>by </b><i><a class="txt" href="pepin.html#E8:64_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</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/>
<a NAME="E5:64_1"/>
<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"><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> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</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></span>)</span></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="txt" href="pepin.html#E1:64_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pepin.html#E4"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>;<br/>
<a NAME="E6:64_1"/><b>then </b>
<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="wsierp_1.html#K2">|^</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</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></span>)</span></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#T14">NEWTON:14</a></i>;<br/>
<a NAME="E7:64_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<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></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</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></span>)</span></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="euler_2.html#T10">EULER_2:10</a></i>;<br/>
<a NAME="E8:64_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</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></span>)</span></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="txt" href="pepin.html#E1:64_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pepin.html#E3:64_1"><i><font color="Green">E51</font></i></a>, </i>;<br/>
<a NAME="E9:64_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</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="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:64_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E1:64_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pepin.html#E4"><i><font color="Green">Lemma54</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:64"/>
 <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>
 <b>by </b><i>, </i>;<br/>


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