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

<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>set </b><font color="Maroon">O</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>  <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> ;<br/>
<a NAME="E1:34_1_1"/>
<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>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1_1_1"/>
<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>  <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> 
 ;<br/>

<a NAME="E2:34_1_1_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="E3:34_1_1_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="E4:34_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">O</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>  <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>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>take </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">O</font> #)
;<br/>

<b>thus </b><a NAME="E3:34_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">O</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<font color="Maroon">O</font> #) <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>  )
 ;<br/>


</div>
