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

<b>let </b><font color="Maroon">P</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>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>;<br/>



<b>thus </b><a NAME="E1:31"/>
 <a href="jordan6.html#K5" target="_self">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> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</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">q</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">q</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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="E1:31_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5" target="_self">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/>

<a NAME="E2:31_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K4" target="_self">R_Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K3" target="_self">L_Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font></span>)</span>
 
;<br/>
<a NAME="E3:31_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K4" target="_self">R_Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K3" target="_self">L_Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:31_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">q</font> where <font color="Olive">q</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">q</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">q</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="E6:31_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>
<a NAME="E7:31_1"/>
<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="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/>;<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="E9:31_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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/>
<b>thus </b><a NAME="E10:31_1"/>
<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">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


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

<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="E2:31"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</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">q</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">q</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">q</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="E4:31"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q</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">q</font>,<font color="Maroon">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>
<a NAME="E5:31"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q3</font> where <font color="Olive">q3</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">q3</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:31"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K4" target="_self">R_Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font>
 
;<br/>
<a NAME="E7:31"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q3</font> where <font color="Olive">q3</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="Olive">q3</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/>;<br/>
<a NAME="E8:31"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K3" target="_self">L_Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font>
 
;<br/>
<a NAME="E9:31"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K4" target="_self">R_Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K3" target="_self">L_Segment</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">q2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E10:31"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5" target="_self">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>
