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

<b>set </b><font color="Maroon">C</font> =  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>;<br/>
<a NAME="E1:58_1_1"/><i><font color="Green">E16</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E2:58_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>set </b><font color="Maroon">Y1</font> = <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font>;<br/>
<a NAME="E4:58_1_1"/><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<a NAME="E6:58_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T105">TMAP_1:105</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E2:58_1_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="tmap_1.html#K5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/><a NAME="E3:58_1_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">p</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span>, <font color="Olive">q</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> : ( <font color="Olive">p</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="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> &amp; <font color="Olive">q</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="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tmap_1.html#D7">TMAP_1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:58_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E6:58_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">P</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="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">Q</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="tex_1.html#K2" target="_self">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_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="E1:58_1_1_1_1_1"/>
( ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> ) )
 </span><b>by </b><i><a class="ref" href="tex_1.html#T7" target="_self">Th7</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:58_1_1_1_1_1_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_1_1_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tex_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:58_1_1_1_1_1_2"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_1_1_1_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:58_1_1_1_1_1_3"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_1_1_1_3"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tex_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:58_1_1_1_1_1_4"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:58_1_1_1_1_1_4"/><b>then </b>
( <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_1_1_1_4"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#T6" target="_self">Th6</a>, <a class="txt" href="tex_1.html#E8"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:58_1_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">A1</font>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:58_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">A1</font>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">Y2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span>
;<br/>

<a NAME="E9:58_1_1"/><i><font color="Green">E52</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A2</font> = <font color="Maroon">A1</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<b>set </b><font color="Maroon">A3</font> = <font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E11:58_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">A2</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A2</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:58_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1_1_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:58_1_1_2"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="tmap_1.html#K5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/><a NAME="E3:58_1_1_2"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span> where <font color="Olive">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span>, <font color="Olive">Q</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> : ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tmap_1.html#D7">TMAP_1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E5:58_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:58_1_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_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="E1:58_1_1_2_1_1"/>
( ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> ) or ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> ) )
 </span><b>by </b><i><a class="ref" href="tex_1.html#T6" target="_self">Th6</a>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:58_1_1_2_1_1_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_2_1_1_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tex_1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:58_1_1_2_1_1_2"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_2_1_1_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tex_1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:58_1_1_2_1_1_3"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_2_1_1_3"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:58_1_1_2_1_1_4"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><a NAME="E2:58_1_1_2_1_1_4"/>
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_2_1_1_4"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tex_1.html#D1" target="_self">Def1</a>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_5"><b>case </b></a><a NAME="E1:58_1_1_2_1_1_5"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><a NAME="E2:58_1_1_2_1_1_5"/>
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E3:58_1_1_2_1_1_5"/><b>then </b>
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_2_1_1_5"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tex_1.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_6"><b>case </b></a><a NAME="E1:58_1_1_2_1_1_6"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> )
 ;<br/><div class="add"><a NAME="E2:58_1_1_2_1_1_6"/>
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E3:58_1_1_2_1_1_6"/><b>then </b>
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_2_1_1_6"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_7"><b>case </b></a><a NAME="E1:58_1_1_2_1_1_7"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_2_1_1_7"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="tex_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tex_1.html#T7" target="_self">Th7</a>, <a class="ref" href="tex_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_8"><b>case </b></a><a NAME="E1:58_1_1_2_1_1_8"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 ;<br/><div class="add"><a NAME="E2:58_1_1_2_1_1_8"/><b>then </b>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tex_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tex_1.html#D1" target="_self">Def1</a>, <a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_2_1_1_8"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_2_1_1_9"><b>case </b></a><a NAME="E1:58_1_1_2_1_1_9"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_2_1_1_9"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tex_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tex_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:58_1_1_2"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">A2</font>,<span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:58_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">A2</font>,<span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span><a href="enumset1.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1_1_3"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">A2</font>,<span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span><a href="enumset1.html#K2">}</a></span>
 ;<br/><a NAME="E2:58_1_1_3"/><b>then </b>
( <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> or <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a>  or <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_3"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tex_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tex_1.html#T7" target="_self">Th7</a>, <a class="txt" href="tex_1.html#E8"><i><font color="Green">Lemma32</font></i></a>, , <a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:58_1_1"/><b>then </b>
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">A2</font>,<span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span><a href="enumset1.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E17:58_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">A2</font>,<span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span><a href="enumset1.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tex_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E18:58_1_1"/>
( 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B0</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">B1</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">B0</font>,<font color="Maroon">B1</font></span><a href="domain_1.html#K7">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="tex_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tex_1.html#T7" target="_self">Th7</a></i>;<br/>
<div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_4"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">A2</font> = <font color="Maroon">A2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_4_1"/>
<font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E2:58_1_1_4_1"/><b>then </b>
<font color="Maroon">B0</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_4_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:58_1_1_4"/>
( <font color="Maroon">A2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tex_1.html#T7" target="_self">Th7</a>, , <a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_5"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">B</font> = <font color="Maroon">B</font>;<br/><a NAME="E1:58_1_1_5"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">A2</font>,<span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E2:58_1_1_5"/><b>then </b><i><font color="Green">E88</font></i>: 
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> or  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a>  or  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_5_1"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E2:58_1_1_5_1"/><b>then </b>
 <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_5_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_5_2"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:58_1_1_5_2"/><b>then </b>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_5_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_5_3"/><i><font color="Green">E91</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><b>reconsider </b><font color="Maroon">I</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="tex_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tex_1.html#T7" target="_self">Th7</a>, <a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/><a NAME="E3:58_1_1_5_3"/>
not <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:58_1_1_5_3"/><b>then </b>
( <font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a>, <a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><a NAME="E5:58_1_1_5_3"/><b>then </b>
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B0</font> or <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E6:58_1_1_5_3"/><b>then </b>
( 1 <a href="hidden.html#R2">in</a> <font color="Maroon">B0</font> or 0 <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:58_1_1_5_3"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:58_1_1_5"/>
<font color="Maroon">B</font> is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i>, , , <a class="ref" href="tops_1.html#T84">TOPS_1:84</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/><b>assume </b><a NAME="E1:58_1_1_6"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:58_1_1_6"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">A2</font>,<span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span><a href="enumset1.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_6_1"/>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 ;<br/><a NAME="E2:58_1_1_6_1"/><b>then </b>
( ( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> ) &amp;  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="tex_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tex_1.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E3:58_1_1_6_1"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/></div><b>end;</b></div><div><i><font color="Green">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_6_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:58_1_1_6_2"/>
<font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:58_1_1_6_2"/><b>then </b>
<font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_6_2"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_6_3"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E2:58_1_1_6_3"/>
<font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p5"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:58_1_1_6_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_6_3"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:58_1_1_6"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, , , <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E24:58_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="tdlat_3.html#V3">almost_discrete</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="tdlat_3.html#V1">discrete</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="tdlat_3.html#V2">anti-discrete</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="pre_topc.html#V1">strict</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" target="_self">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <span class="p4"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">A</font></span>)</span> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i>, , , <a class="ref" href="tdlat_3.html#T20">TDLAT_3:20</a>, <a class="ref" href="tdlat_3.html#T23">TDLAT_3:23</a></i>;<br/>


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