<?xml version="1.0"?>
<div class="add">

<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>assume </b><a NAME="E1:20"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">FX</font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a>
 ;<br/>

<b>set </b><font color="Maroon">GX</font> =  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font>;<br/>
<a NAME="E2:20"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  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="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">W</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>thus </b><a NAME="E1:20_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#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">W</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<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="E2:20_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E3:20_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:20_1_1"/><i><font color="Green">E32</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">CGX</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">CFX</font> = <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> =  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E5:20_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">Y</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/>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FX</font>, <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font><b> such that </b><br/><a NAME="E7:20_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">X</font>)
 <b>from </b><i><a class="ref" href="pcomps_1.html#S1" target="_self">PCOMPS_1:sch 1</a>();<br/></i>
<a NAME="E8:20_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_1"/>
(  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_1_1"/>
 <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:20_1_1_1_1"/><b>then </b>
<font color="Maroon">FX</font> <a href="setfam_1.html#R1">is_finer_than</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:20_1_1_1_1"/>
<font color="Maroon">FX</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="setfam_1.html#T21">SETFAM_1:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FX</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:20_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_2"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  iff <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:20_1_1_2_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  implies <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_2_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_1_1_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:20_1_1_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E5:20_1_1_2_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="txt" href="pcomps_1.html#E1:20"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E7:20_1_1_2_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="txt" href="pcomps_1.html#E1:20"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="pcomps_1.html#E3:20_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E9:20_1_1_2_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">Z</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>, <a class="txt" href="pcomps_1.html#E1:20"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pcomps_1.html#E3:20_1_1"><i><font color="Green">E28</font></i></a>, , </i>;<br/>
<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E11:20_1_1_2_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E12:20_1_1_2_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>and </b><br/><a NAME="E13:20_1_1_2_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E2:20_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E14:20_1_1_2_1_1"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E3:20_1_1"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="pcomps_1.html#T2" target="_self">Th2</a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E15:20_1_1_2_1_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E4:20_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<b>hence </b><a NAME="E16:20_1_1_2_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:20_1_1_2_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  implies <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_1_2_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:20_1_1_2_1_2"/>
 ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:20_1_1_2_1_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E3:20_1_1_2_1_2_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font>
 <b>and </b><br/><a NAME="E4:20_1_1_2_1_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:20"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E5:20_1_1_2_1_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W</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="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:20_1_1_2_1_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:20_1_1_2_1_2_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:20_1_1"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<a NAME="E9:20_1_1_2_1_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_2_1_2_1_1"/>
 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="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> &amp; <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">Q</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">Q</font> = <font color="Maroon">X</font>;<br/>

<b>thus </b><a NAME="E1:20_1_1_2_1_2_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 ;<br/>

<b>thus </b><a NAME="E2:20_1_1_2_1_2_1_1_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E4:20_1_1"><i><font color="Green">E32</font></i></a></i>;<br/>

<a NAME="E3:20_1_1_2_1_2_1_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <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="pcomps_1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E4:20_1_1_2_1_2_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> <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="tops_1.html#T41">TOPS_1:41</a></i>;<br/>
<a NAME="E5:20_1_1_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:20_1_1_2_1_2_1_1_1"/>
<font color="Maroon">Q</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1_2_1_2_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E10:20_1_1_2_1_2_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , <a class="txt" href="pcomps_1.html#E2:20_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pcomps_1.html#T2" target="_self">Th2</a>, <a class="txt" href="pcomps_1.html#E4:20_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pcomps_1.html#E5:20_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_2_1_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1_1_2_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  iff <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="pcomps_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1_2"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>thus </b><a NAME="E10:20_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E11:20_1_1"/>
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E12:20_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> &amp; <font color="Olive">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , <a class="ref" href="finset_1.html#T17">FINSET_1:17</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
 <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a>
 <b>by </b><i/>;<br/>


</div>
