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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> ) </span>}</span>  )
 ;<br/>

<a NAME="E2:33"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></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="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E1:33"><i><font color="Green">E21</font></i></a></i>;<br/><b>thus </b><a NAME="E4:33_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E3:33_1"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
<font color="Maroon">c<sub>3</sub></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="E5:33"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E6:33"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E7:33"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:33_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E4:33_2"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E5:33_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></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="E6:33_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E2:33"><i><font color="Green">E22</font></i></a>, <a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<a NAME="E7:33_2"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T18" target="_self">Th18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E9:33_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E1:33"><i><font color="Green">E21</font></i></a>, <a class="txt" href="borsuk_3.html#E6:33_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a></i>;<br/>
<a NAME="E10:33_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E1:33"><i><font color="Green">E21</font></i></a>, <a class="txt" href="borsuk_3.html#E9:33_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tops_2.html#T18">TOPS_2:18</a></i>;<br/>
<a NAME="E11:33_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E9:33_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>8</sub></font> </span>}</span> ;<br/>
<a NAME="E12:33_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>8</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></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="E1:33_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>8</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:33_2_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>12</sub></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">c<sub>8</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:33_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E3:33_2_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>8</sub></font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E14:33_2"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E10:33_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>
<a NAME="E15:33_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E16:33_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></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="E17:33_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E11:33_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E18:33_2"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></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">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<a NAME="E1:33_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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></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">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></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="E1:33_2_2_1"/>
<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_2_2_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>15</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:33_2_2_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></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">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E3:33_2_2_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E6:33_2_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E5:33_2_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:33_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>15</sub></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="txt" href="borsuk_3.html#E3:33_2_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E8:33_2_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E5:33_2_2_1"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_2_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></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">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E9:33_2"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:33_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E1:33"><i><font color="Green">E21</font></i></a>, <a class="txt" href="borsuk_3.html#E16:33_2"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:33_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E17:33_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:33"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E6:33"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:33"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>


</div>
