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<div class="add">

<b>set </b><font color="Maroon">P</font> =  <a href="moebius1.html#K3" target="_self">NatDivisors</a> <font color="Maroon">n</font>;<br/>
<a NAME="E1:56_1_1"/><i><font color="Green">E41</font></i>: 
not 0 <a href="hidden.html#R2">in</a>  <a href="moebius1.html#K3" target="_self">NatDivisors</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:56_1_1_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="moebius1.html#K3" target="_self">NatDivisors</a> <font color="Maroon">n</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:56_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 ;<br/>
<b>thus </b><a NAME="E4:56_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:56_1_1"/>
 <a href="moebius1.html#K3" target="_self">NatDivisors</a> <font color="Maroon">n</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:56_1_1_2"/>
 <a href="moebius1.html#K3" target="_self">NatDivisors</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E1"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:56_1_1_2"/>
 <a href="moebius1.html#K3" target="_self">NatDivisors</a> <font color="Maroon">n</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:56_1_1"/>
(  <a href="moebius1.html#K3" target="_self">NatDivisors</a> <font color="Maroon">n</font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="moebius1.html#K3" target="_self">NatDivisors</a> <font color="Maroon">n</font> is <a href="setfam_1.html#V1">with_non-empty_elements</a> )
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#D1">PSCOMP_1:def 1</a></i>;<br/>


</div>
