<?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">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="E1:36"/><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> )
 ;<br/>

<b>set </b><font color="Maroon">R</font> = <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="E2:36"/>
<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>  <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">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:36_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</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/>

<b>then consider </b><font color="Maroon">Q1</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="E3:36_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</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">Q1</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> ) )
 ;<br/>
<b>thus </b><a NAME="E4:36_1"/>
<font color="Maroon">s</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">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">R</font> = <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>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:36"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> &amp; <font color="Maroon">C</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">C</font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> 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="zfmisc_1.html#K2">[:</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">a<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Olive">FQ</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">FQ</font> );<br/>
<a NAME="E7:36"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </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="E1:36_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:36_2"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">Q1</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="E4:36_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</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">Q1</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> ) )
 ;<br/>
<b>thus </b><a NAME="E5:36_2"/>
 ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Olive">u</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">t</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:36"/><i><font color="Green">E46</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; ( for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>] ) )
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<a NAME="E10:36"/><i><font color="Green">E47</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><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="E1:36_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">K</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:36_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E7:36_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">FQ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">FQ</font> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</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">x1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FQ</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FQ</font> )
 <b>by </b><i>, <a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<b>thus </b><a NAME="E8:36_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</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>
<a NAME="E11:36"/><i><font color="Green">E56</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>proof </b></a><div class="add">

<a NAME="E1:36_4"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, , <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E2:36_4"/>
for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> holds <br/><font color="Olive">X1</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_4_1"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font>
 ;<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="E3:36_4_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">t</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E5:36_4_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">FQ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">FQ</font> is <a href="finset_1.html#V1">finite</a> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</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">x1</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">FQ</font> &amp; <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FQ</font> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E6:36_4_1"/>
<font color="Maroon">X1</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="txt" href="borsuk_3.html#E6:36"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_4"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="zfmisc_1.html#K2">[:</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">a<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">CX</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span> ;<br/>
<a NAME="E12:36"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</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 <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:36_5"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:36_5"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">f1</font>) &amp; <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>
<a NAME="E4:36_5"/>
<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">f1</font></span><a href="borsuk_1.html#K6">:]</a></span> <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/>
<b>hence </b><a NAME="E5:36_5"/>
<font color="Maroon">s</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 <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="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">CX</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> </span>}</span>  as   <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>reconsider </b><font color="Maroon">CX</font> = <font color="Maroon">CX</font> as   <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>set </b><font color="Maroon">G</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span>;<br/>
<a NAME="E15:36"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</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 <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:36_6"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span>
 ;<br/>

<a NAME="E2:36_6"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:36_6"/>
<font color="Maroon">s</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 <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/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span> as   <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>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <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>take </b>
<font color="Maroon">G</font>
;<br/>

<a NAME="E18:36"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">CX</font> <a href="hidden.html#R1">=</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">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">C</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:36_7_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">CX</font>
 ;<br/><b>then consider </b><font color="Maroon">GG</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_7_1"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GG</font> &amp; <font color="Maroon">GG</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">FF</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:36_7_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">GG</font> <a href="hidden.html#R1">=</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">FF</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">FF</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a></i>;<br/><b>consider </b><font color="Maroon">g1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">g2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:36_7_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E8:36_7_1"/>
<font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="txt" href="borsuk_3.html#E6:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E9:36_7_1"/>
<font color="Maroon">g</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">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">C</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E6:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">g</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="E2:36_7"/>
<font color="Maroon">g</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">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">C</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">g1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">g2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:36_7"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">C</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</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">GG</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:36_7"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GG</font> &amp; <font color="Maroon">GG</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</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/>
<a NAME="E7:36_7"/>
<font color="Maroon">GG</font> <a href="hidden.html#R2">in</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> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Q1</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="E9:36_7"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">GG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</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">Q1</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> ) )
 ;<br/>
<a NAME="E10:36_7"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">g</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">Q1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, , <a class="txt" href="borsuk_3.html#E6:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:36_7"/>
<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">Q1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</font>
 
<b>by </b><i>, <a class="txt" href="borsuk_3.html#E6:36"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:36_7"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">CX</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E115</font></i>:  <a href="setfam_1.html#K5">union</a> <font color="Maroon">CX</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">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
<b>by </b><i>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>
<br/>.= 
 <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/>
;<br/>
<a NAME="E20:36"/><i><font color="Green">E117</font></i>: 
 <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> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</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="E1:36_9"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</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>
 ;<br/>

<b>then consider </b><font color="Maroon">CC</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_9"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CC</font> &amp; <font color="Maroon">CC</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CX</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/>
<b>consider </b><font color="Maroon">Cc</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:36_9"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">CC</font> <a href="hidden.html#R1">=</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">Cc</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">Cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E6:36"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E6:36_9"/>
<font color="Maroon">Cc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Qq</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="E8:36_9"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">Cc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Qq</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">Qq</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> ) )
 ;<br/>
<b>consider </b><font color="Maroon">d1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:36_9"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">d2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cc</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">d1</font>,<font color="Maroon">d2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E6:36"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">FQ1</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="E12:36_9"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">FQ1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">FQ1</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">Qq</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">FQ1</font> &amp; <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Qq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FQ1</font> )
 <b>by </b><i>, , <a class="ref" href="borsuk_3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E13:36_9"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">FQ1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font>
 
<b>by </b><i>, , <a class="ref" href="borsuk_3.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E14:36_9"/>
<font color="Maroon">d</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">Qq</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T4" target="_self">Th4</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">DC</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:36_9"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DC</font> &amp; <font color="Maroon">DC</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FQ1</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E17:36_9"/>
<font color="Maroon">DC</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T5" target="_self">Th5</a>, <a class="ref" href="borsuk_3.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E18:36_9"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:36"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</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="E22:36"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</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="E23:36"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</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="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>thus </b><a NAME="E24:36"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E25:36"/>
<font color="Maroon">G</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>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E6:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>

<b>thus </b><a NAME="E26:36"/>
<font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a></i>;<br/>


</div>
