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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="sprect_1.html#V1">rectangular</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E3:43"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E4:43"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E5:43"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd12.html#T11">GOBRD12:11</a></i>;<br/>
<a NAME="E6:43"/><i><font color="Green">E28</font></i>: 
(  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="sprect_3.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E7:43"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E8:43"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/>
<a NAME="E9:43"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:43"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:43"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E11:43"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E7:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/>
<a NAME="E12:43"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:43"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:43"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E14:43"/>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T96">SPRECT_1:96</a></i>;<br/>
<a NAME="E15:43"/><b>then </b><i><font color="Green">E32</font></i>: 
not  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:43"><i><font color="Green">E27</font></i></a>, <a class="txt" href="sprect_3.html#E6:43"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E10:43"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sprect_3.html#E13:43"><i><font color="Green">E31</font></i></a>, <a class="ref" href="topreal5.html#T4">TOPREAL5:4</a></i>;<br/>
<a NAME="E16:43"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T31">TOPREAL1:31</a></i>;<br/>
<b>hence </b><a NAME="E17:43"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E15:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan6.html#T11">JORDAN6:11</a></i>;<br/>


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