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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V2">compact</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V2">compact</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">Q</font> where <font color="Olive">Q</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">FQ</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">FQ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Olive">FQ</font> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<font color="Olive">Q</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">FQ</font> ) </span>}</span>  )
 ;<br/>

<a NAME="E2:34"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/><b>then consider </b><font color="Maroon">E</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp;  ex <font color="Olive">FQ</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">FQ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Olive">FQ</font> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">E</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">FQ</font> ) )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:34_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:34"/>
<font color="Maroon">R</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>

<a NAME="E5:34"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E6:34"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E7:34"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</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="E1:34_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k'</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E4:34_2"/>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E5:34_2"/><b>then </b>
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E6:34_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T2" target="_self">Th2</a>, <a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<a NAME="E7:34_2"/>
<font color="Maroon">Z</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E1:34"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E9:34_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">G</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, , <a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a></i>;<br/>
<a NAME="E10:34_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">G</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="E11:34_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">uR</font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">Q</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> </span>}</span> ;<br/>
<a NAME="E12:34_2"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</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="E1:34_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:34_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> )
 ;<br/>
<b>thus </b><a NAME="E4:34_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Q</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E14:34_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E2:34"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>
<a NAME="E15:34_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:34_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<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#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E17:34_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:34_2"/>
 ex <font color="Olive">FQ</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">FQ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Olive">FQ</font> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Q</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">FQ</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">G</font>
;<br/>

<a NAME="E1:34_2_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Q</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_2_1"><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="E1:34_2_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Q</font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_2_2_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">x2'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:34_2_2_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">x2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2'</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:34_2_2_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2'</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:34_2_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2'</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E8:34_2_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_2_2"/>
( <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">Q</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E19:34_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:34_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:34"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:34"/>
<font color="Maroon">R</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>


</div>
