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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x0</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">T</font> =  <a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font>;<br/>
<b>set </b><font color="Maroon">O1</font> = <span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">O2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">F</font> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">F</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font></span>)</span>;<br/>

<a NAME="E1:37"/><i><font color="Green">E29</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">F</font> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">F</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:37"/>
(  not <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> or  not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="finset_1.html#V1">finite</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>

<a NAME="E2:37_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> </span>}</span>  or <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">F</font> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">F</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:37_1"/><b>then </b>
(  ex <font color="Olive">U</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">U</font> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> ) or  ex <font color="Olive">F</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) )
 
;<br/>
<b>hence </b><a NAME="E4:37_1"/>
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:37"/>
(  not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="finset_1.html#V1">finite</a> )
 ;<br/>

<a NAME="E4:37"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> </span>}</span>  or <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">F</font> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">F</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:37"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


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