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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:31"/>
<font color="Maroon" title="c1">p</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">n</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="ref" href="moebius1.html#D1" target="_self" title="MOEBIUS1:def.1">Def1</a></i>;<br/>
<a NAME="E4:31"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2
 
<b>by </b><i><a class="ref" href="nat_3.html#T3" title="NAT_3:th.3">NAT_3:3</a></i>;<br/>
<a NAME="E5:31"/><b>then </b><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/>
<a NAME="E6:31"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</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" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> )
 </span><b>by </b><i><a class="txt" href="moebius1.html#E5:31"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">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" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="INT_2:def.5">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" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 ;<br/><div class="add"><a NAME="E2:31_1_2"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2
 <b>by </b><i><a class="txt" href="moebius1.html#E3:31"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="moebius1.html#E4:31"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="nat_d.html#T5" title="NAT_D:th.5">NAT_D:5</a></i>;<br/><a NAME="E3:31_1_2"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2
 <b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/><a NAME="E4:31_1_2"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E6:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="pepin.html#T31" title="PEPIN:th.31">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" title="INT_2:def.5">INT_2:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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