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

<b>set </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<b>set </b><font color="Maroon">X</font> = <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E1:45"/><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E2:45"/>
<font color="Maroon">C</font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T73">JORDAN2C:73</a></i>;<br/>
<a NAME="E3:45"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T98">TOPREAL6:98</a></i>;<br/>
<a NAME="E4:45"/><b>then </b><i><font color="Green">E53</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="jct_misc.html#T23">JCT_MISC:23</a></i>;<br/>
<b>set </b><font color="Maroon">u</font> =  <a href="jordan6.html#K11">Upper_Middle_Point</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">l</font> =  <a href="jordan6.html#K10">Lower_Middle_Point</a> <font color="Maroon">C</font>;<br/>
<a NAME="E5:45"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K11">Upper_Middle_Point</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan6.html#K10">Lower_Middle_Point</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:45"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K11">Upper_Middle_Point</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="jordan6.html#T80">JORDAN6:80</a></i>;<br/>
<a NAME="E7:45"/><b>then </b>
(  <a href="jordan6.html#K11">Upper_Middle_Point</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="jordan6.html#K11">Upper_Middle_Point</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T83">JORDAN6:83</a>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<a NAME="E8:45"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="jordan6.html#K11">Upper_Middle_Point</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:45"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K11">Upper_Middle_Point</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="jordan6.html#K11">Upper_Middle_Point</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<a NAME="E10:45"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K11">Upper_Middle_Point</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan21.html#E2"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E11:45"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K10">Lower_Middle_Point</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="jordan6.html#T79">JORDAN6:79</a></i>;<br/>
<a NAME="E12:45"/><b>then </b>
(  <a href="jordan6.html#K10">Lower_Middle_Point</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="jordan6.html#K10">Lower_Middle_Point</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6">Vertical_Line</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T34">JORDAN6:34</a></i>;<br/>
<a NAME="E13:45"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="jordan6.html#K10">Lower_Middle_Point</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:45"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K10">Lower_Middle_Point</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="jordan6.html#K10">Lower_Middle_Point</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/>
<a NAME="E15:45"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan6.html#K10">Lower_Middle_Point</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E16:45"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="txt" href="jordan21.html#E1"><i><font color="Green">Lemma35</font></i></a>, , <a class="ref" href="seq_4.html#T25">SEQ_4:25</a></i>;<br/>


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