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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></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="E1:72"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="moebius1.html#V1" target="_self">square-containing</a> ) </span>}</span> 
 ;<br/>

<a NAME="E2:72"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:72_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">meets</a>  <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:72_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="moebius1.html#K1" target="_self">SCNAT</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:72_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="moebius1.html#V1" target="_self">square-containing</a> )
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:72"><i><font color="Green">E46</font></i></a>, <a class="txt" href="moebius1.html#E3:72_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:72_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E3:72_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="moebius1.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:72"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:72_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
 <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">c<sub>2</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a> 
<b>by </b><i><a class="ref" href="moebius1.html#D5" target="_self">Def5</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="moebius1.html#E2:72"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/>
<a NAME="E4:72_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></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/>
<b>hence </b><a NAME="E5:72_2"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K16">EmptyBag</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:72"/>
 <a href="moebius1.html#K4" target="_self">SMoebius</a> <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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