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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E50">A1</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:64"/><b>then </b><i><font color="Green" title="E51">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="pepin.html#D2" target="_self" title="PEPIN:def.2">Def2</a></i>;<br/>
<a NAME="E3:64"/><i><font color="Green" title="E52">A3</font></i>: 
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="pepin.html#E1:64"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="pepin.html#D2" target="_self" title="PEPIN:def.2">Def2</a></i>;<br/>
<a NAME="E4:64"/>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E53">A4</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c5">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green" title="E54">A5</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="txt" href="pepin.html#E3:64"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c5">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">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" title="c3">i</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E1:64"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="pepin.html#E3:64_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="newton.html#T13" title="NEWTON:th.13">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" title="c3">i</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p4">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T14" title="NEWTON:th.14">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" title="c3">i</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p5">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c6">t</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="euler_2.html#T10" title="EULER_2:th.10">EULER_2:10</a></i>;<br/>
<a NAME="E8:64_1"/><b>then </b><i><font color="Green" title="E55">A6</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E1:64"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="pepin.html#E2:64"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="pepin.html#T36" target="_self" title="PEPIN:th.36">Th36</a></i>;<br/>
<a NAME="E9:64_1"/>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="pepin.html#E1:64_1"><i><font color="Green" title="E53">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:64_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E1:64"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="pepin.html#E3:64_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="txt" href="pepin.html#E8:64_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="pepin.html#D2" target="_self" title="PEPIN:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:64"/>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="pepin.html#E3:64"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="pepin.html#T6" target="_self" title="PEPIN:th.6">Th6</a></i>;<br/>


</div>
