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



<b>set </b><font color="Maroon">PR</font> = <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:85"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="nat_1.html#R1">divides</a>  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:85"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp;  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E3:85"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> is  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:85_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> is  not  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E2:85_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 0 <a href="nat_1.html#R1">divides</a>  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a>  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="nat_3.html#D7">NAT_3:def 7</a>, </i>;<br/>
<a NAME="E3:85_1"/><b>then </b><i><font color="Green">E51</font></i>: 
not <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a>  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<b>thus </b><a NAME="E4:85_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:85"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:85_2"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2
 ;<br/>

<a NAME="E2:85_2"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T71">NAT_1:71</a></i>;<br/>
<a NAME="E3:85_2"/><b>then </b>
not <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a>  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#D7">NAT_3:def 7</a>, </i>;<br/>
<b>hence </b><a NAME="E4:85_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:85"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E6:85"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
