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<b>let </b><font color="Maroon">c<sub>1</sub></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>assume </b><a NAME="E1:99"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="topreal4.html#V1">special_polygonal</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</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="E2:99"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E3:99"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sppol_2.html#E1:99"><i><font color="Green">E75</font></i></a>, <a class="ref" href="sppol_2.html#T64" target="_self">Th64</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a><b> such that </b><br/><a NAME="E5:99"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E6:99"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>and </b><br/><a NAME="E7:99"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</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>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E8:99"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</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>5</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E9:99"/><i><font color="Green">E80</font></i>: 
( <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>4</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>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> =  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>take </b><font color="Maroon">c<sub>7</sub></font> =  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>thus </b><a NAME="E10:99"/>
<font color="Maroon">c<sub>6</sub></font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E5:99"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sppol_2.html#E7:99"><i><font color="Green">E78</font></i></a>, <a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a></i>;<br/>

<b>thus </b><a NAME="E11:99"/>
<font color="Maroon">c<sub>7</sub></font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="sppol_2.html#E6:99"><i><font color="Green">E77</font></i></a>, <a class="txt" href="sppol_2.html#E8:99"><i><font color="Green">E79</font></i></a>, <a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a></i>;<br/>

<b>thus </b><a NAME="E12:99"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="sppol_2.html#E9:99"><i><font color="Green">E80</font></i></a></i>;<br/>


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