<?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="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:71"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:71"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 
;<br/>
<b>thus </b><a NAME="E3:71"/>
( <font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a>  implies  ex <font color="Olive">C1</font>, <font color="Olive">C2</font>, <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Olive">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">C2</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1"/>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">C1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:71_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E4:71_1"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:71_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T9" target="_self">Th9</a></i>;<br/>
<b>take </b>
<font color="Maroon">C1</font>
;<br/>

<b>take </b>
<font color="Maroon">C2</font>
;<br/>

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

<b>thus </b><a NAME="E6:71_1"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="tsep_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


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

<b>given </b><font color="Maroon">C1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E4:71"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:71"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E6:71"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<a NAME="E7:71"/>
 ex <font color="Olive">C1</font>, <font color="Olive">C2</font>, <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Olive">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">C2</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">C1</font>
;<br/>

<b>take </b>
<font color="Maroon">C2</font>
;<br/>

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

<b>thus </b><a NAME="E1:71_2"/>
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="tsep_1.html#T15" target="_self">Th15</a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:71"/>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T9" target="_self">Th9</a></i>;<br/>


</div>
