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<div class="add">

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>set </b><font color="Maroon">MM</font> = <font color="Maroon">M</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">M</font></span>)</span>;<br/>
<a NAME="E1:9"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:9"/>
 <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:9_1"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">M</font> holds <br/><font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:9_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:9_1_1"/>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>
<b>thus </b><a NAME="E6:9_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:9_1"/>
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">M</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:9_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:9_1_2"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E3:9_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:9_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:9"/><b>then </b>
<font color="Maroon">M</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<b>hence </b><a NAME="E4:9"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="closure3.html#K1" target="_self">supp</a> <font color="Maroon">M</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
