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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:71"/>
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:71"/><b>then </b><i><font color="Green" title="E49">A1</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font>
 
;<br/>
<b>thus </b><a NAME="E3:71"/>
( <font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a>  implies  ex <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font>, <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">C2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b1">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Olive" title="b2">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Olive" title="b3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Olive" title="b1">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b3">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">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" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">C1</font>, <font color="Maroon" title="c5">C2</font>, <font color="Maroon" title="c6">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:71_1"/><i><font color="Green" title="E50">A2</font></i>: 
( <font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c6">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> )
 <b>and </b><br/><a NAME="E4:71_1"/><i><font color="Green" title="E51">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font>
 <b>and </b><br/><a NAME="E5:71_1"/><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c4">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c5">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T59" target="_self" title="TSEP_1:th.59">Th59</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">C1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">C2</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">C</font>
;<br/>

<b>thus </b><a NAME="E6:71_1"/>
( <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c4">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c5">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c4">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c5">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E2:71"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="tsep_1.html#E3:71_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="tsep_1.html#E4:71_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="tsep_1.html#E5:71_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c4">C1</font>, <font color="Maroon" title="c5">C2</font>, <font color="Maroon" title="c6">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E4:71"/><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font>
 <b>and </b><br/><a NAME="E5:71"/><i><font color="Green" title="E51">A6</font></i>: 
( <font color="Maroon" title="c4">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c5">C2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> )
 <b>and </b><br/><a NAME="E6:71"/><i><font color="Green" title="E52">A7</font></i>: 
( <font color="Maroon" title="c4">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c5">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>

<a NAME="E7:71"/>
 ex <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font>, <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Olive" title="b2">C2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Olive" title="b3">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">C2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">C</font> &amp; <font color="Olive" title="b1">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b3">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">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" title="c4">C1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">C2</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">C</font>
;<br/>

<b>thus </b><a NAME="E1:71_2"/>
( <font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c5">C2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c6">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">C2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c4">C1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c5">C2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E2:71"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="tsep_1.html#E4:71"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="tsep_1.html#E5:71"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="tsep_1.html#E6:71"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:71"/>
<font color="Maroon" title="c2">A1</font>,<font color="Maroon" title="c3">A2</font> <a href="tsep_1.html#R2" title="TSEP_1:pred.2">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T59" target="_self" title="TSEP_1:th.59">Th59</a></i>;<br/>


</div>
