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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">G</font> be   <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">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> ex <font color="Olive">GS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ex <font color="Olive">GT</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">GT</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">GS</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Olive">GT</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">GT</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> ) </span>}</span> ;<br/>
<a NAME="E2:8"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">GS</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Olive">GT</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">GT</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</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">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">GS</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Olive">GT</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">GT</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">GS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Maroon">GT</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">GS</font>,<font color="Maroon">GT</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E4:8_1"/>
( <font color="Maroon">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">GT</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">GS</font>,<font color="Maroon">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 ;<br/>
<b>thus </b><a NAME="E5:8_1"/>
<font color="Maroon">e</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">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">GS</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Olive">GT</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">GT</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</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">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</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">S</font>,<font color="Maroon">T</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E5:8"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> iff  ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:8_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> implies  ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon">GS</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>, <font color="Maroon">GT</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:8_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">GT</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">GS</font>,<font color="Maroon">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">GS</font>,<font color="Maroon">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">Y</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">GS</font>,<font color="Maroon">GT</font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>

<b>thus </b><a NAME="E4:8_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:8_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:8_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E4:8_2"/>
 ex <font color="Olive">GS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ex <font color="Olive">GT</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">GT</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:8"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:8"/>
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">A</font> holds <br/> ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><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="E1:8_3"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:8_3"/><b>then </b>
 ex <font color="Olive">GS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ex <font color="Olive">GT</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">GT</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 
;<br/>
<b>hence </b><a NAME="E3:8_3"/>
 ex <font color="Olive">GS</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ex <font color="Olive">GT</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">GS</font>,<font color="Olive">GT</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">GS</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">GT</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:8"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>


</div>
