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<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>

<b>assume </b><a NAME="E1:149"/><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">C</font>
 ;<br/>

<b>set </b><font color="Maroon">U</font> =  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>;<br/>
<a NAME="E2:149"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E3:149"/><i><font color="Green">E74</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T43">JORDAN21:43</a></i>;<br/>
<a NAME="E4:149"/>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T44">JORDAN21:44</a></i>;<br/>
<a NAME="E5:149"/><b>then </b><i><font color="Green">E78</font></i>: 
(  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> or  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:149"/><i><font color="Green">E79</font></i>: 
(  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</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"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="jordan.html#T52" target="_self">Th52</a>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="149_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:149_1"/>
(  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> or  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> )
 </span><b>by </b><i>, <a class="ref" href="jordan.html#T57" target="_self">Th57</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="149_1_1"><b>suppose </b></a><a NAME="E1:149_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>take </b>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
;<br/><b>take </b>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
;<br/><b>thus </b><a NAME="E2:149_1_1"/>
(  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a> <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> &amp;  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a> <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> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp;  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> &amp;  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="jordan6.html#T65">JORDAN6:65</a>, <a class="ref" href="jordan21.html#T63">JORDAN21:63</a>, <a class="ref" href="jordan21.html#T26">JORDAN21:26</a>, <a class="ref" href="jordan21.html#T27">JORDAN21:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="149_1_2"><b>suppose </b></a><a NAME="E1:149_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>take </b>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
;<br/><b>take </b>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
;<br/><b>thus </b><a NAME="E2:149_1_2"/>
(  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a> <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> &amp;  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a> <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> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> &amp;  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> &amp;  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="jordan6.html#T65">JORDAN6:65</a>, <a class="ref" href="jordan21.html#T62">JORDAN21:62</a>, <a class="ref" href="jordan21.html#T28">JORDAN21:28</a>, <a class="ref" href="jordan21.html#T29">JORDAN21:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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