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

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:31"/>
<font color="Maroon">p</font> is <a href="moebius1.html#V1" target="_self">square-containing</a>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:31"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> 2
 
<b>by </b><i><a class="ref" href="nat_3.html#T3">NAT_3:3</a></i>;<br/>
<a NAME="E5:31"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 </span><b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>suppose </b></a><a NAME="E1:31_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_1_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><b>suppose </b></a><a NAME="E1:31_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:31_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> 2
 <b>by </b><i>, <a class="ref" href="nat_1.html#T52">NAT_1:52</a>, </i>;<br/><a NAME="E3:31_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="wsierp_1.html#K2">|^</a> 2
 <b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/><a NAME="E4:31_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="pepin.html#T31">PEPIN:31</a></i>;<br/><b>hence </b><a NAME="E5:31_1_2"/>
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></div><b>end;</b></div>

</div>
