<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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>5</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>assume </b><a NAME="E1:68"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E2:68"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T36">JGRAPH_3:36</a></i>;<br/>
<a NAME="E3:68"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</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>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8">JORDAN6:def 8</a></i>;<br/>
<a NAME="E4:68"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E2:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<a NAME="E5:68"/><i><font color="Green">E68</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jgraph_5.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E6:68"/>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></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>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E7:68"/><b>then </b><i><font color="Green">E69</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E8:68"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:68"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jgraph_5.html#E5:68"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E9:68"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jgraph_5.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E10:68"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/>
<a NAME="E11:68"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E8:68"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jgraph_5.html#E9:68"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E12:68"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E2:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/>
<a NAME="E13:68"/><i><font color="Green">E73</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>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E9:68"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jgraph_5.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E14:68"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/>
<a NAME="E15:68"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E8:68"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jgraph_5.html#E13:68"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E16:68"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E2:68"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/>
<a NAME="E17:68"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E15:68"><i><font color="Green">E74</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/>
<a NAME="E18:68"/><i><font color="Green">E76</font></i>: 
<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>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E13:68"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jgraph_5.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E19:68"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>5</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E3:68"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/>
<a NAME="E20:68"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E8:68"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jgraph_5.html#E18:68"><i><font color="Green">E76</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E21:68"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E2:68"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jgraph_5.html#E17:68"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/>
<b>hence </b><a NAME="E22:68"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">b<sub>6</sub></font> being  <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> holds <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>6</sub></font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">b<sub>6</sub></font></span><a href="toprns_1.html#K5">.|</a></span> ) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">b<sub>5</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">b<sub>5</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:68"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jgraph_5.html#E7:68"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jgraph_5.html#E12:68"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jgraph_5.html#E17:68"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jgraph_5.html#T62" target="_self">Th62</a></i>;<br/>


</div>
