<?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">p</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>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">q1</font> <a href="jordan20.html#R2" target="_self">is_Rin</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font> &amp; <font color="Maroon">q2</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> )
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green">E28</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; <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp;  ex <font color="Olive">p4</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 <br/>( <font color="Olive">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p4</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; ( for <font color="Olive">p5</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  <a href="jordan5c.html#R1">LE</a> <font color="Olive">p4</font>,<font color="Olive">p5</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">p5</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font> ) ) )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p4</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="E4:28"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">p4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">e</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; ( for <font color="Olive">p5</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  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Olive">p5</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">p5</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p5</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<a NAME="E5:28"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> )
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E6:28"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:28"><i><font color="Green">E30</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="28_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1"/>
(  <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> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</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#E2:28"><i><font color="Green">E28</font></i></a>, , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>case </b></a><a NAME="E1:28_1_1_1"/><i><font color="Green">E37</font></i>: 
 <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>
 ;<br/><div class="add"><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_1_1_1"/>
( <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> or <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:28_1_1_1_1_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_1_1_1_1"/>
 <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> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="jordan20.html#E6:28"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan20.html#E1:28"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:28_1_1_1_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_1_1_1_2"/>
 <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> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:28_1_1_1"/>
 <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> <a href="hidden.html#R1">=</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><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="E5:28_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p</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">q2</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:28_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/><a NAME="E7:28_1_1_1"/><i><font color="Green">E46</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:28"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E8:28_1_1_1"/>
for <font color="Olive">p6</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  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Olive">p6</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">p6</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p6</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p6</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="E1:28_1_1_1_2"/><i><font color="Green">E47</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p6</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">p6</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:28_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">p6</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_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="E1:28_1_1_1_2_1_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p6</font>,<font color="Maroon">q1</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">q1</font>,<font color="Maroon">p6</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#E2:28_1_1_1"><i><font color="Green">E39</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:28_1_1_1_2_1_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p6</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_1_1_1_2_1_1_1"/>
<font color="Maroon">p6</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:28_1_1_1_2_1_1_2"/><i><font color="Green">E71</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">p6</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_2_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p6</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="jordan20.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan5c.html#T13">JORDAN5C:13</a></i>;<br/><a NAME="E3:28_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">p6</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">qq</font> where <font color="Olive">qq</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">qq</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">qq</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="E4:28_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">p6</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>hence </b><a NAME="E5:28_1_1_1_2_1_1_2"/>
<font color="Maroon">p6</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_1_1_1_2"/>
<font color="Maroon">p6</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:28_1_1_1"/>
<font color="Maroon">p</font> <a href="jordan20.html#R2" target="_self">is_Rin</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="jordan20.html#E4:28"><i><font color="Green">E30</font></i></a>, , <a class="txt" href="jordan20.html#E1:28_1_1_1"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2"><b>case </b></a><a NAME="E1:28_1_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><div class="add"><div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_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="E1:28_1_1_2_1_1"/>
( <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> or <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:28_1_1_2_1_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15">TOPREAL1:15</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_2_1_1_1"/>
 <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">q1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="jordan20.html#E6:28"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan20.html#E1:28"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:28_1_1_2_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><b>hence </b><a NAME="E3:28_1_1_2_1_1_2"/>
 <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">q1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:28_1_1_2"/>
 <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">q1</font> <a href="hidden.html#R1">=</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">q2</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">q1</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="E5:28_1_1_2"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p3</font> &amp;  <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;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">q1</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:28_1_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/><a NAME="E7:28_1_1_2"/><i><font color="Green">E82</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p4</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">p4</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="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, , , <a class="txt" href="jordan20.html#E5:28"><i><font color="Green">E32</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_1_1_2_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/><a NAME="E2:28_1_1_2_2"/><b>then </b>
<font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">qq</font> where <font color="Olive">qq</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">qq</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">qq</font>,<font color="Maroon">q1</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#E4:28"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E3:28_1_1_2_2"/><b>then </b>
<font color="Maroon">p4</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">q1</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29">JORDAN6:29</a></i>;<br/><b>hence </b><a NAME="E4:28_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="jordan20.html#E4:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self">Th2</a>, <a class="ref" href="goboard7.html#T5">GOBOARD7:5</a></i>;<br/></div><b>end;</b></div><a NAME="E9:28_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p</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="E10:28_1_1_2"/>
for <font color="Olive">p6</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  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Olive">p6</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">p6</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> holds <br/><font color="Olive">p6</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p6</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="E1:28_1_1_2_3"/><i><font color="Green">E85</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p6</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">p6</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:28_1_1_2_3"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p6</font>,<font color="Maroon">q1</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/>
<b>hence </b><a NAME="E3:28_1_1_2_3"/>
<font color="Maroon">p6</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:28"><i><font color="Green">E30</font></i></a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:28_1_1_2"/>
<font color="Maroon">p</font> <a href="jordan20.html#R2" target="_self">is_Rin</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:28"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="jordan20.html#E4:28"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="jordan20.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:28"/>
<font color="Maroon">p</font> <a href="jordan20.html#R2" target="_self">is_Rin</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">e</font>
 ;<br/>


</div>
