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

<b>let </b><font color="Maroon">C</font> be  <a href="connsp_1.html#V2">connected</a> <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<a NAME="E1:20"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K4" target="_self">BDD-Family</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K4" target="_self">BDD-Family</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 ;<br/>
<a NAME="E4:20_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:20_1"/>
 <a href="jordan2c.html#K1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="jordan10.html#K4" target="_self">BDD-Family</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>


</div>
