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

<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>assume </b><a NAME="E1:144"/><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>
 ;<br/>

<b>set </b><font color="Maroon">m</font> =  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E2:144"/><i><font color="Green">E70</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"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="jordan.html#T49" target="_self">Th49</a>, <a class="ref" href="jordan.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E3:144"/><i><font color="Green">E74</font></i>: 
<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>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:144"/><i><font color="Green">E78</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="E5:144"/><i><font color="Green">E79</font></i>: 
 <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/>
<b>thus </b><a NAME="E6:144"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span></span><a href="domain_1.html#K6">}</a></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="144_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="E1:144_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:144_1"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">x</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> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <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>by </b><i><a class="ref" href="jordan.html#T57" target="_self">Th57</a></i>;<br/>
<a NAME="E4:144_1"/>
 <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#T51" target="_self">Th51</a>, <a class="ref" href="jordan.html#T50" target="_self">Th50</a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E5:144_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <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="jordan.html#T52" target="_self">Th52</a>, , <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/>
<a NAME="E6:144_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan.html#T51" target="_self">Th51</a>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<a NAME="E7:144_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:144_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">x</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#K1">UMP</a> <font color="Maroon">D</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan21.html#T41">JORDAN21:41</a></i>;<br/>
<a NAME="E9:144_1"/>
<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">x</font> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan.html#T49" target="_self">Th49</a>, , <a class="ref" href="jordan.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E10:144_1"/><b>then </b>
<font color="Maroon">x</font> <a href="euclid.html#K22">`2</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#K22">`2</a> 
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E11:144_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>
<b>hence </b><a NAME="E12:144_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></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="E7:144"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E8:144"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:144"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="jordan.html#T52" target="_self">Th52</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
