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

<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><a NAME="E1:70"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>

<a NAME="E2:70"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M2">a_neighborhood</a> of <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:70_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D2">CONNSP_2:def 2</a></i>;<br/><a NAME="E3:70_1_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K10" target="_self">Base-Appr</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E21:70_1_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E4:70_1_1_1"/><b>then </b>
 <a href="borsuk_1.html#K10" target="_self">Base-Appr</a> <font color="Maroon">G</font> <a href="compts_1.html#R1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">K</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span><b> such that </b><br/><a NAME="E6:70_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a>  <a href="borsuk_1.html#K10" target="_self">Base-Appr</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E7:70_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">K</font> <a href="compts_1.html#R1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <b>and </b><br/><a NAME="E8:70_1_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> holds <br/><font color="Olive">c</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T16" target="_self">Th16</a></i>;<br/><b>reconsider </b><font color="Maroon">PK</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" target="_self">Pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E10:70_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">PK</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:70_1_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="borsuk_1.html#K10" target="_self">Base-Appr</a> <font color="Maroon">G</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E16:70_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E12:70_1_1_1"/><b>then </b>
<font color="Maroon">K</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T18">TOPS_2:18</a></i>;<br/><a NAME="E13:70_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" target="_self">Pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font> is <a href="tops_2.html#V1">open</a> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" target="_self">Pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font> is <a href="tops_2.html#V1">open</a> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">M</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E15:70_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" target="_self">Pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E16:70_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">M</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E17:70_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">M</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T17" target="_self">Th17</a>, <a class="ref" href="borsuk_1.html#T18" target="_self">Th18</a>, <a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a></i>;<br/><b>consider </b><font color="Maroon">N</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span><b> such that </b><br/><a NAME="E19:70_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E20:70_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">N</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E21:70_1_1_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K11" target="_self">Pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T19" target="_self">Th19</a>, , </i>;<br/><a NAME="E22:70_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">N</font> <a href="compts_1.html#R1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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>,<a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a><br/>

<b>assume </b><a NAME="E1:70_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 ;<br/>

