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<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="E1:10"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">FX</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:10"/>
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> <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="10_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>take </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
;<br/>

<a NAME="E1:10_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>hence </b><a NAME="E2:10_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/>

<b>thus </b><a NAME="E3:10_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E4:10_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>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">FX</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:10_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>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</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>
<b>hence </b><a NAME="E3:10"/>
<font color="Maroon">FX</font> is <a href="pcomps_1.html#V1" target="_self">locally_finite</a>
 <b>by </b><i/>;<br/>


</div>
