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<div class="add">

<b>let </b><font color="Maroon" title="c22">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/>

<b>assume </b><a NAME="E1:148"/><i><font color="Green" title="E172">A1</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="jordan24.html#R1" title="JORDAN24:pred.1">realize-max-dist-in</a> <font color="Maroon" title="c22">C</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c23">U</font> =  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font>;<br/>
<b>set </b><font color="Maroon" title="c24">L</font> =  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font>;<br/>
<a NAME="E2:148"/><i><font color="Green" title="E173">A2</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E3:148"/><i><font color="Green" title="E174">A3</font></i>: 
 <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">C</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T43" title="JORDAN21:th.43">JORDAN21:43</a></i>;<br/>
<a NAME="E4:148"/>
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">C</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T44" title="JORDAN21:th.44">JORDAN21:44</a></i>;<br/>
<a NAME="E5:148"/><b>then </b><i><font color="Green" title="E175">A4</font></i>: 
(  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font> or  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font> )
 
<b>by </b><i><a class="txt" href="jordan.html#E2:148"><i><font color="Green" title="E173">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:148"/><i><font color="Green" title="E176">A5</font></i>: 
(  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp;  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="jordan.html#E1:148"><i><font color="Green" title="E172">A1</font></i></a>, <a class="ref" href="jordan.html#T79" target="_self" title="JORDAN:th.79">Th79</a>, <a class="ref" href="jordan.html#T80" target="_self" title="JORDAN:th.80">Th80</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:148_1"/>
(  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font> or  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font> )
 </span><b>by </b><i><a class="txt" href="jordan.html#E2:148"><i><font color="Green" title="E173">A2</font></i></a>, <a class="txt" href="jordan.html#E3:148"><i><font color="Green" title="E174">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_1"><b>suppose </b></a><a NAME="E1:148_1_1"/><i><font color="Green" title="E177">A6</font></i>: 
 <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font>
 ;<br/><div class="add"><b>take </b>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font>
;<br/><b>take </b>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font>
;<br/><b>thus </b><a NAME="E2:148_1_1"/>
(  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp;  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font> &amp;  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font> &amp;  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> &amp;  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan.html#E6:148"><i><font color="Green" title="E176">A5</font></i></a>, <a class="txt" href="jordan.html#E1:148_1_1"><i><font color="Green" title="E177">A6</font></i></a>, <a class="txt" href="jordan.html#E5:148"><i><font color="Green" title="E175">A4</font></i></a>, <a class="ref" href="jordan6.html#T65" title="JORDAN6:th.65">JORDAN6:65</a>, <a class="ref" href="jordan21.html#T63" title="JORDAN21:th.63">JORDAN21:63</a>, <a class="ref" href="jordan21.html#T26" title="JORDAN21:th.26">JORDAN21:26</a>, <a class="ref" href="jordan21.html#T27" title="JORDAN21:th.27">JORDAN21:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="148_1_2"><b>suppose </b></a><a NAME="E1:148_1_2"/><i><font color="Green" title="E177">A7</font></i>: 
 <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font>
 ;<br/><div class="add"><b>take </b>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font>
;<br/><b>take </b>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font>
;<br/><b>thus </b><a NAME="E2:148_1_2"/>
(  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp;  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font> &amp;  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c22">C</font> &amp;  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> &amp;  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c22">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c22">C</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan.html#E6:148"><i><font color="Green" title="E176">A5</font></i></a>, <a class="txt" href="jordan.html#E1:148_1_2"><i><font color="Green" title="E177">A7</font></i></a>, <a class="txt" href="jordan.html#E5:148"><i><font color="Green" title="E175">A4</font></i></a>, <a class="ref" href="jordan6.html#T65" title="JORDAN6:th.65">JORDAN6:65</a>, <a class="ref" href="jordan21.html#T62" title="JORDAN21:th.62">JORDAN21:62</a>, <a class="ref" href="jordan21.html#T28" title="JORDAN21:th.28">JORDAN21:28</a>, <a class="ref" href="jordan21.html#T29" title="JORDAN21:th.29">JORDAN21:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
