<?xml version="1.0"?>
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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="sprect_2.html#V1" target="_self">clockwise_oriented</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:89"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:89"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:89"/><i><font color="Green">E84</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T50" target="_self">Th50</a></i>;<br/>
<a NAME="E4:89"/><i><font color="Green">E85</font></i>: 
 <a href="pscomp_1.html#K33">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E5:89"/><i><font color="Green">E86</font></i>: 
 <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E6:89"/><i><font color="Green">E87</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E7:89"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E3:89"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E8:89"/><b>then </b><i><font color="Green">E89</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:89"/><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E5:89"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E10:89"/><b>then </b><i><font color="Green">E91</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green">E92</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E9:89"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E5:89"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E12:89"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>
<i><font color="Green">E94</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E3:89"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E14:89"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="sprect_2.html#T57" target="_self">Th57</a></i>;<br/>
<a NAME="E15:89"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E2:89"><i><font color="Green">E83</font></i></a>, <a class="txt" href="sprect_2.html#E11:89"><i><font color="Green">E92</font></i></a>, <a class="txt" href="sprect_2.html#E13:89"><i><font color="Green">E94</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E16:89"/><b>then </b><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E10:89"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:89"/><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E4:89"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E18:89"/><b>then </b><i><font color="Green">E98</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E19:89"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E8:89"><i><font color="Green">E89</font></i></a>, <a class="txt" href="sprect_2.html#E15:89"><i><font color="Green">E95</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<i><font color="Green">E100</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E17:89"><i><font color="Green">E97</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K33">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E4:89"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E21:89"/><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sprect_2.html#E87"><i><font color="Green">Lemma80</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sprect_2.html#E16:89"><i><font color="Green">E96</font></i></a>, <a class="txt" href="sprect_2.html#E18:89"><i><font color="Green">E98</font></i></a>, <a class="ref" href="sprect_2.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E23:89"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sprect_2.html#E17:89"><i><font color="Green">E97</font></i></a>, <a class="ref" href="sprect_2.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E24:89"/>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K33">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sprect_2.html#E17:89"><i><font color="Green">E97</font></i></a>, <a class="txt" href="sprect_2.html#E20:89"><i><font color="Green">E100</font></i></a>, <a class="ref" href="sprect_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E25:89"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E26:89"/><i><font color="Green">E104</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E8:89"><i><font color="Green">E89</font></i></a>, <a class="txt" href="sprect_2.html#E18:89"><i><font color="Green">E98</font></i></a>, <a class="ref" href="jordan4.html#T47">JORDAN4:47</a></i>;<br/>
<a NAME="E27:89"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sprect_2.html#E17:89"><i><font color="Green">E97</font></i></a>, <a class="ref" href="sprect_2.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E28:89"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="sprect_2.html#T55" target="_self">Th55</a></i>;<br/>
<a NAME="E29:89"/><i><font color="Green">E107</font></i>: 
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal5.html#T22">TOPREAL5:22</a></i>;<br/>
<a NAME="E30:89"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E31:89"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E12:89"><i><font color="Green">E93</font></i></a>, <a class="txt" href="sprect_2.html#E18:89"><i><font color="Green">E98</font></i></a>, <a class="txt" href="sprect_2.html#E20:89"><i><font color="Green">E100</font></i></a>, <a class="txt" href="sprect_2.html#E29:89"><i><font color="Green">E107</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E32:89"/>
 <a href="pscomp_1.html#K30">N-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T13">SPRECT_1:13</a></i>;<br/>
<a NAME="E33:89"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E34:89"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E28:89"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:89"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E36:89"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sprect_2.html#E10:89"><i><font color="Green">E91</font></i></a>, <a class="txt" href="sprect_2.html#E11:89"><i><font color="Green">E92</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,1,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sprect_2.html#E19:89"><i><font color="Green">E99</font></i></a>, <a class="ref" href="sprect_2.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E38:89"/><i><font color="Green">E110</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E10:89"><i><font color="Green">E91</font></i></a>, <a class="txt" href="sprect_2.html#E15:89"><i><font color="Green">E95</font></i></a>, <a class="txt" href="sprect_2.html#E21:89"><i><font color="Green">E101</font></i></a>, <a class="txt" href="sprect_2.html#E31:89"><i><font color="Green">E108</font></i></a>, <a class="ref" href="sprect_2.html#T52" target="_self">Th52</a></i>;<br/>
