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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <a href="numbers.html#K5">NAT</a>  implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <a href="numbers.html#K5">NAT</a>  implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  =  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = 0;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a> ;<br/>
<a NAME="E1:58_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] implies <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] implies <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) )
 
<b>from </b><i><a class="ref" href="pre_circ.html#S2">PRE_CIRC:sch 2</a>();<br/></i>
<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:58_1_1"/><i><font color="Green">E38</font></i>: 
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 ;<br/>
<a NAME="E4:58_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><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> ; <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="E1:58_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:58_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <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="E3:58_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:58_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_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_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>] or  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:58_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_1_1_1"/>
<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#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:58_1_1_1_1_2"/>
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_1_1_2"/>
<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="txt" href="moebius1.html#E2:58_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="moebius1.html#E3:58_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="moebius1.html#E4:58_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="moebius1.html#E1:58_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:58_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_1_1_2"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a> 
  <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="58_1_1_2_1"><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> ; <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="E1:58_1_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:58_1_1_2_1"/><b>then </b>
<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="txt" href="moebius1.html#E4:58_1_1"><i><font color="Green">E39</font></i></a></i>;<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/>
<b>assume </b><a NAME="E4:58_1_1_2_1"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 ;<br/>

<a NAME="E5:58_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_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="txt" href="moebius1.html#E3:58_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:58_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:58_1_1_2_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>3</sub></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="E2:58_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <a href="moebius1.html#K1" target="_self">SCNAT</a> 
 ;<br/>

<a NAME="E3:58_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<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#D3">XBOOLE_0:def 3</a></i>;<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 NAME="E5:58_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E2:58_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="moebius1.html#E3:58_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E6:58_1_1_2"/>
 <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:58_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="moebius1.html#T38" target="_self">Th38</a></i>;<br/>
<b>hence </b><a NAME="E7:58_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:58_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E7:58_1_1"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <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#E5:58_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E8:58_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>hence </b><a NAME="E9:58_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="moebius1.html#K2" target="_self">Moebius</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:58_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="moebius1.html#E5:58_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div>
