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<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<a NAME="E1:9"/><i><font color="Green">E18</font></i>: 
 <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> :  ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:9"/><i><font color="Green">E19</font></i>: 
 <a href="tdlat_2.html#NK1" target="_self">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">D</font> where <font color="Olive">D</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> :  ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:9"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font> iff <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

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

<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><b>consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:9_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#D3">PCOMPS_1:def 3</a></i>;<br/><a NAME="E6:9_1_1"/>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , <a class="ref" href="pcomps_1.html#D3">PCOMPS_1:def 3</a>, <a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/><b>hence </b><a NAME="E7:9_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><a NAME="E3:9_1_2"/>
 ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  ex <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font> ) )
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:9_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E6:9_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E7:9_1_2"/>
 ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">R</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">R</font> &amp; <font color="Olive">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) )
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">R</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E9:9_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E10:9_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E11:9_1_2"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/><b>hence </b><a NAME="E12:9_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tdlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="pcomps_1.html#D3">PCOMPS_1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font> iff <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
 <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
