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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">d</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">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:52"/><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">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font> &amp; not <font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">p</font> )
 ;<br/>

<a NAME="E2:52"/><i><font color="Green">E45</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/>
<a NAME="E3:52"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:52_1"/><b>then </b>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E3:52_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:52"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E5:52"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:52"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:52"/>
not <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:52_2"/><b>then </b>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
;<br/>
<a NAME="E3:52_2"/><b>then </b>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E1:52"><i><font color="Green">E44</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E4:52_2"/><b>then </b>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:52_2"/><b>then </b>
<font color="Maroon">d</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="pepin.html#E2:52"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:52_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="pepin.html#E1"><i><font color="Green">Lemma43</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:52"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="pepin.html#E2:52"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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