<?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="E1:25"/>
for <font color="Olive">P</font> being  <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> holds <br/> ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K4" target="_self">BDD-Family</a> <font color="Maroon">C</font> iff <font color="Olive">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K3" target="_self">UBD-Family</a> <font color="Maroon">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <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/>

<b>thus </b><a NAME="E1:25_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K4" target="_self">BDD-Family</a> <font color="Maroon">C</font> implies <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K3" target="_self">UBD-Family</a> <font color="Maroon">C</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon">P</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="E3:25_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">P</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="E4:25_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</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>, <a class="ref" href="gobrd14.html#T47">GOBRD14:47</a></i>;<br/>
<a NAME="E5:25_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</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="subset_1.html#K4">\/</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><a class="ref" href="gobrd14.html#T31">GOBRD14:31</a></i>;<br/>
<a NAME="E6:25_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T43">SPRECT_3:43</a></i>;<br/>
<a NAME="E7:25_1_1"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/>
<a NAME="E8:25_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard9.html#K2">LeftComp</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> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="goboard9.html#K2">LeftComp</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> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i>, <a class="ref" href="gobrd14.html#T25">GOBRD14:25</a></i>
<br/>.= 
 <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
;<br/>
<a NAME="E10:25_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</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="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E11:25_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T46">SUBSET_1:46</a></i>;<br/>
<a NAME="E12:25_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2">UBD</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><a class="ref" href="gobrd14.html#T46">GOBRD14:46</a></i>;<br/>
<b>hence </b><a NAME="E13:25_1_1"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K3" target="_self">UBD-Family</a> <font color="Maroon">C</font>
 ;<br/>


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

<b>assume </b><a NAME="E2:25_1"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K3" target="_self">UBD-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="E4:25_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2">UBD</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>
 ;<br/>
<a NAME="E5:25_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="gobrd14.html#T46">GOBRD14:46</a></i>;<br/>
<a NAME="E6:25_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/>
<a NAME="E7:25_1"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="sprect_3.html#T43">SPRECT_3:43</a></i>;<br/>
<a NAME="E8:25_1"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</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="subset_1.html#K4">\/</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>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>
<a NAME="E9:25_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</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><a class="ref" href="gobrd14.html#T31">GOBRD14:31</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p4">(<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> <a href="subset_1.html#K3">`</a> </span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p4">(<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></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T53">XBOOLE_1:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p4">(<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="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p4">(<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></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="gobrd14.html#T31">GOBRD14:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
<b>by </b><i>, <a class="ref" href="gobrd14.html#T25">GOBRD14:25</a></i>
<br/>.= 
 <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
;<br/>
<a NAME="E11:25_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3">RightComp</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> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E12:25_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</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>, <a class="ref" href="subset_1.html#T46">SUBSET_1:46</a></i>;<br/>
<a NAME="E13:25_1"/><b>then </b>
<font color="Maroon">P</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>
 
<b>by </b><i><a class="ref" href="gobrd14.html#T47">GOBRD14:47</a></i>;<br/>
<b>hence </b><a NAME="E14:25_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan10.html#K4" target="_self">BDD-Family</a> <font color="Maroon">C</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:25"/>
 <a href="setfam_1.html#K7">COMPLEMENT</a> <span class="p1">(<span class="default"><a href="jordan10.html#K3" target="_self">UBD-Family</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan10.html#K4" target="_self">BDD-Family</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D8">SETFAM_1:def 8</a></i>;<br/>


</div>
