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<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><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>set </b><font color="Maroon">m</font> =  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:142"/><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>
 <b>and </b><br/><a NAME="E2:142"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">P</font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">C</font>
 ;<br/>

<b>consider </b><font color="Maroon">VP</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:142"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">VP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E5:142"/>
<font color="Maroon">VP</font> <a href="connsp_1.html#R3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E6:142"/>
<font color="Maroon">VP</font> is  <a href="tbsp_1.html#V6">bounded</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="jordan.html#T48" target="_self">Th48</a>, <a class="ref" href="jordan2c.html#T17">JORDAN2C:17</a></i>;<br/>
<a NAME="E7:142"/>
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E8:142"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:142"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<a NAME="E10:142"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K6">north_halfline</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E11:142"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E12:142"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K6">north_halfline</a> <span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:142"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T51" target="_self">Th51</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:142"/>
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T48" target="_self">Th48</a>, </i>;<br/>
<a NAME="E15:142"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T52" target="_self">Th52</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E16:142"/>
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T43">JORDAN21:43</a></i>;<br/>
<a NAME="E17:142"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan.html#T49" target="_self">Th49</a>, </i>;<br/>
<b>hence </b><a NAME="E18:142"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,3</span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan.html#T50" target="_self">Th50</a>, , <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>


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