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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</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/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:110"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd12.html#T11">GOBRD12:11</a></i>;<br/>
<a NAME="E2:110"/><i><font color="Green">E103</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T102" target="_self">Th102</a></i>;<br/>
<a NAME="E3:110"/><i><font color="Green">E104</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T99" target="_self">Th99</a></i>;<br/>
<div><i><font color="Green">E105</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_1"/>
not  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:110_1"/><b>then </b>
not  <a href="jordan2c.html#K1" target="_self">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:110"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:110_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:110_1"/>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E6:110_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:110"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan2c.html#E4:110_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:110_1"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:110"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan2c.html#E4:110_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E8:110_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2"/>
not  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:110_2"/><b>then </b>
not  <a href="jordan2c.html#K2" target="_self">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:110"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:110_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E5:110_2"/>
 <a href="jordan2c.html#K2" target="_self">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E6:110_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:110"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan2c.html#E4:110_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:110_2"/><b>then </b>
 <a href="jordan2c.html#K1" target="_self">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:110"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan2c.html#E4:110_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E8:110_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:110"/>
(  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2" target="_self">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp;  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:110"><i><font color="Green">E105</font></i></a></i>;<br/>


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