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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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>let </b><font color="Maroon">c<sub>4</sub></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>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:41"/><i><font color="Green">E27</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> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:41"/><i><font color="Green">E28</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E5:41"/><i><font color="Green">E29</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="topreal5.html#T25">TOPREAL5:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</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> <b>by </b><i><a class="txt" href="jordan16.html#E3:41"><i><font color="Green">E27</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1"/>
(  <a href="jordan5c.html#R1">LE</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>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or  <a href="jordan5c.html#R1">LE</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>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan16.html#E2:41"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan16.html#E3:41"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan16.html#E4:41"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan16.html#E5:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan5c.html#T14">JORDAN5C:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>suppose </b></a><a NAME="E1:41_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="jordan5c.html#R1">LE</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>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</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> <b>by </b><i><a class="txt" href="jordan16.html#E2:41"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan16.html#E1:41_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan16.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:41_1_1"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E4:41_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan16.html#E1:41"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:41_1_1"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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>
 <b>by </b><i><a class="txt" href="jordan16.html#E2:41"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan16.html#E5:41"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan16.html#E1:41_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan16.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E6:41_1_1"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</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="jordan16.html#E4:41_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="jordan16.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E7:41_1_1"/>
(  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="jordan16.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2"><b>suppose </b></a><a NAME="E1:41_1_2"/><i><font color="Green">E30</font></i>: 
 <a href="jordan5c.html#R1">LE</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>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</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> <b>by </b><i><a class="txt" href="jordan16.html#E2:41"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan16.html#E1:41_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan16.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:41_1_2"/><i><font color="Green">E31</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E4:41_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan16.html#E1:41"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:41_1_2"/>
 <a href="jordan6.html#K5">Segment</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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="jordan16.html#E2:41"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan16.html#E5:41"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan16.html#E1:41_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan16.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E6:41_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</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>
 <b>by </b><i><a class="ref" href="jordan5b.html#T14">JORDAN5B:14</a></i>;<br/><b>hence </b><a NAME="E7:41_1_2"/>
(  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="jordan16.html#E3:41_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan16.html#E4:41_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan16.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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