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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/>





<b>assume </b><b>that </b><br/><a NAME="E1:9"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">FX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E2:9"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">GX</font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </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="E1:9_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> <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="9_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:9_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:9_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:9_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">W</font> = <font color="Maroon">Y</font>;<br/>

<a NAME="E5:9_1_1"/><i><font color="Green">E48</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>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1_1_1"/><i><font color="Green">E49</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> <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="E2:9_1_1_1_1"/>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1_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="E2:9_1_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E3:9_1_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font>
 <b>and </b><br/><a NAME="E4:9_1_1_1_1_1"/><i><font color="Green">E52</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/>;<br/>
<b>take </b><font color="Maroon">X</font> = <font color="Maroon">V</font>;<br/>

<b>thus </b><a NAME="E5:9_1_1_1_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E6:9_1_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E7:9_1_1_1_1_1"/>
<font color="Maroon">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:9_1_1_1_1"/>
<font color="Maroon">Z</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="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font> ) </span>}</span> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:9_1_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> <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="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> &amp; <font color="Olive">X</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Y</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:9_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="E7:9_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="E8:9_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> <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#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
<font color="Maroon">FX</font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a>
 <b>by </b><i/>;<br/>


</div>
