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

<b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>

<b>assume </b><a NAME="E1:151"/><i><font color="Green">E174</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>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></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">E175</font></i>: 
<font color="Maroon">c<sub>23</sub></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">E176</font></i>: 
<font color="Maroon">c<sub>24</sub></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">E177</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E5:151"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>24</sub></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">E179</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E7:151"/><i><font color="Green">E180</font></i>: 
 <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>24</sub></font>
 <b>and </b><br/><a NAME="E8:151"/><i><font color="Green">E181</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E9:151"/><i><font color="Green">E182</font></i>: 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> =  <a href="connsp_1.html#K1">skl</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">c<sub>23</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></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>24</sub></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">c<sub>23</sub></font></span>)</span></span>)</span></span>)</span>,<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> 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">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>22</sub></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">c<sub>25</sub></font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="jordan.html#T95" target="_self">Th95</a>, <a class="txt" href="jordan.html#E1:151"><i><font color="Green">E174</font></i></a>, <a class="txt" href="jordan.html#E2:151"><i><font color="Green">E175</font></i></a>, <a class="txt" href="jordan.html#E3:151"><i><font color="Green">E176</font></i></a>, <a class="txt" href="jordan.html#E4:151"><i><font color="Green">E177</font></i></a>, <a class="txt" href="jordan.html#E5:151"><i><font color="Green">E178</font></i></a>, <a class="txt" href="jordan.html#E6:151"><i><font color="Green">E179</font></i></a>, <a class="txt" href="jordan.html#E7:151"><i><font color="Green">E180</font></i></a>, <a class="txt" href="jordan.html#E8:151"><i><font color="Green">E181</font></i></a>, <a class="txt" href="jordan.html#E9:151"><i><font color="Green">E182</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:151_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>22</sub></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">c<sub>26</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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<sub>1</sub></font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> ;<br/>
<b>let </b><font color="Maroon">c<sub>27</sub></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">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:151_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E6:151_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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<sub>1</sub></font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">c<sub>22</sub></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<sub>1</sub></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">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">c<sub>22</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan.html#E6:151_1"><i><font color="Green">E184</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:151_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="jordan.html#E5:151_1"><i><font color="Green">E183</font></i></a>, <a class="ref" href="jordan.html#T95" target="_self">Th95</a>, <a class="txt" href="jordan.html#E1:151"><i><font color="Green">E174</font></i></a>, <a class="txt" href="jordan.html#E2:151"><i><font color="Green">E175</font></i></a>, <a class="txt" href="jordan.html#E3:151"><i><font color="Green">E176</font></i></a>, <a class="txt" href="jordan.html#E4:151"><i><font color="Green">E177</font></i></a>, <a class="txt" href="jordan.html#E5:151"><i><font color="Green">E178</font></i></a>, <a class="txt" href="jordan.html#E6:151"><i><font color="Green">E179</font></i></a>, <a class="txt" href="jordan.html#E7:151"><i><font color="Green">E180</font></i></a>, <a class="txt" href="jordan.html#E8:151"><i><font color="Green">E181</font></i></a>, <a class="txt" href="jordan.html#E9:151"><i><font color="Green">E182</font></i></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<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_1.html#K1">skl</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">c<sub>23</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></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>24</sub></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">c<sub>23</sub></font></span>)</span></span>)</span></span>)</span>,<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>
 ;<br/>


</div>
