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

<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</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">p1</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>, <font color="Maroon">p2</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>, <font color="Maroon">q1</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>, <font color="Maroon">q2</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>, <font color="Maroon">q3</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="E1:23"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:23"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E3:23"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E2:23"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Olive">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then consider </b><font color="Maroon">p</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><br/><a NAME="E4:23_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><a NAME="E5:23_1_1_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i>, , <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E6:23_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p3</font> where <font color="Olive">p3</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Olive">p3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p3</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:23_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Olive">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>then consider </b><font color="Maroon">p</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><br/><a NAME="E4:23_1_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><a NAME="E5:23_1_1_2"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i>, , <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E6:23_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p3</font> where <font color="Olive">p3</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Olive">p3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p3</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:23_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:23"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font>
 ;<br/>

<a NAME="E2:23_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p3</font> where <font color="Olive">p3</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Olive">p3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p3</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">p3</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><br/><a NAME="E4:23_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>
<a NAME="E5:23_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:23"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_2_1_1"/>
( <font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> or <font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_1"><b>suppose </b></a><a NAME="E1:23_2_1_1_1"/>
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 ;<br/><div class="add"><a NAME="E2:23_2_1_1_1"/><b>then </b>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i>, , <a class="ref" href="jordan5c.html#T9">JORDAN5C:9</a></i>;<br/><a NAME="E3:23_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p31</font> where <font color="Olive">p31</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Olive">p31</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p31</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan20.html#E2:23"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E4:23_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font> )
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_2"><b>suppose </b></a><a NAME="E1:23_2_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_2_1_1_2_1_1"/>
( (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) or (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) )
 </span><b>by </b><i>, , , <a class="txt" href="jordan20.html#E3:23"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan5c.html#T14">JORDAN5C:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:23_2_1_1_2_1_1_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><div class="add"><a NAME="E2:23_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p31</font> where <font color="Olive">p31</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Olive">p31</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p31</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan20.html#E2:23"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_2_1_1_2_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font> )
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:23_2_1_1_2_1_1_2"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; not  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><div class="add"><a NAME="E2:23_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p31</font> where <font color="Olive">p31</font> is   <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> : (  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Olive">p31</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p31</font>,<font color="Maroon">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan20.html#E2:23"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_2_1_1_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font> )
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:23_2_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:23_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:23"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>,<font color="Maroon">q3</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>,<font color="Maroon">q3</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:23"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
