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

<b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> , <font color="Maroon">Y</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:62"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 ;<br/>

<b>thus </b><a NAME="E2:62"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:62_1"/><i><font color="Green">E50</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:62_1"/><i><font color="Green">E51</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:62_1"/><i><font color="Green">E56</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:62"><i><font color="Green">E41</font></i></a>, <a class="txt" href="moebius1.html#E1:62_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="xboole_1.html#T76">XBOOLE_1:76</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_1"/>
( <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> <font color="Maroon">X</font></span>)</span> or <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> <font color="Maroon">Y</font></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><b>suppose </b></a><a NAME="E1:62_1_1_1"/><i><font color="Green">E72</font></i>: 
<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> <font color="Maroon">X</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:62_1_1_1"/><b>then </b><i><font color="Green">E73</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> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E3:62_1_1_1"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:62_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E5:62_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:62_1_1_1"/>
<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"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2"><b>suppose </b></a><a NAME="E1:62_1_1_2"/><i><font color="Green">E74</font></i>: 
<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> <font color="Maroon">Y</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:62_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> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E3:62_1_1_2"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:62_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E5:62_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:62_1_1_2"/>
<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"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E3:62"/>
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T91">POLYNOM1:91</a></i>;<br/>


</div>
