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

<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>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="E1:35_1_1"/><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/>
<a NAME="E2:35_1_1"/>
<span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E3:35_1_1"/><b>then </b>
<font color="Maroon">X</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> 
 
;<br/>
<b>hence </b><a NAME="E4:35_1_1"/>
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#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="topgen_2.html#T1" target="_self">Th1</a>, <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#D1">PRE_TOPC:def 1</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM3">Subset-Family</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/><b>assume </b><a NAME="E1:35_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</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>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_1_1"/>
(  not <font color="Maroon">a</font> <a href="tarski.html#R1">c=</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="tarski.html#R1">c=</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1_1_1"/>
not <font color="Maroon">a</font> <a href="tarski.html#R1">c=</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> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; not <font color="Maroon">b</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>  )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:35_1_1_1_1_1"/>
<font color="Maroon">b</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="topgen_2.html#T1" target="_self">Th1</a>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">F</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:35_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E7:35_1_1_1_1_1"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E8:35_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="topgen_2.html#T1" target="_self">Th1</a>, , <a class="ref" href="subset_1.html#T36">SUBSET_1:36</a></i>;<br/><a NAME="E9:35_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E10:35_1_1_1_1_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <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="topgen_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E11:35_1_1_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <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="topgen_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</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> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_1_1_2_1"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_1_1_2_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:35_1_1_1_1_2_1"/>
<font color="Maroon">b</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> 
 <b>by </b><i>, </i>;<br/><a NAME="E5:35_1_1_1_1_2_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">b</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> )
 ;<br/><b>hence </b><a NAME="E6:35_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E3:35_1_1_1_1_2"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <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> 
 <b>by </b><i><a class="ref" href="topgen_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E4:35_1_1_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <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="topgen_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<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>, <font color="Maroon">b</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/>

<b>assume </b><a NAME="E6:35_1_1"/><i><font color="Green">E41</font></i>: 
( <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> &amp; <font color="Maroon">b</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> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_2"/>
( ( <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>  &amp; <font color="Maroon">b</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>  ) or <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">b</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>  )
 </span><b>by </b><i><a class="ref" href="topgen_2.html#T1" target="_self">Th1</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_1"><b>suppose </b></a><a NAME="E1:35_1_1_2_1"/><i><font color="Green">E42</font></i>: 
( <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>  &amp; <font color="Maroon">b</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>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F1</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:35_1_1_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">F1</font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/><b>consider </b><font color="Maroon">F2</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:35_1_1_2_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">F2</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i/>;<br/><a NAME="E6:35_1_1_2_1"/>
( <font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">F2</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">F1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">F2</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a>, <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/><a NAME="E7:35_1_1_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</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> 
 ;<br/><b>hence </b><a NAME="E8:35_1_1_2_1"/>
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</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="topgen_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2_2"><b>suppose </b></a><a NAME="E1:35_1_1_2_2"/>
( <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">b</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>  )
 ;<br/><div class="add"><a NAME="E2:35_1_1_2_2"/><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">U</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">b</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> ) )
 ;<br/><a NAME="E3:35_1_1_2_2"/><b>then </b>
not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:35_1_1_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</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> 
 <b>by </b><i><a class="ref" href="topgen_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E5:35_1_1_2_2"/>
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</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="topgen_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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