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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:63"/><i><font color="Green" title="E47">A2</font></i>: 
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 1,<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E3:63"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64" title="NEWTON:th.64">NEWTON:64</a></i>;<br/>
<a NAME="E4:63"/><b>then </b><i><font color="Green" title="E48">A3</font></i>: 
1,<font color="Maroon" title="c1">p</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<span class="p1">(<span class="default">1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> = 
1 <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="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="pepin.html#E1:63"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
;<br/>
<a NAME="E6:63"/><b>then </b>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 1,<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E1:63"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="pepin.html#E4:63"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="pepin.html#D2" target="_self" title="PEPIN:def.2">Def2</a></i>;<br/>
<a NAME="E7:63"/><b>then </b>
(  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 1,<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 1,<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="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E8:63"/><b>then </b>
(  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 1,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 1,<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="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>hence </b><a NAME="E9:63"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E1:63"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="pepin.html#E2:63"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="pepin.html#E4:63"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="pepin.html#D2" target="_self" title="PEPIN:def.2">Def2</a></i>;<br/>


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