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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">n</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="E1:50"/><i><font color="Green" title="E38">A1</font></i>: 
( <font color="Maroon" title="c1">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">n</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="E2:50"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E3:50"/><i><font color="Green" title="E40">A3</font></i>: 
<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#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E4:50"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:50_1"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a></i>;<br/>
<a NAME="E3:50_1"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:50_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:50"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">p</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> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="pepin.html#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="euler_2.html#T34" title="EULER_2:th.34">EULER_2:34</a></i>;<br/>
<a NAME="E6:50"/><b>then </b><i><font color="Green" title="E42">A5</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_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> <font color="Maroon" title="c2">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="pepin.html#E3:50"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="pepin.html#E4:50"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="pepin.html#T27" target="_self" title="PEPIN:th.27">Th27</a></i>;<br/>
<a NAME="E7:50"/>
<font color="Maroon" title="c2">n</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="pepin.html#E4:50"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/>
<b>hence </b><a NAME="E8:50"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</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#E1:50"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="pepin.html#E2:50"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="pepin.html#E4:50"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="pepin.html#E6:50"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="euler_2.html#T27" title="EULER_2:th.27">EULER_2:27</a></i>;<br/>


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