<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">k</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">j</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="E1:26"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">k</font> <a href="pepin.html#K1" target="_self">^2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>suppose </b></a><a NAME="E1:26_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:26_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><a NAME="E4:26_1_1"/>
<font color="Maroon">i</font> <a href="pepin.html#K1" target="_self">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <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">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="pepin.html#K1" target="_self">^2</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:26_1_1"/><b>then </b>
<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">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="pepin.html#K1" target="_self">^2</a> </span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>;<br/><b>hence </b><a NAME="E6:26_1_1"/>
<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">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="pepin.html#K1" target="_self">^2</a> 
 <b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>suppose </b></a><a NAME="E1:26_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_1_2"/>
<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">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="pepin.html#K1" target="_self">^2</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
