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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">FX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">GX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>





<a NAME="E1:16"/>
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="pcomps_1.html#K2" target="_self">clf</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font></span>)</span> iff <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_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">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">X'</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">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:16_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E5:16_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_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="E1:16_1_1_1_1"/>
( <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> or <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_1"/><b>then </b>
<font color="Maroon">X'</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:16_1_1_1_1_1"/>
<font color="Maroon">X'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_1_2"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_2"/><b>then </b>
<font color="Maroon">X'</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:16_1_1_1_1_2"/>
<font color="Maroon">X'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:16_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_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="E1:16_1_2_1_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> or <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><a NAME="E3:16_1_2_1_1_1"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:16_1_2_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E3:16_1_2_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">Z</font>
;<br/>

<b>thus </b><a NAME="E4:16_1_2_1_1_1_1"/>
( <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:16_1_2_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><a NAME="E3:16_1_2_1_1_2"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:16_1_2_1_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E3:16_1_2_1_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">Z</font>
;<br/>

<b>thus </b><a NAME="E4:16_1_2_1_1_2_1"/>
( <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:16_1_2_1_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:16_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font></span>)</span> iff <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16"/>
 <a href="pcomps_1.html#K2" target="_self">clf</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">GX</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">GX</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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