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

<b>let </b><font color="Maroon">S</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/><a NAME="E2:18_1"/>
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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><a NAME="E3:18_1"/><b>then </b>
 ex <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x1</font>,<font color="Olive">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>then reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">S</font>, <font color="Maroon">T1</font> = <font color="Maroon">T</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <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">S1</font>,<font color="Maroon">T1</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">K</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">p</font><b> such that </b><br/><a NAME="E7:18_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">Q</font> being  <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">S1</font>,<font color="Maroon">T1</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> holds <br/><span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">Q</font>
 <b>by </b><i>, <a class="ref" href="yellow13.html#T17">YELLOW13:17</a></i>;<br/><b>consider </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S1</font>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E9:18_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/><a NAME="E10:18_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/><font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:18_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:18_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">G</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</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/>
<a NAME="E4:18_1_1"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">G</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being   <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">S1</font>,<font color="Maroon">T1</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E6:18_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E7:18_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">G</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E8:18_1_1"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:18_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E11:18_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">G</font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:18_1_1"/><b>then </b>
<font color="Maroon">a</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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/>
<a NAME="E13:18_1_1"/><b>then </b>
 ex <font color="Olive">a1</font>, <font color="Olive">a2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> &amp; <font color="Olive">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T1</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">a2</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>hence </b><a NAME="E14:18_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:18_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/><a NAME="E12:18_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/><font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:18_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:18_1_2"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span> <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">S1</font></span>)</span>,<font color="Maroon">G</font></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/>
<a NAME="E4:18_1_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">S1</font></span>)</span>,<font color="Maroon">G</font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being   <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">S1</font>,<font color="Maroon">T1</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E6:18_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E7:18_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</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">S1</font></span>)</span>,<font color="Maroon">G</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E8:18_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:18_1_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E11:18_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</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">S1</font></span>)</span>,<font color="Maroon">G</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:18_1_2"/><b>then </b>
<font color="Maroon">a</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"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/>
<a NAME="E13:18_1_2"/><b>then </b>
 ex <font color="Olive">a1</font>, <font color="Olive">a2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S1</font></span>)</span> &amp; <font color="Olive">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a1</font>,<font color="Olive">a2</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>hence </b><a NAME="E14:18_1_2"/>
<font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:18_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/><b>hence </b><a NAME="E14:18_1"/>
<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"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</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/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</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="E2:18"/>
<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"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span></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="E4:18"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><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>reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">S</font>, <font color="Maroon">T1</font> = <font color="Maroon">T</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T1</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">K1</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x1</font><b> such that </b><br/><a NAME="E11:18"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font>  st <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font> holds <br/><font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">Q</font>
 <b>by </b><i>, <a class="ref" href="yellow13.html#T17">YELLOW13:17</a></i>;<br/>
<b>consider </b><font color="Maroon">K2</font> being    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x2</font><b> such that </b><br/><a NAME="E13:18"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>  st <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font> holds <br/><font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">Q</font>
 <b>by </b><i>, <a class="ref" href="yellow13.html#T17">YELLOW13:17</a></i>;<br/>
<a NAME="E14:18"/>
for <font color="Olive">G</font> being  <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">S1</font>,<font color="Maroon">T1</font></span><a href="borsuk_1.html#K5">:]</a></span>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/><span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<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">S1</font>,<font color="Maroon">T1</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>consider </b><font color="Maroon">F</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">S1</font>,<font color="Maroon">T1</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E4:18_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:18_2"/><i><font color="Green">E85</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">F</font> holds <br/> ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</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> )
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:18_2"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E8:18_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</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="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font>, <font color="Maroon">Y1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E10:18_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E11:18_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E12:18_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:18_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">V1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font><b> such that </b><br/><a NAME="E15:18_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">V1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K1</font>
 <b>and </b><br/><a NAME="E16:18_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">V1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 <b>by </b><i>, <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E17:18_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V1</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E19:18_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E20:18_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E21:18_2"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">V2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E23:18_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">V2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K2</font>
 <b>and </b><br/><a NAME="E24:18_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font>
 <b>by </b><i>, <a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E25:18_2"/>
<font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V2</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E27:18_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E28:18_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E29:18_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i>, , , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E30:18_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E31:18_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></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="E32:18_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:18"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>


</div>
