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<b>let </b><font color="Maroon" title="c1">k</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:80"/><i><font color="Green" title="E51">A1</font></i>: 
not <font color="Maroon" title="c1">k</font> is <a href="moebius1.html#V1" title="MOEBIUS1:attr.1">square-containing</a>
 ;<br/>

<a NAME="E2:80"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>  holds <br/><span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">i</font> as  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:80_1_1"/>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">k</font></span>)</span> or  not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">k</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_1"><b>suppose </b></a><a NAME="E1:80_1_1_1"/><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:80_1_1_1"/><i><font color="Green" title="E53">A3</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</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#E1:80"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="moebius1.html#T24" target="_self" title="MOEBIUS1:th.24">Th24</a></i>;<br/><a NAME="E3:80_1_1_1"/><i><font color="Green" title="E54">A4</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E4:80_1_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1_1_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:80_1_1_1_1"/><b>then </b>
not <font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:80_1_1_1"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="nat_3.html#T27" title="NAT_3:th.27">NAT_3:27</a></i>;<br/>
<b>hence </b><a NAME="E3:80_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:80_1_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="nat_3.html#T36" title="NAT_3:th.36">NAT_3:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:80_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E6:80_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E2:80_1_1_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">p</font> = 
<font color="Maroon" title="c3">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1
<b>by </b><i><a class="txt" href="moebius1.html#E1:80_1_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
;<br/><b>hence </b><a NAME="E8:80_1_1_1"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:80_1_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="moebius1.html#D6" target="_self" title="MOEBIUS1:def.6">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1_2"><b>suppose </b></a><a NAME="E1:80_1_1_2"/><i><font color="Green" title="E52">A5</font></i>: 
not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:80_1_1_2"/><b>then </b>
not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="moebius1.html#D6" target="_self" title="MOEBIUS1:def.6">Def6</a></i>;<br/><a NAME="E3:80_1_1_2"/><b>then </b><i><font color="Green" title="E53">A6</font></i>: 
<span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:80_1_1_2"/>
not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:80_1_1_2"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="nat_3.html#D9" title="NAT_3:def.9">NAT_3:def 9</a></i>;<br/><b>hence </b><a NAME="E5:80_1_1_2"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:80_1_1_2"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:80"/><b>then </b>
 <a href="moebius1.html#K5" title="MOEBIUS1:func.5">PFactors</a> <font color="Maroon" title="c1">k</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E4:80"/>
 <a href="moebius1.html#K6" title="MOEBIUS1:func.6">Radical</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T61" title="NAT_3:th.61">NAT_3:61</a></i>;<br/>


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