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<b>let </b><font color="Maroon">q1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">q1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">q</font> is <a href="int_2.html#V1">prime</a> )
 ;<br/>

<b>assume </b><a NAME="E2:19"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">q1</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E3:19"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">q1</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/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:19"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="pepin.html#T33">PEPIN:33</a></i>;<br/>
<a NAME="E6:19"/>
<font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E7:19"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> )
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E8:19"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T7">XCMPLX_1:7</a>, , </i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:19"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 <b>by </b><i>, </i>;<br/>


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