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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E17">A1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 ;<br/>

<b>assume </b><a NAME="E2:16"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 ;<br/>

<b>assume </b><a NAME="E3:16"/><i><font color="Green" title="E19">A3</font></i>: 
not <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c3">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="E5:16"/>
<font color="Maroon" title="c4">a</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="nat_4.html#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="pepin.html#T2" title="PEPIN:th.2">PEPIN:2</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:16"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E9:16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="nat_4.html#E8:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="pepin.html#T12" title="PEPIN:th.12">PEPIN:12</a></i>;<br/>
<a NAME="E10:16"/><b>then </b>
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="nat_d.html#T13" title="NAT_D:th.13">NAT_D:13</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/>
<b>hence </b><a NAME="E12:16"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_d.html#T13" title="NAT_D:th.13">NAT_D:13</a></i>;<br/>


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