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<b>let </b><font color="Maroon">f</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">g</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:58_1_2"/><i><font color="Green">E81</font></i>: 
(  <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font> <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>  &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Olive">k</font> ) )
 <b>and </b><br/><a NAME="E2:58_1_2"/><i><font color="Green">E91</font></i>: 
(  <a href="polynom1.html#K11">support</a> <font color="Maroon">g</font> <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>  &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Olive">k</font> ) )
 ;<br/>

<a NAME="E3:58_1_2"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:58_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_2_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font> or  not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:58_1_2_1_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">j</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:58_1_2_1_1_2"/><i><font color="Green">E93</font></i>: 
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
0
<b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:58_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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