<?xml version="1.0"?>
<div class="add">

<b>thus </b><a NAME="E1:86_1_1"/>
( <font color="Maroon">P</font> is <a href="topreal4.html#V1">special_polygonal</a> implies  ex <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <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> st <font color="Olive">p1</font>,<font color="Olive">p2</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">p1</font> being   <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>, <font color="Maroon">p2</font> being   <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>, <font color="Maroon">P1</font> being   <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>, <font color="Maroon">P2</font> being   <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><b> such that </b><a NAME="E1:86_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E2:86_1_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>and </b><br/><a NAME="E3:86_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">P1</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E4:86_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">P2</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E5:86_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal4.html#D2">TOPREAL4:def 2</a><br/>

<b>take </b>
<font color="Maroon">p1</font>
;<br/>

<b>take </b>
<font color="Maroon">p2</font>
;<br/>

<b>thus </b><a NAME="E6:86_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="sppol_2.html#D1" target="_self">SPPOL_2:def 1</a><br/>

<b>consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:86_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f1</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <b>and </b><br/><a NAME="E9:86_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</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">f1</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E11:86_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f2</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a> 
 <b>and </b><br/><a NAME="E12:86_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</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">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, <a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f1</font>, <font color="Maroon">f2</font> = <font color="Maroon">f2</font> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="ref" href="sppol_2.html#T20" target="_self">Th20</a>, <a class="txt" href="sppol_2.html#E7:86_1_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">f1</font>
;<br/>

<b>take </b>
<font color="Maroon">f2</font>
;<br/>

<b>thus </b><a NAME="E14:86_1_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</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">f1</font></span>)</span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</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">f2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</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">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</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">f2</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">p1</font> being   <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>, <font color="Maroon">p2</font> being   <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><b> such that </b><a NAME="E2:86_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="sppol_2.html#R1" target="_self">split</a> <font color="Maroon">P</font>
 ;<br/>

<b>consider </b><font color="Maroon">f1</font> being   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a>, <font color="Maroon">f2</font> being   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a><b> such that </b><br/><a NAME="E4:86_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</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">f1</font></span>)</span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</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">f2</font></span>)</span> )
 <b>and </b><br/><a NAME="E5:86_1_1"/><i><font color="Green">E52</font></i>: 
( <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</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">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</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">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, <a class="ref" href="sppol_2.html#T12" target="_self">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon">p1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal4.html#D2">TOPREAL4:def 2</a><br/>

<b>take </b>
<font color="Maroon">p2</font>
;<br/>

<b>take </b><font color="Maroon">P1</font> =  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font>;<br/>

<b>take </b><font color="Maroon">P2</font> =  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font>;<br/>

<b>thus </b><a NAME="E6:86_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="sppol_2.html#T18" target="_self">Th18</a>, <a class="ref" href="sppol_2.html#T12" target="_self">Th12</a></i>;<br/>

<a NAME="E7:86_1_1"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_1.html#T29">XBOOLE_1:29</a></i>;<br/>
<a NAME="E8:86_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:86_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E10:86_1_1"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">f</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">f</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a><br/>

<b>thus </b><a NAME="E11:86_1_1"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">f</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">f</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a><br/>

<b>thus </b><a NAME="E12:86_1_1"/>
( <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> )
 <b>by </b><i><a class="ref" href="sppol_2.html#T19" target="_self">Th19</a></i>;<br/>


</div>