<a NAME="E39:89"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E10:89"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E40:89"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E10:89"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jordan4.html#T47">JORDAN4:47</a></i>;<br/>
<i><font color="Green">E112</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sprect_2.html#E9:89"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sprect_2.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E5:89"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_5.html#T41">FINSEQ_5:41</a></i>
;<br/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="sprect_2.html#E10:89"><i><font color="Green">E91</font></i></a>, <a class="ref" href="jordan4.html#T20">JORDAN4:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="sprect_2.html#E10:89"><i><font color="Green">E91</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/>
<a NAME="E43:89"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E36:89"><i><font color="Green">E109</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E44:89"/><i><font color="Green">E114</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sprect_2.html#E17:89"><i><font color="Green">E97</font></i></a>, <a class="ref" href="sprect_2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E45:89"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sprect_2.html#E17:89"><i><font color="Green">E97</font></i></a>, <a class="txt" href="sprect_2.html#E21:89"><i><font color="Green">E101</font></i></a>, <a class="ref" href="sprect_2.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E46:89"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E44:89"><i><font color="Green">E114</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E47:89"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E48:89"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sprect_2.html#E9:89"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sprect_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E49:89"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sprect_2.html#E9:89"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sprect_2.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E50:89"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E41:89"><i><font color="Green">E112</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_6"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_6"/>
( (  <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) or (  <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) or (  <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) or (  <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_6_1"><b>suppose </b></a><b>that </b><a NAME="E1:89_6_1"/><i><font color="Green">E119</font></i>: 
 <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:89_6_1"/><i><font color="Green">E120</font></i>: 
 <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E3:89_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E13:89"><i><font color="Green">E94</font></i></a>, <a class="txt" href="sprect_2.html#E27:89"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sprect_2.html#E2:89_6_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:89_6_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="sprect_2.html#R3" target="_self">is_a_v.c._for</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E23:89"><i><font color="Green">E102</font></i></a>, <a class="txt" href="sprect_2.html#E25:89"><i><font color="Green">E103</font></i></a>, <a class="ref" href="sprect_2.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E5:89_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sprect_2.html#E48:89"><i><font color="Green">E116</font></i></a>, <a class="txt" href="sprect_2.html#E1:89_6_1"><i><font color="Green">E119</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E6:89_6_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="sprect_2.html#R2" target="_self">is_a_h.c._for</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E49:89"><i><font color="Green">E117</font></i></a>, <a class="txt" href="sprect_2.html#E50:89"><i><font color="Green">E118</font></i></a>, <a class="ref" href="sprect_2.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E7:89_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_2.html#E38:89"><i><font color="Green">E110</font></i></a>, <a class="txt" href="sprect_2.html#E43:89"><i><font color="Green">E113</font></i></a>, <a class="txt" href="sprect_2.html#E47:89"><i><font color="Green">E115</font></i></a>, <a class="txt" href="sprect_2.html#E4:89_6_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="sprect_2.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_6_2"><b>suppose </b></a><b>that </b><a NAME="E1:89_6_2"/><i><font color="Green">E119</font></i>: 
 <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:89_6_2"/><i><font color="Green">E120</font></i>: 
 <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E3:89_6_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E13:89"><i><font color="Green">E94</font></i></a>, <a class="txt" href="sprect_2.html#E27:89"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sprect_2.html#E2:89_6_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:89_6_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="sprect_2.html#R3" target="_self">is_a_v.c._for</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E23:89"><i><font color="Green">E102</font></i></a>, <a class="txt" href="sprect_2.html#E25:89"><i><font color="Green">E103</font></i></a>, <a class="ref" href="sprect_2.html#D3" target="_self">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sprect_2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sprect_2.html#E9:89"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sprect_2.html#E1:89_6_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="sprect_2.html#T70" target="_self">Th70</a></i>;<br/><a NAME="E6:89_6_2"/><i><font color="Green">E122</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E7:89_6_2"/>
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E8:89_6_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E49:89"><i><font color="Green">E117</font></i></a>, <a class="ref" href="sprect_2.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E9:89_6_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T16">FINSEQ_5:16</a></i>;<br/><a NAME="E10:89_6_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E11:89_6_2"/><i><font color="Green">E125</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E12:89_6_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>then </b><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E11:89_6_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sprect_2.html#E9:89"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sprect_2.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E5:89"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_5.html#T41">FINSEQ_5:41</a></i>
