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

<b>let </b><font color="Maroon">z</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">i1</font> = <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">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">i2</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:85"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">z</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">z</font></span>)</span>
 <b>and </b><br/><a NAME="E2:85"/><i><font color="Green">E26</font></i>: 
<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">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E3:85"/><i><font color="Green">E27</font></i>: 
 <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">z</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E10:85"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E4:85"/><i><font color="Green">E28</font></i>: 
 <a href="pscomp_1.html#K34">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:85"/><i><font color="Green">E42</font></i>: 
<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">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T22" target="_self">Th22</a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E6:85"/><b>then </b><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&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">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> &amp; <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">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:85"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E8:85"/><b>then </b><i><font color="Green">E45</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green">E46</font></i>: <font color="Maroon">z</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#K34">S-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">z</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#K34">S-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K34">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E10:85"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">z</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">z</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">z</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>
<i><font color="Green">E63</font></i>: <font color="Maroon">z</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#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span> = 
<font color="Maroon">z</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#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <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">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T22" target="_self">Th22</a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E12:85"/>
( <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">z</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">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</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">z</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E13:85"/><b>then </b>
<font color="Maroon">z</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#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</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#K34">S-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T26" target="_self">Th26</a>, <a class="ref" href="sprect_2.html#T27" target="_self">Th27</a>, <a class="ref" href="topreal5.html#T22">TOPREAL5:22</a></i>;<br/>
<a NAME="E14:85"/><b>then </b><i><font color="Green">E64</font></i>: 
<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">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E15:85"/><b>then </b><i><font color="Green">E65</font></i>: 
<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">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="sprect_2.html#T25" target="_self">Th25</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:85"/><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E17:85"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">z</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">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>
<a NAME="E18:85"/><i><font color="Green">E68</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="realset1.html#T13">REALSET1:13</a></i>;<br/>
<a NAME="E19:85"/><b>then </b><i><font color="Green">E69</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E20:85"/>
<font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K24">N-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><i><font color="Green">E70</font></i>: <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  = 
<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">z</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#T98">PSCOMP_1:98</a></i>
<br/>.= 
 <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">z</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>
<a NAME="E22:85"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">z</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#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E23:85"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="sprect_2.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E24:85"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</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">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T26" target="_self">Th26</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E25:85"/>
 <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">z</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">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal5.html#T22">TOPREAL5:22</a></i>;<br/>
<a NAME="E26:85"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 
<b>by </b><i><a class="ref" href="sprect_2.html#D3" target="_self">Def3</a>, <a class="ref" href="sprect_2.html#T33" target="_self">Th33</a>, , , <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K34">S-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span>,2 as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="ref" href="sprect_2.html#T25" target="_self">Th25</a>, <a class="txt" href="sprect_2.html#E7:85"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E28:85"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</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="E29:85"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">h</font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#D1" target="_self">Def1</a>, <a class="ref" href="sprect_2.html#D2" target="_self">Def2</a>, <a class="txt" href="sprect_2.html#E3:85"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E30:85"/>
<font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K34">S-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#D1" target="_self">Def1</a>, <a class="ref" href="sprect_2.html#T26" target="_self">Th26</a>, <a class="ref" href="sprect_2.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E31:85"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</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">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E32:85"/><i><font color="Green">E106</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T25" target="_self">Th25</a>, <a class="ref" href="sprect_2.html#T32" target="_self">Th32</a>, <a class="ref" href="jordan4.html#T47">JORDAN4:47</a></i>;<br/>
<a NAME="E33:85"/>
<font color="Maroon">h</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">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> 2
 
