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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>thus </b><a NAME="E1:83"/>
(  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E41</font></i>: 
 <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:83_1"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E4:83_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:83_1"/><i><font color="Green">E50</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:83_1"/><i><font color="Green">E51</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">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_3.html#T37">NAT_3:37</a></i>;<br/>
<a NAME="E7:83_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E9:83_1"/><i><font color="Green">E72</font></i>: 
(  <a href="nat_3.html#K8">Product</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K17">Product</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_3.html#D5">NAT_3:def 5</a></i>;<br/>
<a NAME="E10:83_1"/><i><font color="Green">E73</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E11:83_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>
<a NAME="E12:83_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E14:83_1"/><i><font color="Green">E74</font></i>: 
 <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">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<a NAME="E15:83_1"/><i><font color="Green">E75</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="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E7:83_1"><i><font color="Green">E56</font></i></a>, </i>;<br/>
<a NAME="E16:83_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="uproots.html#T5">UPROOTS:5</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:83_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E19:83_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, , <a class="txt" href="moebius1.html#E5"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="moebius1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E20:83_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="moebius1.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, </i>;<br/>
<a NAME="E21:83_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="moebius1.html#T5" target="_self">Th5</a>, <a class="ref" href="moebius1.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E22:83_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#T7">NAT_3:7</a></i>;<br/>
<b>hence </b><a NAME="E23:83_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T54">NAT_1:54</a>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:83"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 implies  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="nat_3.html#T20">NAT_3:20</a>, </i>;<br/>


</div>
