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<b>let </b><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">D</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">x</font> =  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:138"/><i><font color="Green">E69</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">D</font>
 <b>and </b><br/><a NAME="E2:138"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<a NAME="E3:138"/><i><font color="Green">E74</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E4:138"/><i><font color="Green">E78</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="ref" href="jordan.html#T46" target="_self">Th46</a>, </i>;<br/>
<a NAME="E5:138"/><i><font color="Green">E79</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</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">0,3</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">0,3</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#K1">UMP</a> <font color="Maroon">D</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#K1">UMP</a> <font color="Maroon">D</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#K1">UMP</a> <font color="Maroon">D</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="E6:138"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</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"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E7:138"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</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#K1">UMP</a> <font color="Maroon">D</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E8:138"/>
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T43">JORDAN21:43</a></i>;<br/>
<a NAME="E9:138"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan.html#T46" target="_self">Th46</a>, , </i>;<br/>
<b>hence </b><a NAME="E10:138"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan.html#T47" target="_self">Th47</a>, , <a class="ref" href="jordan.html#T48" target="_self">Th48</a>, <a class="ref" href="jgraph_6.html#T9">JGRAPH_6:9</a></i>;<br/>


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