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<b>let </b><font color="Maroon" title="c1">C</font> be  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> non <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> non <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<a NAME="E1:20"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan10.html#K4" title="JORDAN10:func.4">BDD-Family</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">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" title="c2">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan10.html#K4" title="JORDAN10:func.4">BDD-Family</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
 ;<br/>
<a NAME="E4:20_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan10.html#T12" target="_self" title="JORDAN10:th.12">Th12</a></i>;<br/>
<a NAME="E5:20_1"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p3">(<span class="default"><a href="jordan9.html#K1" title="JORDAN9:func.1">Cage</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1"/>
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Z</font>
 <b>by </b><i><a class="txt" href="jordan10.html#E3:20_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="jordan10.html#E4:20_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20"/>
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="jordan10.html#K4" title="JORDAN10:func.4">BDD-Family</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/>


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