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<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>

<b>assume </b><a NAME="E1:151"/><i><font color="Green">E69</font></i>: 
<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>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span> <a href="jordan24.html#R1">realize-max-dist-in</a> <font color="Maroon">C</font>
 ;<br/>

<b>let </b><font color="Maroon">Jc</font> be  <a href="compts_1.html#V6">compact</a> <a href="jordan21.html#V1">with_the_max_arc</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>, <font color="Maroon">Jd</font> be  <a href="compts_1.html#V6">compact</a> <a href="jordan21.html#V1">with_the_max_arc</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><b>that </b><br/><a NAME="E2:151"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Jc</font> <a href="topreal1.html#R1">is_an_arc_of</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>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E3:151"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">Jd</font> <a href="topreal1.html#R1">is_an_arc_of</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>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E4:151"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Jc</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Jd</font>
 <b>and </b><br/><a NAME="E5:151"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">Jc</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>and </b><br/><a NAME="E6:151"/><i><font color="Green">E80</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jc</font>
 <b>and </b><br/><a NAME="E7:151"/><i><font color="Green">E81</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Jd</font>
 <b>and </b><br/><a NAME="E8:151"/><i><font color="Green">E82</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">Jc</font>
 <b>and </b><br/><a NAME="E9:151"/><i><font color="Green">E83</font></i>: 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">Jc</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">Ux</font> =  <a href="connsp_1.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> as   <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="ref" href="pre_topc.html#T39">PRE_TOPC:39</a></i>;<br/>
<a NAME="E11:151"/>
<font color="Maroon">Ux</font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="151_1"><b>proof </b></a><div class="add">

<a NAME="E1:151_1"/>
<font color="Maroon">Ux</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T52" target="_self">Th52</a>, <a class="ref" href="jordan.html#T57" target="_self">Th57</a>, , , , , , , , </i>;<br/>
<b>hence </b><a NAME="E2:151_1"/>
<font color="Maroon">Ux</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T26">JORDAN2C:26</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <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> : <font color="Olive">B</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font> </span>}</span> ;<br/>
<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:151_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:151_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E6:151_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <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> : <font color="Olive">B</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:151_1"/>
 ex <font color="Olive">B</font> being  <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> st <br/>( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:151_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ux</font>
 <b>by </b><i>, <a class="ref" href="jordan.html#T52" target="_self">Th52</a>, <a class="ref" href="jordan.html#T57" target="_self">Th57</a>, , , , , , , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:151"/>
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="connsp_1.html#K1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K2">Down</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan21.html#K1">UMP</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 3</span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Jd</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">Jc</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 ;<br/>


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