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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</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>let </b><font color="Maroon">c<sub>2</sub></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="E1:6"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></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<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/>
<a NAME="E3:6"/><i><font color="Green">E9</font></i>: 
(  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/>
<a NAME="E4:6"/><i><font color="Green">E10</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E2:6"><i><font color="Green">E8</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="6_1_1"><b>suppose </b></a><a NAME="E1:6_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:6_1_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:6"><i><font color="Green">E9</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:6_1_1"/>
 <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:6"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan7.html#E1:6_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2"><b>suppose </b></a><a NAME="E1:6_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_2_1"/>
(  not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_1"><b>suppose </b></a><a NAME="E1:6_1_2_1_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:6_1_2_1_1"/>
 <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:6"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan7.html#E1:6_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_2"><b>suppose </b></a><a NAME="E1:6_1_2_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:6_1_2_1_2"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:6"><i><font color="Green">E10</font></i></a>, <a class="ref" href="jordan5c.html#T9">JORDAN5C:9</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_1_2"/>
 <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:6"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan7.html#E1:6_1_2_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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