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<b>let </b><font color="Maroon">P</font> be   <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>let </b><font color="Maroon">q</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>assume </b><a NAME="E1:67"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<a NAME="E2:67"/><b>then </b><i><font color="Green">E47</font></i>: 
(  <a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" target="_self">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" target="_self">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:67"/><i><font color="Green">E49</font></i>: 
 <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E3:67"><i><font color="Green">E49</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font> or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan6.html#E3:67"><i><font color="Green">E49</font></i></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="67_1_1_1"><b>case </b></a><a NAME="E1:67_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:67_1_1_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">q</font>, <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan5c.html#T9">JORDAN5C:9</a></i>;<br/><b>hence </b><a NAME="E3:67_1_1_1"/>
 <a href="jordan6.html#R1" target="_self">LE</a> <font color="Maroon">q</font>,<font color="Maroon">q</font>,<font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1_2"><b>case </b></a><a NAME="E1:67_1_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font> )
 ;<br/><div class="add"><a NAME="E2:67_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">q</font>, <a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan5c.html#T9">JORDAN5C:9</a></i>;<br/><a NAME="E3:67_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan6.html#E1:67_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/><b>hence </b><a NAME="E4:67_1_1_2"/>
 <a href="jordan6.html#R1" target="_self">LE</a> <font color="Maroon">q</font>,<font color="Maroon">q</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:67_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:67"/>
 <a href="jordan6.html#R1" target="_self">LE</a> <font color="Maroon">q</font>,<font color="Maroon">q</font>,<font color="Maroon">P</font>
 ;<br/>


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