<?xml version="1.0"?>
<div class="add">

<a NAME="E1:61"/><i><font color="Green">E41</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>,1 <a href="uproots.html#K2">-bag</a>  as   <a href="uproots.html#NM1">Rbag</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:61"/><i><font color="Green">E42</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="uproots.html#T10">UPROOTS:10</a></i>;<br/>
<a NAME="E4:61"/><i><font color="Green">E50</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:61"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E6:61"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">J</font> = 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E8:61"/>
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:61"/><b>then </b>
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:61"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:61"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:61"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="seq_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:61"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="seq_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:61_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  implies <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="uproots.html#T9">UPROOTS:9</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_2"/><i><font color="Green">E74</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:61_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:61"><i><font color="Green">E42</font></i></a>, </i>;<br/><font color="Maroon">M</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> = 
0
<b>by </b><i><a class="ref" href="uproots.html#T8">UPROOTS:8</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a>, </i>
;<br/><b>hence </b><a NAME="E4:61_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  implies <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:61"/><b>then </b>
 <a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> <a href="pboole.html#R6">=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>then </b> <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" target="_self">NatDivisors</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">M</font> <a href="seq_1.html#K1">.</a> 1
<b>by </b><i>, , </i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="uproots.html#T9">UPROOTS:9</a>, </i>
;<br/>
<b>hence </b><a NAME="E15:61"/>
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <span class="p2">(<span class="default"><a href="moebius1.html#K3" target="_self">NatDivisors</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/>


</div>