;<br/><a NAME="E14:89_6_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E15:89_6_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="sprect_2.html#R2" target="_self">is_a_h.c._for</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E8:89_6_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="sprect_2.html#E10:89_6_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="sprect_2.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E16:89_6_2"/><i><font color="Green">E127</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></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="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T20">SPPOL_2:20</a></i>;<br/><a NAME="E17:89_6_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E26:89"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E18:89_6_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sprect_2.html#E48:89"><i><font color="Green">E116</font></i></a>, <a class="ref" href="pscomp_1.html#T102">PSCOMP_1:102</a></i>;<br/><a NAME="E19:89_6_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E43:89"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/><a NAME="E20:89_6_2"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E38:89"><i><font color="Green">E110</font></i></a>, <a class="txt" href="sprect_2.html#E16:89_6_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="sprect_2.html#E18:89_6_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="sprect_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E21:89_6_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_2.html#E47:89"><i><font color="Green">E115</font></i></a>, <a class="txt" href="sprect_2.html#E4:89_6_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="sprect_2.html#E6:89_6_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="sprect_2.html#E15:89_6_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="sprect_2.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_6_3"><b>suppose </b></a><b>that </b><a NAME="E1:89_6_3"/><i><font color="Green">E119</font></i>: 
 <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:89_6_3"/><i><font color="Green">E120</font></i>: 
 <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E3:89_6_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="txt" href="sprect_2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sprect_2.html#E9:89"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sprect_2.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E4:89_6_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sprect_2.html#E1:89_6_3"><i><font color="Green">E119</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E5:89_6_3"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="sprect_2.html#R2" target="_self">is_a_h.c._for</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E49:89"><i><font color="Green">E117</font></i></a>, <a class="txt" href="sprect_2.html#E50:89"><i><font color="Green">E118</font></i></a>, <a class="ref" href="sprect_2.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sprect_2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sprect_2.html#E13:89"><i><font color="Green">E94</font></i></a>, <a class="txt" href="sprect_2.html#E17:89"><i><font color="Green">E97</font></i></a>, <a class="txt" href="sprect_2.html#E2:89_6_3"><i><font color="Green">E120</font></i></a>, <a class="ref" href="sprect_2.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E7:89_6_3"/><i><font color="Green">E122</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E8:89_6_3"/>
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E9:89_6_3"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E23:89"><i><font color="Green">E102</font></i></a>, <a class="ref" href="sprect_2.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E10:89_6_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E11:89_6_3"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E25:89"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E13:89_6_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><a NAME="E14:89_6_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E15:89_6_3"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="sprect_2.html#R3" target="_self">is_a_v.c._for</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E9:89_6_3"><i><font color="Green">E123</font></i></a>, <a class="txt" href="sprect_2.html#E11:89_6_3"><i><font color="Green">E124</font></i></a>, <a class="ref" href="sprect_2.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E16:89_6_3"/><i><font color="Green">E126</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></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="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T19">SPPOL_2:19</a></i>;<br/><a NAME="E17:89_6_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E40:89"><i><font color="Green">E111</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E18:89_6_3"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E13:89"><i><font color="Green">E94</font></i></a>, <a class="txt" href="sprect_2.html#E27:89"><i><font color="Green">E105</font></i></a>, <a class="ref" href="pscomp_1.html#T112">PSCOMP_1:112</a></i>;<br/><a NAME="E19:89_6_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E47:89"><i><font color="Green">E115</font></i></a>, <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/><a NAME="E20:89_6_3"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E38:89"><i><font color="Green">E110</font></i></a>, <a class="txt" href="sprect_2.html#E16:89_6_3"><i><font color="Green">E126</font></i></a>, <a class="txt" href="sprect_2.html#E18:89_6_3"><i><font color="Green">E127</font></i></a>, <a class="ref" href="sprect_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E21:89_6_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_2.html#E5:89_6_3"><i><font color="Green">E121</font></i></a>, <a class="txt" href="sprect_2.html#E7:89_6_3"><i><font color="Green">E122</font></i></a>, <a class="txt" href="sprect_2.html#E15:89_6_3"><i><font color="Green">E125</font></i></a>, <a class="ref" href="sprect_2.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_6_4"><b>suppose </b></a><b>that </b><a NAME="E1:89_6_4"/><i><font color="Green">E119</font></i>: 
 <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:89_6_4"/><i><font color="Green">E120</font></i>: 
 <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sprect_2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sprect_2.html#E9:89"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sprect_2.html#E1:89_6_4"><i><font color="Green">E119</font></i></a>, <a class="ref" href="sprect_2.html#T70" target="_self">Th70</a></i>;<br/><a NAME="E4:89_6_4"/>
