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<b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">c<sub>23</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>set </b><font color="Maroon">c<sub>24</sub></font> =  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:143"/><i><font color="Green">E168</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<sub>22</sub></font>
 <b>and </b><br/><a NAME="E2:143"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>26</sub></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<sub>22</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:143"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E5:143"/>
<font color="Maroon">c<sub>26</sub></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<sub>22</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>and </b><br/><a NAME="E6:143"/>
<font color="Maroon">c<sub>26</sub></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="txt" href="jordan.html#E2:143"><i><font color="Green">E169</font></i></a>, <a class="ref" href="jordan2c.html#T17">JORDAN2C:17</a></i>;<br/>
<a NAME="E7:143"/>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E8:143"/><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#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:143"/><b>then </b><i><font color="Green">E171</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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> <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">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p4">(<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#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#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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="E10:143"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K8">south_halfline</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:143"><i><font color="Green">E168</font></i></a>, <a class="ref" href="jordan.html#T88" target="_self">Th88</a></i>;<br/>
<a NAME="E11:143"/><b>then </b><i><font color="Green">E172</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<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#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#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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#K8">south_halfline</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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="E12:143"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="jordan.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E13:143"/><b>then </b><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<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#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#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E11:143"><i><font color="Green">E172</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E14:143"/>
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>22</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E2:143"><i><font color="Green">E169</font></i></a>, <a class="ref" href="jordan.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E15:143"/><b>then </b><i><font color="Green">E174</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,<span class="p3">(<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#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#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E13:143"><i><font color="Green">E173</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E16:143"/>
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T44">JORDAN21:44</a></i>;<br/>
<a NAME="E17:143"/><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#K2">LMP</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="jordan.html#E4:143"><i><font color="Green">E170</font></i></a>, <a class="txt" href="jordan.html#E21"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:143"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>22</sub></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> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E9:143"><i><font color="Green">E171</font></i></a>, <a class="txt" href="jordan.html#E15:143"><i><font color="Green">E174</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>


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