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



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> 1 )
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E3:27"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E5:27"/>
<font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>then </b><font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, <a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/>
<a NAME="E7:27"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="square_1.html#K3">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
;<br/>
<b>reconsider </b><font color="Maroon">ii</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="pepin.html#K1" target="_self">^2</a> </span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ii</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
1 <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<b>hence </b><a NAME="E11:27"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="pepin.html#K1" target="_self">^2</a> </span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


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