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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:87"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<i><font color="Green">E45</font></i>: 5 <a href="binarith.html#K3">-'</a> 1 = 
5 <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
4 <a href="nat_1.html#K1">+</a> 0

;<br/>
<i><font color="Green">E49</font></i>: 2 <a href="series_1.html#K3">to_power</a> 1 = 
2 <a href="wsierp_1.html#K2">|^</a> 1
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
;<br/>
<i><font color="Green">E50</font></i>: 2 <a href="binarith.html#K3">-'</a> 1 = 
2 <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
1 <a href="nat_1.html#K1">+</a> 0

;<br/>
2 <a href="series_1.html#K3">to_power</a> 2 = 
2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> 2
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
2 <a href="nat_1.html#K2">*</a> 2
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
;<br/>
<b>then </b>4 <a href="nat_1.html#K5">div</a> 2 = 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">2 <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pepin.html#E21"><i><font color="Green">Lemma74</font></i></a>, <a class="ref" href="nat_2.html#T18">NAT_2:18</a></i>
<br/>.= 
2 <a href="wsierp_1.html#K2">|^</a> 1
<b>by </b><i><a class="txt" href="pepin.html#E22"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
;<br/>
<a NAME="E7:87"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="txt" href="pepin.html#E20"><i><font color="Green">Lemma73</font></i></a>, </i>;<br/>
<a NAME="E8:87"/><b>then </b>
 <a href="pepin.html#K5" target="_self">Fermat</a> 1 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E19"><i><font color="Green">Lemma72</font></i></a>, </i>;<br/>
<a NAME="E9:87"/><b>then </b>
 <a href="pepin.html#K5" target="_self">Fermat</a> 1 <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:87"/>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4">-</a> 1 <a href="int_1.html#R1">are_congruent_mod</a>  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="pepin.html#E19"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="int_2.html#T19">INT_2:19</a></i>;<br/>


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