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E5:89_6_4"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E49:89"><i><font color="Green">E117</font></i></a>, <a class="ref" href="sprect_2.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E6:89_6_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T16">FINSEQ_5:16</a></i>;<br/><a NAME="E7:89_6_4"/><b>then </b><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E8:89_6_4"/><i><font color="Green">E123</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E9:89_6_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>then </b><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E8:89_6_4"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E6:89"><i><font color="Green">E87</font></i></a>, <a class="txt" href="sprect_2.html#E9:89"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sprect_2.html#T13" target="_self">Th13</a></i>
<br/>.= 
 <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_2.html#E5:89"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_5.html#T41">FINSEQ_5:41</a></i>
;<br/><a NAME="E11:89_6_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E12:89_6_4"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="sprect_2.html#R2" target="_self">is_a_h.c._for</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E5:89_6_4"><i><font color="Green">E121</font></i></a>, <a class="txt" href="sprect_2.html#E7:89_6_4"><i><font color="Green">E122</font></i></a>, <a class="ref" href="sprect_2.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="txt" href="sprect_2.html#E7:89"><i><font color="Green">E88</font></i></a>, <a class="txt" href="sprect_2.html#E13:89"><i><font color="Green">E94</font></i></a>, <a class="txt" href="sprect_2.html#E17:89"><i><font color="Green">E97</font></i></a>, <a class="txt" href="sprect_2.html#E2:89_6_4"><i><font color="Green">E120</font></i></a>, <a class="ref" href="sprect_2.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E14:89_6_4"/><i><font color="Green">E125</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/><a NAME="E15:89_6_4"/>
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E16:89_6_4"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E23:89"><i><font color="Green">E102</font></i></a>, <a class="ref" href="sprect_2.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E17:89_6_4"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E18:89_6_4"/><b>then </b><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E25:89"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E20:89_6_4"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><a NAME="E21:89_6_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E22:89_6_4"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="sprect_2.html#R3" target="_self">is_a_v.c._for</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E16:89_6_4"><i><font color="Green">E126</font></i></a>, <a class="txt" href="sprect_2.html#E18:89_6_4"><i><font color="Green">E127</font></i></a>, <a class="ref" href="sprect_2.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E23:89_6_4"/><i><font color="Green">E129</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></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="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T19">SPPOL_2:19</a></i>;<br/><a NAME="E24:89_6_4"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E40:89"><i><font color="Green">E111</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E25:89_6_4"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E13:89"><i><font color="Green">E94</font></i></a>, <a class="txt" href="sprect_2.html#E27:89"><i><font color="Green">E105</font></i></a>, <a class="ref" href="pscomp_1.html#T112">PSCOMP_1:112</a></i>;<br/><a NAME="E26:89_6_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E47:89"><i><font color="Green">E115</font></i></a>, <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/><a NAME="E27:89_6_4"/><b>then </b><i><font color="Green">E131</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E38:89"><i><font color="Green">E110</font></i></a>, <a class="txt" href="sprect_2.html#E23:89_6_4"><i><font color="Green">E129</font></i></a>, <a class="txt" href="sprect_2.html#E25:89_6_4"><i><font color="Green">E130</font></i></a>, <a class="ref" href="sprect_2.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E28:89_6_4"/><i><font color="Green">E132</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font></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="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T20">SPPOL_2:20</a></i>;<br/><a NAME="E29:89_6_4"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></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="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sprect_2.html#E13:89"><i><font color="Green">E94</font></i></a>, <a class="txt" href="sprect_2.html#E27:89"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sprect_2.html#E48:89"><i><font color="Green">E116</font></i></a>, <a class="txt" href="sprect_2.html#E88"><i><font color="Green">Lemma81</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="sprect_2.html#E23:89_6_4"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

;<br/><a NAME="E31:89_6_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E26:89"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E32:89_6_4"/><b>then </b><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E1:89"><i><font color="Green">E82</font></i></a>, <a class="txt" href="sprect_2.html#E48:89"><i><font color="Green">E116</font></i></a>, <a class="ref" href="pscomp_1.html#T102">PSCOMP_1:102</a></i>;<br/><a NAME="E33:89_6_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E43:89"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/><a NAME="E34:89_6_4"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E27:89_6_4"><i><font color="Green">E131</font></i></a>, <a class="txt" href="sprect_2.html#E28:89_6_4"><i><font color="Green">E132</font></i></a>, <a class="txt" href="sprect_2.html#E32:89_6_4"><i><font color="Green">E133</font></i></a>, <a class="ref" href="sprect_2.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E35:89_6_4"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_2.html#E12:89_6_4"><i><font color="Green">E124</font></i></a>, <a class="txt" href="sprect_2.html#E14:89_6_4"><i><font color="Green">E125</font></i></a>, <a class="txt" href="sprect_2.html#E22:89_6_4"><i><font color="Green">E128</font></i></a>, <a class="ref" href="sprect_2.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
