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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:80"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:80"/><i><font color="Green">E66</font></i>: 
(  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font>,3 <a href="int_2.html#R2">are_relative_prime</a>  or <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> 3 <a href="hidden.html#R1">=</a> 3 )
 
<b>by </b><i><a class="ref" href="pepin.html#T2" target="_self">Th2</a>, <a class="ref" href="pepin.html#T44" target="_self">Th44</a></i>;<br/>
<a NAME="E3:80"/>
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1"/>
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> 3 <a href="hidden.html#R1">=</a> 3
 ;<br/>

<a NAME="E2:80_1"/><b>then </b>
3 <a href="nat_1.html#R1">divides</a>  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:80_1"/><i><font color="Green">E67</font></i>: 
3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="pepin.html#T44" target="_self">Th44</a>, <a class="ref" href="pepin.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E5:80_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E4:80_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E6:80_1"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> 2
 
<b>by </b><i><a class="ref" href="nat_2.html#T5">NAT_2:5</a></i>;<br/>
<a NAME="E7:80_1"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pepin.html#E19"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E8:80_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; 2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/>
<a NAME="E9:80_1"/><b>then </b>
2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="wsierp_1.html#K2">|^</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E10:80_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E1:80"><i><font color="Green">E65</font></i></a>, <a class="ref" href="pepin.html#T31" target="_self">Th31</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:80"/>
3, <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="pepin.html#E2:80"><i><font color="Green">E66</font></i></a></i>;<br/>


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