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

<b>set </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">T</font> = <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> ;<br/>
<a NAME="E1:36_1_1"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span>  <a href="xboole_0.html#R1">misses</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #);<br/>
<b>thus </b><a NAME="E2:36_1_1"/>
 <a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font> is <a href="pre_topc.html#V1">strict</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<a NAME="E4:36_1_1"/>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>  holds <br/>(  not <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> or  not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> or  not <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 
;<br/>
<a NAME="E5:36_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
not <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> 
 
;<br/>
<a NAME="E6:36_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #) <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #);<br/><b>assume </b><a NAME="E1:36_1_1_1"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 ;<br/><a NAME="E2:36_1_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="tex_1.html#E1:36_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E3:36_1_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_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="E1:36_1_1_1_1_1"/>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> or  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_1_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 <b>by </b><i><a class="txt" href="tex_1.html#E5:36_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_1_1_1_2"/><i><font color="Green">E26</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:36_1_1_1_1_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>assume </b><a NAME="E2:36_1_1_1_1_1_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E3:36_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="tex_1.html#E6:36_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_1_1_1_1_2_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> ;<br/><a NAME="E3:36_1_1_1_1_1_2_2"/>
not <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="tex_1.html#E6:36_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:36_1_1_1_1_1_2_2"/><b>then </b>
(  not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or  not <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 ;<br/><b>hence </b><a NAME="E5:36_1_1_1_1_1_2_2"/>
not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="txt" href="tex_1.html#E6:36_1_1"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:36_1_1_1_1_1_2"/><b>then </b>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> or  not <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/><a NAME="E5:36_1_1_1_1_1_2"/><b>then </b>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>  holds <br/>(  not <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> or  not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> or  not <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:36_1_1_1_1_1_2"/><b>then </b>
not  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E7:36_1_1_1_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:36_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #), <font color="Maroon">E</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #);<br/><b>assume </b><a NAME="E1:36_1_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #) &amp; <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #) )
 ;<br/><div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>;<br/><b>assume </b><a NAME="E1:36_1_1_2_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #) &amp; <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 ;<br/><a NAME="E2:36_1_1_2_1"/><b>then </b>
not <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_2_1"/>
not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_2_2_1"/>
( ( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> ) or <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> or <font color="Maroon">E</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:36_1_1_2_2_1_1"/>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_2_2_1_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 <b>by </b><i><a class="txt" href="tex_1.html#E5:36_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:36_1_1_2_2_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> or <font color="Maroon">E</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_2_2_1_2_1_1"/>
( <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> or <font color="Maroon">E</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:36_1_1_2_2_1_2_1_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_2_2_1_2_1_1_1"/><b>then </b>
( not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E3:36_1_1_2_2_1_2_1_1_1"/><b>then </b>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>  holds <br/>(  not <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> or  not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> or  not <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 ;<br/><a NAME="E4:36_1_1_2_2_1_2_1_1_1"/><b>then </b>
not <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:36_1_1_2_2_1_2_1_1_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:36_1_1_2_2_1_2_1_1_2"/>
<font color="Maroon">E</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_2_2_1_2_1_1_2"/><b>then </b>
( not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E3:36_1_1_2_2_1_2_1_1_2"/><b>then </b>
for <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>  holds <br/>(  not <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> or  not <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> or  not <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> )
 ;<br/><a NAME="E4:36_1_1_2_2_1_2_1_1_2"/><b>then </b>
not <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:36_1_1_2_2_1_2_1_1_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:36_1_1_2_2_1_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:36_1_1_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> </span>)</span> #)
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:36_1_1"/>
 <a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font> is <a href="pre_topc.html#V2">TopSpace-like</a>
 <b>by </b><i><a class="ref" href="tex_1.html#T3" target="_self">Th3</a>, <a class="ref" href="tex_1.html#T4" target="_self">Th4</a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>


</div>
