<?xml version="1.0"?>
<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="E1:59"/><i><font color="Green">E46</font></i>: 
(  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> &amp;  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:59"/><i><font color="Green">E53</font></i>: 
(  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">n</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">s</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">S1</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">S2</font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<a NAME="E3:59"/><i><font color="Green">E54</font></i>: 
 <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:59"/><i><font color="Green">E56</font></i>: 
 <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:59"/><i><font color="Green">E57</font></i>: 
 <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:59"/><i><font color="Green">E59</font></i>: 
 <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:59"/><i><font color="Green">E60</font></i>: 
 <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:59"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E9:59"/><i><font color="Green">E65</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T76">JORDAN6:76</a></i>;<br/>
<a NAME="E10:59"/><i><font color="Green">E66</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T76">JORDAN6:76</a></i>;<br/>
<a NAME="E11:59"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E12:59"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan5b.html#T14">JORDAN5B:14</a></i>;<br/>
<a NAME="E13:59"/><i><font color="Green">E113</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8">JORDAN6:def 8</a></i>;<br/>
<div><i><font color="Green">E114</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1"/>
 <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:59_1"/><b>then </b>
 <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:59_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan21.html#E1:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E115</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_2"/>
 <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:59_2"/><b>then </b>
 <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:59_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan21.html#E4"><i><font color="Green">Lemma44</font></i></a>, <a class="txt" href="jordan21.html#E1:59"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:59"/><i><font color="Green">E116</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#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#K15">W-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#K17">E-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="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E17:59"/><i><font color="Green">E119</font></i>: 
<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#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#K15">W-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#K17">E-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="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E18:59"/><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></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#K15">W-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#K17">E-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="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E19:59"/><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></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#K15">W-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#K17">E-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="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E20:59"/><i><font color="Green">E122</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E21:59"/><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E22:59"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E23:59"/><i><font color="Green">E125</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="txt" href="jordan21.html#E2:59"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E24:59"/><i><font color="Green">E126</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> is <a href="sppol_1.html#V2">vertical</a>
 
<b>by </b><i><a class="txt" href="jordan21.html#E3:59"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan21.html#E4:59"><i><font color="Green">E56</font></i></a>, <a class="ref" href="sppol_1.html#T37">SPPOL_1:37</a></i>;<br/>
<a NAME="E25:59"/><i><font color="Green">E127</font></i>: 
 <a href="jordan21.html#K1" target="_self">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">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="E26:59"/><i><font color="Green">E128</font></i>: 
 <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></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="E27:59"/><i><font color="Green">E129</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T33">SPRECT_1:33</a></i>;<br/>
<a NAME="E28:59"/><i><font color="Green">E130</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E29:59"/><i><font color="Green">E131</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> holds <br/>( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="ref" href="jordan21.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan21.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E30:59"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> holds <br/>( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i><a class="ref" href="jordan21.html#T2" target="_self">Th2</a>, , <a class="txt" href="jordan21.html#E6:59"><i><font color="Green">E59</font></i></a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E31:59"/><i><font color="Green">E133</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_3"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:59_3"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:59_3"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">x</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:59_3"/><i><font color="Green">E136</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</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" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:59_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T2" target="_self">Th2</a>, , <a class="ref" href="jordan21.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:59_3"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:59_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:59_3"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan21.html#E1:59"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:59"/><i><font color="Green">E138</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_4"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:59_4"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:59_4"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">x</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:59_4"/><i><font color="Green">E141</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:59_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan21.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan21.html#T4" target="_self">Th4</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:59_4"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:59_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan21.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:59_4"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan21.html#E4"><i><font color="Green">Lemma44</font></i></a>, <a class="txt" href="jordan21.html#E1:59"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_5"/>
