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<span class="kw">set </span><font color="Maroon" title="c2">P</font> =  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E1:55_1_1"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:55_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:55_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">k</font> being   <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>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">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>  &amp; <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:55_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:55_1_1"/>
 <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="moebius1.html#T40" target="_self" title="MOEBIUS1:th.40">Th40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:55_1_1"/>
(  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c1">n</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp;  <a href="moebius1.html#K3" title="MOEBIUS1:func.3">NatDivisors</a> <font color="Maroon" title="c1">n</font> is  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="measure6.html#D7" title="MEASURE6:def.7">MEASURE6:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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