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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><a NAME="E1:8_1"/><i><font color="Green" title="E5">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D10" title="BORSUK_1:def.10">BORSUK_1:def 10</a></i>;<br/><a NAME="E2:8_1"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p1">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:8_1"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:8_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="eqrel_1.html#T51" title="EQREL_1:th.51">EQREL_1:51</a></i>;<br/><a NAME="E5:8_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:8_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="eqrel_1.html#D11" title="EQREL_1:def.11">EQREL_1:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">q</font> = <font color="Maroon" title="c2">p</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E4:8_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/><a NAME="E7:8_1"/><i><font color="Green" title="E9">A5</font></i>: 
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c5">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:8_1_1"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c5">Y</font> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E2:8_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="eqrel_1.html#D10" title="EQREL_1:def.10">EQREL_1:def 10</a></i>;<br/>
<a NAME="E4:8_1_1"/>
 <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E2:8_1_1"><i><font color="Green" title="E10">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:8_1_1"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> </span>}</span>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:8_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> </span>}</span>  or <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )</span><br/>

<b>assume </b><a NAME="E1:8_1_2"/>
<font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">Y</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:8_1_2"/><i><font color="Green" title="E10">A7</font></i>: 
( <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">Y</font> &amp; <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> )
 ;<br/>
<b>thus </b><a NAME="E4:8_1_2"/>
<font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="#E3:8_1_2"><i><font color="Green" title="E10">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">m</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">S</font></span>)</span> where <font color="Olive" title="b1">S</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="#E7:8_1"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c5">m</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E11:8_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> holds <br/><font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> implies <font color="Maroon" title="c7">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<b>assume </b><a NAME="E1:8_1_3"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<b>then consider </b><font color="Maroon" title="c8">S</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:8_1_3"/><i><font color="Green" title="E10">A8</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c8">S</font> &amp; <font color="Maroon" title="c8">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">B</font> being    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:8_1_3"/><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font> &amp; <font color="Maroon" title="c9">B</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> )
 <b>by </b><i><a class="txt" href="#E3:8_1_3"><i><font color="Green" title="E10">A8</font></i></a></i>;<br/>
<a NAME="E6:8_1_3"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S</font>
 
<b>by </b><i><a class="txt" href="#E3:8_1_3"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:8_1_3"/>
<font color="Maroon" title="c7">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E5:8_1_3"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E12:8_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E5:8_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="pre_topc.html#T44" title="PRE_TOPC:th.44">PRE_TOPC:44</a></i>;<br/><a NAME="E13:8_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/><a NAME="E14:8_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E15:8_1"/><b>then </b><i><font color="Green" title="E10">A10</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="#E2:8_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="eqrel_1.html#T53" title="EQREL_1:th.53">EQREL_1:53</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">I</font> = <span class="p1">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E17:8_1"/><i><font color="Green" title="E11">A11</font></i>: 
<font color="Maroon" title="c7">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E15:8_1"><i><font color="Green" title="E10">A10</font></i></a></i>;<br/><a NAME="E18:8_1"/><i><font color="Green" title="E12">A12</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#D10" title="BORSUK_1:def.10">BORSUK_1:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c7">I</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K12" title="EQREL_1:func.12">Intersection</a> <span class="p3">(<span class="default"><a href="t_1topsp.html#K1" title="T_1TOPSP:func.1">Closed_Partitions</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#D10" title="BORSUK_1:def.10">BORSUK_1:def 10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">I</font> = <font color="Maroon" title="c8">I</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span> ;<br/><a NAME="E21:8_1"/>
<font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D10" title="BORSUK_1:def.10">BORSUK_1:def 10</a></i>;<br/><a NAME="E22:8_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E17:8_1"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="#E18:8_1"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E23:8_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:8"/>
 <a href="t_1topsp.html#K2" title="T_1TOPSP:func.2">T_1-reflex</a> <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V7" title="PRE_TOPC:attr.7">T_1</a>
 <b>by </b><i><a class="ref" href="urysohn1.html#T27" title="URYSOHN1:th.27">URYSOHN1:27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
