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<div class="add">

<b>let </b><font color="Maroon">C</font> be  <a href="compts_1.html#V6">compact</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/><b>let </b><font color="Maroon">WH</font> be  <a href="connsp_1.html#V2">connected</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/>



<b>assume </b><a NAME="E1:145"/><i><font color="Green">E47</font></i>: 
( not <font color="Maroon">WH</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a> &amp; <font color="Maroon">WH</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:145"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">WH</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="145_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:145_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">WH</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:145_1"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:145_1"/>
 <a href="subset_1.html#K2">[#]</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> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T25">SUBSET_1:25</a></i>;<br/>
<a NAME="E4:145_1"/><b>then </b>
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> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
;<br/>
<a NAME="E5:145_1"/><b>then </b>
<font color="Maroon">WH</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="jordan2c.html#T39" target="_self">Th39</a>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E6:145_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">WH</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E7:145_1"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <font color="Maroon">C</font> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E8:145_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:145"/><i><font color="Green">E60</font></i>: 
 <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">C</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E4:145"/>
<font color="Maroon">WH</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T36" target="_self">Th36</a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<b>hence </b><a NAME="E5:145"/>
<font color="Maroon">WH</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T37" target="_self">Th37</a>, <a class="ref" href="jordan2c.html#T38" target="_self">Th38</a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/>


</div>
