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<div class="add">

<b>let </b><font color="Maroon">k</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:80"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">k</font> is <a href="moebius1.html#V1" target="_self">square-containing</a>
 ;<br/>

<a NAME="E2:80"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a>  holds <br/><span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">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">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">i</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="newton.html#D6">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">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">k</font></span>)</span> or  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">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">E42</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:80_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="txt" href="moebius1.html#E3:80_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E3:80_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E4:80_1_1_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <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="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">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:80_1_1_1_1"/><b>then </b>
not <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="nat_3.html#T27">NAT_3:27</a></i>;<br/>
<b>hence </b><a NAME="E3:80_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_3.html#T36">NAT_3:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:80_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <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="E6:80_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font> = 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1
<b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="newton.html#T10">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" target="_self">PFactors</a> <font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="moebius1.html#T1" target="_self">Th1</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">E56</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:80_1_1_2"/><b>then </b>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:80_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:80_1_1_2"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_3.html#D9">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" target="_self">PFactors</a> <font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">k</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">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" target="_self">PFactors</a> <font color="Maroon">k</font> <a href="pboole.html#R6">=</a>  <a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E4:80"/>
 <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T61">NAT_3:61</a></i>;<br/>


</div>
