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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies 3 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> )</span><br/>

<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E65">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 3 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span><br/>

<i><font color="Green" title="E66">A2</font></i>: 5 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
5 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></i>
<br/>.= 
4 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<i><font color="Green" title="E67">A3</font></i>: 2 <a href="power.html#K4" title="POWER:func.4">to_power</a> 1 = 
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1

<br/>.= 
2
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
;<br/>
<i><font color="Green" title="E68">A4</font></i>: 2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></i>
<br/>.= 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
2 <a href="power.html#K4" title="POWER:func.4">to_power</a> 2 = 
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
;<br/>
<b>then </b>4 <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 = 
2 <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">2 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E3:81"><i><font color="Green" title="E67">A3</font></i></a>, <a class="ref" href="nat_2.html#T18" title="NAT_2:th.18">NAT_2:18</a></i>
<br/>.= 
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1
<b>by </b><i><a class="txt" href="#E4:81"><i><font color="Green" title="E68">A4</font></i></a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
;<br/>
<a NAME="E7:81"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="txt" href="#E2:81"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="pepin.html#T56" target="_self" title="PEPIN:th.56">Th56</a></i>;<br/>
<a NAME="E8:81"/><b>then </b>
 <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default">3 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="pepin.html#E89"><i><font color="Green" title="E63">Lm25</font></i></a></i>;<br/>
<a NAME="E9:81"/><b>then </b>
 <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default">3 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
;<br/>
<b>hence </b><a NAME="E10:81"/>
3 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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