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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>;<br/><a NAME="E1:24_1"/><i><font color="Green">E19</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/><a NAME="E2:24_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:24_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:24_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="t_1topsp.html#T2" target="_self">Th2</a></i>;<br/><b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/><a NAME="E7:24_1"/><i><font color="Green">E24</font></i>: 
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:24_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="t_1topsp.html#E5:24_1"><i><font color="Green">E23</font></i></a>, </i>;<br/>
<a NAME="E4:24_1_1"/>
 <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E5:24_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:24_1_1"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div><a NAME="E8:24_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:24_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> )
 ;<br/>
<b>thus </b><a NAME="E4:24_1_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="t_1topsp.html#E5:24_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">m</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Olive">S</font></span>)</span> where <font color="Olive">S</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">S</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="t_1topsp.html#E4:24_1"><i><font color="Green">E22</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/><a NAME="E11:24_1"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font> holds <br/><font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:24_1_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon">S</font> being    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:24_1_3"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="t_1topsp.html#K1" target="_self">EqClass</a> <font color="Maroon">x</font>,<font color="Maroon">S</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font> )
 ;<br/>
<a NAME="E4:24_1_3"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">B</font> is <a href="tops_2.html#V2">closed</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E5:24_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:24_1_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> is <a href="tops_2.html#V2">closed</a> )
 ;<br/>
<a NAME="E7:24_1_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="t_1topsp.html#E5:24_1"><i><font color="Green">E23</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E8:24_1_3"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="t_1topsp.html#E7:24_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:24_1"/><b>then </b>
<font color="Maroon">q</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="t_1topsp.html#E2:24_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pre_topc.html#T44">PRE_TOPC:44</a></i>;<br/><a NAME="E13:24_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">q</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><a NAME="E14:24_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E15:24_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 ;<br/><a NAME="E16:24_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="t_1topsp.html#E1:24_1"><i><font color="Green">E19</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">I</font> = <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E18:24_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">I</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 <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span> :  <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="t_1topsp.html#E5:24_1"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E19:24_1"/>
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14">space</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</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 <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span> :  <a href="tarski.html#K3">union</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/><a NAME="E20:24_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">I</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="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="t_1topsp.html#E7:24_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14">space</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" target="_self">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K3" target="_self">Closed_Partitions</a> <font color="Maroon">T</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span> ;<br/><a NAME="E23:24_1"/>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/><a NAME="E24:24_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> 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="E25:24_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24"/>
 <a href="t_1topsp.html#K4" target="_self">T_1-reflex</a> <font color="Maroon">T</font> is <a href="urysohn1.html#V1">being_T1</a>
 <b>by </b><i><a class="ref" href="urysohn1.html#T27">URYSOHN1:27</a></i>;<br/>


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