<a NAME="E2:70_1_1_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
( <font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E3:70_1_1_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:70_1_1_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:70_1_1_1_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">XY</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span><b> such that </b><br/><a NAME="E8:70_1_1_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">XY</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E9:70_1_1_1_1"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K11" target="_self">Pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">XY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T21" target="_self">Th21</a>, <a class="ref" href="borsuk_1.html#T26" target="_self">Th26</a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E10:70_1_1_1_1"/>
<font color="Maroon">XY</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E11:70_1_1_1_1"/><b>then </b>
<font color="Maroon">XY</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K10" target="_self">Base-Appr</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">X2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">Y2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E13:70_1_1_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">XY</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <b>and </b><br/><a NAME="E14:70_1_1_1_1"/>
( <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">X2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y2</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>
<a NAME="E15:70_1_1_1_1"/>
<font color="Maroon">XY</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T26" target="_self">Th26</a>, , <a class="ref" href="funct_3.html#T9">FUNCT_3:9</a></i>;<br/>
<a NAME="E16:70_1_1_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T26" target="_self">Th26</a>, , , </i>;<br/>
<a NAME="E17:70_1_1_1_1"/>
<font color="Maroon">XY</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="borsuk_1.html#T27" target="_self">Th27</a></i>;<br/>
<b>then consider </b><font color="Maroon">xy</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:70_1_1_1_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">xy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">XY</font> &amp; <font color="Maroon">xy</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E20:70_1_1_1_1"/>
<font color="Maroon">xy</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E21:70_1_1_1_1"/><b>then </b>
<font color="Maroon">xy</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E22:70_1_1_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E23:70_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T23" target="_self">Th23</a>, <a class="ref" href="borsuk_1.html#T25" target="_self">Th25</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<a NAME="E24:70_1_1_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D1" target="_self">Def1</a>, <a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E25:70_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T27" target="_self">Th27</a>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:70_1_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="borsuk_1.html#K10" target="_self">Base-Appr</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/><a NAME="E2:70_1_1_1_2"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K10" target="_self">Base-Appr</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E3:70_1_1_1_2"/><b>then </b>
 ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/><b>hence </b><a NAME="E4:70_1_1_1_2"/>
( <font color="Maroon">e</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> )
 ;<br/></div><b>end;</b></div><a NAME="E25:70_1_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">N</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tops_2.html#T18">TOPS_2:18</a></i>;<br/><a NAME="E26:70_1_1_1"/>
( <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">N</font> )
 <b>by </b><i>, , <a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><a NAME="E27:70_1_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" target="_self">Pr1</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E29:70_1_1_1"/>
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T25" target="_self">Th25</a>, </i>;<br/><a NAME="E30:70_1_1_1"/><b>then </b><i><font color="Green">E148</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/><a NAME="E31:70_1_1_1"/>
<font color="Maroon">F</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E32:70_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/><a NAME="E33:70_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon">V</font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> as    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="connsp_2.html#D2">CONNSP_2:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">H</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" target="_self">Pr2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">N</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E36:70_1_1_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">H</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T20" target="_self">Th20</a>, <a class="ref" href="finset_1.html#T17">FINSET_1:17</a></i>;<br/><a NAME="E37:70_1_1_1"/>
<font color="Maroon">H</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T25" target="_self">Th25</a>, </i>;<br/><a NAME="E38:70_1_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T27">TOPS_2:27</a></i>;<br/><a NAME="E39:70_1_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T26" target="_self">Th26</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1_1_1_3"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 ;<br/><b>then consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span><b> such that </b><br/><a NAME="E3:70_1_1_1_3"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E4:70_1_1_1_3"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <b>by </b><i>, , </i>;<br/><a NAME="E5:70_1_1_1_3"/><b>then </b>
<font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span><b> such that </b><br/><a NAME="E7:70_1_1_1_3"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E8:70_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:70_1_1_1_3"/><b>then </b><i><font color="Green">E154</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">t</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E10:70_1_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E11:70_1_1_1_3"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><a NAME="E12:70_1_1_1_3"/><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self">Th30</a>, <a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>;<br/><a NAME="E13:70_1_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E14:70_1_1_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#T32" target="_self">Th32</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a>, <a class="ref" href="borsuk_1.html#T31" target="_self">Th31</a></i>;<br/></div><b>end;</b></div><a NAME="E41:70_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D1" target="_self">Def1</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E42:70_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon">W</font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">H</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">t</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><b>take </b><font color="Maroon">V</font> = <font color="Maroon">V</font>;<br/><b>take </b><font color="Maroon">W</font> = <font color="Maroon">W</font>;<br/><b>thus </b><a NAME="E44:70_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K9" target="_self">[:</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">W</font></span><a href="borsuk_1.html#K9" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T24" target="_self">Th24</a>, <a class="txt" href="borsuk_1.html#E22:70_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:70_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">V</font> =  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> as    <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="connsp_2.html#D2">CONNSP_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">W</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">t</font>;<br/><b>take </b><font color="Maroon">V</font> = <font color="Maroon">V</font>;<br/><b>take </b><font color="Maroon">W</font> = <font color="Maroon">W</font>;<br/><a NAME="E5:70_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K9" target="_self">[:</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">W</font></span><a href="borsuk_1.html#K9" target="_self">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><b>hence </b><a NAME="E6:70_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K9" target="_self">[:</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">W</font></span><a href="borsuk_1.html#K9" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:70"/>
 ex <font color="Olive">V</font> being   <a href="connsp_2.html#M2">a_neighborhood</a> of <font color="Maroon">A</font> ex <font color="Olive">W</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">t</font> st <span class="p1"><a href="borsuk_1.html#K9" target="_self">[:</a><span class="default"><font color="Olive">V</font>,<font color="Olive">W</font></span><a href="borsuk_1.html#K9" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 ;<br/>


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