(  <a href="jordan6.html#R1">LE</a>  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>,<font color="Maroon">C</font> or  <a href="jordan6.html#R1">LE</a>  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>,<font color="Maroon">C</font> )
 </span><b>by </b><i><a class="txt" href="jordan21.html#E3"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="jordan16.html#T11">JORDAN16:11</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_1"><b>suppose </b></a><a NAME="E1:59_5_1"/>
 <a href="jordan6.html#R1">LE</a>  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>,<font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:59_5_1"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="jordan5c.html#R1">LE</a>  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>set </b><font color="Maroon">S</font> =  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>;<br/><a NAME="E3:59_5_1"/><i><font color="Green">E145</font></i>: 
 <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan21.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan16.html#T9">JORDAN16:9</a></i>;<br/><a NAME="E4:59_5_1"/><i><font color="Green">E146</font></i>: 
 <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan21.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan16.html#T9">JORDAN16:9</a></i>;<br/><a NAME="E5:59_5_1"/><i><font color="Green">E147</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5">JORDAN16:5</a></i>;<br/><a NAME="E6:59_5_1"/><b>then </b><i><font color="Green">E148</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:59_5_1"/><i><font color="Green">E149</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:59_5_1_1"/>
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</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" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:59_5_1_1"/>
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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" target="_self">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="jordan21.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</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>let </b><font color="Maroon">x</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="E3:59_5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:59_5_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:59_5_1_1"/>
 <a href="jordan21.html#K1" target="_self">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">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/>
<b>hence </b><a NAME="E6:59_5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:59_5_1"/><i><font color="Green">E151</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:59_5_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:59_5_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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" target="_self">LMP</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="jordan21.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</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>let </b><font color="Maroon">x</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="E3:59_5_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:59_5_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:59_5_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan21.html#E8:59"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E152</font></i>: <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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan21.html#E9:59"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/><a NAME="E10:59_5_1"/>
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan21.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan16.html#T36">JORDAN16:36</a></i>;<br/><a NAME="E11:59_5_1"/><b>then </b>
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan5b.html#T14">JORDAN5B:14</a></i>;<br/><a NAME="E12:59_5_1"/><b>then </b>
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>, <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T17">TOPREAL1:17</a></i>;<br/><a NAME="E13:59_5_1"/><b>then </b>
<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, <a class="ref" href="jordan21.html#T11" target="_self">Th11</a>, <a class="ref" href="topreal1.html#T16">TOPREAL1:16</a></i>;<br/><a NAME="E14:59_5_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jordan5b.html#T14">JORDAN5B:14</a></i>;<br/><a NAME="E15:59_5_1"/><i><font color="Green">E154</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:59_5_1_4"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_5_1_4_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:59_5_1_4_1"/><b>then </b>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> or <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:59_5_1_4_1"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_5_1_4"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:59_5_1"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_5_1_5"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:59_5_1_5"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:59_5_1_5"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:59_5_1_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:59_5_1_5"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:59_5_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan21.html#E1:59"><i><font color="Green">E46</font></i></a>, , , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:59_5_1"/><b>then </b><i><font color="Green">E158</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R1">misses</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan21.html#T3" target="_self">Th3</a>, , </i>;<br/><div><i><font color="Green">E159</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:59_5_1_6"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_5_1_6_1"/>
( <font color="Maroon">p</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> or <font color="Maroon">p</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="ref" href="jordan21.html#D1" target="_self">Def1</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_1_6_1_1"><b>suppose </b></a><a NAME="E1:59_5_1_6_1_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">p</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_5_1_6_1_1"/><i><font color="Green">E162</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="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan21.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan21.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/><a NAME="E3:59_5_1_6_1_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></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>  &amp; <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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan21.html#E17"><i><font color="Green">Lemma83</font></i></a>, , <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><b>hence </b><a NAME="E4:59_5_1_6_1_1"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E17"><i><font color="Green">Lemma83</font></i></a>, <a class="ref" href="jordan21.html#T12" target="_self">Th12</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_1_6_1_2"><b>suppose </b></a><a NAME="E1:59_5_1_6_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_5_1_6_1_2"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan21.html#T5" target="_self">Th5</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_1_6_1_3"><b>suppose </b></a><a NAME="E1:59_5_1_6_1_3"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">p</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:59_5_1_6_1_3"/><i><font color="Green">E164</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></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#K2" target="_self">LMP</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#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan21.html#E2"><i><font color="Green">Lemma36</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_5_1_6_1_3"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan21.html#E17"><i><font color="Green">Lemma83</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E4:59_5_1_6_1_3"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">p</font> <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#K2" target="_self">LMP</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#E17"><i><font color="Green">Lemma83</font></i></a>, <a class="ref" href="jordan21.html#T12" target="_self">Th12</a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E5:59_5_1_6_1_3"/>
