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

<b>let </b><font color="Maroon">c<sub>22</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>let </b><font color="Maroon">c<sub>23</sub></font> be  <a href="compts_1.html#V6">compact</a> <a href="jordan21.html#V1">with_the_max_arc</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>set </b><font color="Maroon">c<sub>24</sub></font> =  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:139"/><i><font color="Green">E164</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="jordan24.html#R1">realize-max-dist-in</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E2:139"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/>

<a NAME="E3:139"/><i><font color="Green">E166</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E4:139"/><i><font color="Green">E167</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:139"><i><font color="Green">E164</font></i></a>, <a class="ref" href="jordan.html#T82" target="_self">Th82</a></i>;<br/>
<a NAME="E5:139"/><i><font color="Green">E168</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E6:139"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E7:139"/><b>then </b><i><font color="Green">E169</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E3:139"><i><font color="Green">E166</font></i></a>, <a class="txt" href="jordan.html#E4:139"><i><font color="Green">E167</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E8:139"/>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T44">JORDAN21:44</a></i>;<br/>
<a NAME="E9:139"/><b>then </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan.html#E1:139"><i><font color="Green">E164</font></i></a>, <a class="txt" href="jordan.html#E102"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="jordan.html#T84" target="_self">Th84</a></i>;<br/>
<b>hence </b><a NAME="E10:139"/>
<font color="Maroon">c<sub>22</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan.html#E2:139"><i><font color="Green">E165</font></i></a>, <a class="txt" href="jordan.html#E5:139"><i><font color="Green">E168</font></i></a>, <a class="txt" href="jordan.html#E7:139"><i><font color="Green">E169</font></i></a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>


</div>