<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"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan21.html#E5:59"><i><font color="Green">E57</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:59_5_1_6_1_3"/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan21.html#T13" target="_self">Th13</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:59_5_1_7"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_5_1_7_1"/>
( <font color="Maroon">p</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> or <font color="Maroon">p</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="ref" href="jordan21.html#D1" target="_self">Def1</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_1_7_1_1"><b>suppose </b></a><a NAME="E1:59_5_1_7_1_1"/>
<font color="Maroon">p</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_5_1_7_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</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#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan21.html#E7:59"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:59_5_1_7_1_1"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E2:59"><i><font color="Green">E53</font></i></a>, , <a class="txt" href="jordan21.html#E6:59"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_1_7_1_2"><b>suppose </b></a><a NAME="E1:59_5_1_7_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_5_1_7_1_2"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="jordan21.html#E6:59"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan21.html#T5" target="_self">Th5</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_1_7_1_3"><b>suppose </b></a><a NAME="E1:59_5_1_7_1_3"/>
<font color="Maroon">p</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:59_5_1_7_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <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#K21">`1</a> 
 <b>by </b><i>, <a class="txt" href="jordan21.html#E8:59"><i><font color="Green">E63</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:59_5_1_7_1_3"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E3:59"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan21.html#E6:59"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E20:59_5_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan21.html#E10:59"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan21.html#E12:59"><i><font color="Green">E67</font></i></a>, , , , <a class="ref" href="jgraph_1.html#T62">JGRAPH_1:62</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_2"><b>suppose </b></a><a NAME="E1:59_5_2"/>
 <a href="jordan6.html#R1">LE</a>  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>,<font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:59_5_2"/><b>then </b><i><font color="Green">E167</font></i>: 
 <a href="jordan5c.html#R1">LE</a>  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>set </b><font color="Maroon">S</font> =  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>;<br/><a NAME="E3:59_5_2"/><i><font color="Green">E168</font></i>: 
 <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan21.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan16.html#T9">JORDAN16:9</a></i>;<br/><a NAME="E4:59_5_2"/><i><font color="Green">E169</font></i>: 
 <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan21.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan16.html#T9">JORDAN16:9</a></i>;<br/><a NAME="E5:59_5_2"/><i><font color="Green">E170</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5">JORDAN16:5</a></i>;<br/><a NAME="E6:59_5_2"/><b>then </b><i><font color="Green">E171</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:59_5_2"/><i><font color="Green">E172</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:59_5_2_1"/>
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</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" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:59_5_2_1"/>
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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" target="_self">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="jordan21.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</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>let </b><font color="Maroon">x</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="E3:59_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:59_5_2_1"/><b>then </b><i><font color="Green">E173</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:59_5_2_1"/>
 <a href="jordan21.html#K1" target="_self">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">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/>
<b>hence </b><a NAME="E6:59_5_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:59_5_2"/><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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:59_5_2_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:59_5_2_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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" target="_self">LMP</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="jordan21.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</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>let </b><font color="Maroon">x</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="E3:59_5_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:59_5_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:59_5_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan21.html#E8:59"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E175</font></i>: <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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan21.html#E9:59"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/><a NAME="E10:59_5_2"/>
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>, <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan21.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan16.html#T36">JORDAN16:36</a></i>;<br/><a NAME="E11:59_5_2"/><b>then </b>
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>, <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="topreal1.html#T17">TOPREAL1:17</a></i>;<br/><a NAME="E12:59_5_2"/><b>then </b>
<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, <a class="ref" href="jordan21.html#T11" target="_self">Th11</a>, <a class="ref" href="topreal1.html#T16">TOPREAL1:16</a></i>;<br/><a NAME="E13:59_5_2"/><b>then </b><i><font color="Green">E176</font></i>: 
<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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jordan5b.html#T14">JORDAN5B:14</a></i>;<br/><a NAME="E14:59_5_2"/><i><font color="Green">E177</font></i>: 
 <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</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="E1:59_5_2_4"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_5_2_4_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:59_5_2_4_1"/><b>then </b>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> or <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:59_5_2_4_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan21.html#E4"><i><font color="Green">Lemma44</font></i></a>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_5_2_4"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:59_5_2"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_5_2_5"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:59_5_2_5"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:59_5_2_5"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:59_5_2_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:59_5_2_5"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:59_5_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan21.html#E1:59"><i><font color="Green">E46</font></i></a>, , , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:59_5_2"/><b>then </b><i><font color="Green">E181</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="subset_1.html#R1">misses</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan21.html#T3" target="_self">Th3</a>, , </i>;<br/><div><i><font color="Green">E182</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:59_5_2_6"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_5_2_6_1"/>
( <font color="Maroon">p</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> or <font color="Maroon">p</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="ref" href="jordan21.html#D1" target="_self">Def1</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_2_6_1_1"><b>suppose </b></a><a NAME="E1:59_5_2_6_1_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">p</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_5_2_6_1_1"/><i><font color="Green">E185</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="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan21.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan21.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/><a NAME="E3:59_5_2_6_1_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></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>  &amp; <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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan21.html#E17"><i><font color="Green">Lemma83</font></i></a>, , <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><b>hence </b><a NAME="E4:59_5_2_6_1_1"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E17"><i><font color="Green">Lemma83</font></i></a>, <a class="ref" href="jordan21.html#T12" target="_self">Th12</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_2_6_1_2"><b>suppose </b></a><a NAME="E1:59_5_2_6_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_5_2_6_1_2"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E5:59"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan21.html#T5" target="_self">Th5</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_2_6_1_3"><b>suppose </b></a><a NAME="E1:59_5_2_6_1_3"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">p</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:59_5_2_6_1_3"/><i><font color="Green">E187</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></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#K2" target="_self">LMP</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#E5:59"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan21.html#E2"><i><font color="Green">Lemma36</font></i></a></i>;<br/><b>hence </b><a NAME="E3:59_5_2_6_1_3"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan21.html#E17"><i><font color="Green">Lemma83</font></i></a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E4:59_5_2_6_1_3"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">p</font> <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#K2" target="_self">LMP</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#E17"><i><font color="Green">Lemma83</font></i></a>, <a class="ref" href="jordan21.html#T12" target="_self">Th12</a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/><a NAME="E5:59_5_2_6_1_3"/>
<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"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan21.html#E5:59"><i><font color="Green">E57</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:59_5_2_6_1_3"/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="jordan21.html#T13" target="_self">Th13</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>assume </b><a NAME="E1:59_5_2_7"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5">Segment</a> <span class="p3">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_5_2_7_1"/>
( <font color="Maroon">p</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> or <font color="Maroon">p</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="ref" href="jordan21.html#D1" target="_self">Def1</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_2_7_1_1"><b>suppose </b></a><a NAME="E1:59_5_2_7_1_1"/>
<font color="Maroon">p</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_5_2_7_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</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#K21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan21.html#E7:59"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:59_5_2_7_1_1"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E2:59"><i><font color="Green">E53</font></i></a>, , <a class="txt" href="jordan21.html#E6:59"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_2_7_1_2"><b>suppose </b></a><a NAME="E1:59_5_2_7_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K5">Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_5_2_7_1_2"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="jordan21.html#E6:59"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan21.html#T5" target="_self">Th5</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_5_2_7_1_3"><b>suppose </b></a><a NAME="E1:59_5_2_7_1_3"/>
<font color="Maroon">p</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" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K15">W-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#K17">E-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:59_5_2_7_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <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#K21">`1</a> 
 <b>by </b><i>, <a class="txt" href="jordan21.html#E8:59"><i><font color="Green">E63</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:59_5_2_7_1_3"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="txt" href="jordan21.html#E3:59"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan21.html#E6:59"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="topreal3.html#T3">TOPREAL3:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:59_5_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan21.html#E10:59"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan21.html#E12:59"><i><font color="Green">E67</font></i></a>, , , , <a class="ref" href="jgraph_1.html#T62">JGRAPH_1:62</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