<b>by </b><i><a class="ref" href="sprect_2.html#D1" target="_self">Def1</a>, <a class="ref" href="sprect_2.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E34:85"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">h</font> <a href="sprect_2.html#R3" target="_self">is_a_v.c._for</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#D3" target="_self">Def3</a>, <a class="txt" href="sprect_2.html#E28:85"><i><font color="Green">E103</font></i></a>, <a class="txt" href="sprect_2.html#E29:85"><i><font color="Green">E104</font></i></a>, </i>;<br/>
<a NAME="E35:85"/>
 <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">z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E36:85"/><b>then </b><i><font color="Green">E134</font></i>: 
<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">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="sprect_2.html#T27" target="_self">Th27</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="ref" href="sprect_2.html#T29" target="_self">Th29</a>, <a class="txt" href="sprect_2.html#E7:85"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E38:85"/><i><font color="Green">E135</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">M</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="sprect_2.html#T28" target="_self">Th28</a>, , </i>;<br/>
<i><font color="Green">E136</font></i>: <font color="Maroon">M</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">M</font></span>)</span> = 
<font color="Maroon">z</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#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, </i>
<br/>.= 
 <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">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T22" target="_self">Th22</a>, <a class="ref" href="finseq_5.html#T41">FINSEQ_5:41</a></i>
;<br/>
<a NAME="E40:85"/><i><font color="Green">E137</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan4.html#T21">JORDAN4:21</a></i>;<br/>
<a NAME="E41:85"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#D4" target="_self">Def4</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E42:85"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E43:85"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<a NAME="E44:85"/><b>then </b><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E45:85"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">M</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">M</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/>
<a NAME="E46:85"/><i><font color="Green">E140</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E47:85"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">M</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <a class="ref" href="sprect_2.html#T31" target="_self">Th31</a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_5"/>
( (  <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">z</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">z</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">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <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">z</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">z</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">z</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">z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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">z</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">z</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">z</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">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <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">z</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">z</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">z</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">z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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">z</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_5_1"><b>suppose </b></a><b>that </b><a NAME="E1:85_5_1"/><i><font color="Green">E142</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">z</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">z</font></span>)</span>
 <b>and </b><br/><a NAME="E2:85_5_1"/><i><font color="Green">E143</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">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <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">z</font></span>)</span>
 ;<br/><div class="add"><a NAME="E3:85_5_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">M</font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <a class="txt" href="sprect_2.html#E3:85"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E4:85_5_1"/>
<font color="Maroon">M</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">z</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">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, </i>;<br/><a NAME="E5:85_5_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</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">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="sprect_2.html#T31" target="_self">Th31</a>, <a class="txt" href="sprect_2.html#E43"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E6:85_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">M</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">M</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">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_2.html#T35" target="_self">Th35</a>, <a class="txt" href="sprect_2.html#E44"><i><font color="Green">Lemma116</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E7:85_5_1"/><b>then </b>
<font color="Maroon">M</font> <a href="sprect_2.html#R2" target="_self">is_a_h.c._for</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T39" target="_self">Th39</a>, <a class="ref" href="sprect_2.html#T41" target="_self">Th41</a>, </i>;<br/><b>hence </b><a NAME="E8:85_5_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="sprect_2.html#E32:85"><i><font color="Green">E106</font></i></a>, <a class="ref" href="sprect_2.html#T34" target="_self">Th34</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_5_2"><b>suppose </b></a><b>that </b><a NAME="E1:85_5_2"/><i><font color="Green">E146</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">z</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">z</font></span>)</span>
 <b>and </b><br/><a NAME="E2:85_5_2"/><i><font color="Green">E147</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">z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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">z</font></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">M</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T27" target="_self">Th27</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <a class="txt" href="sprect_2.html#E44"><i><font color="Green">Lemma116</font></i></a>, <a class="txt" href="sprect_2.html#E18:85"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E4:85_5_2"/><i><font color="Green">E148</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</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="E5:85_5_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">M</font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <a class="txt" href="sprect_2.html#E3:85"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E6:85_5_2"/>
<span class="p1"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:85_5_2"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">g</font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T41" target="_self">Th41</a>, <a class="txt" href="sprect_2.html#E4:85"><i><font color="Green">E28</font></i></a></i>;<br/><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">M</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="sprect_2.html#T36" target="_self">Th36</a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <a class="ref" href="sprect_2.html#T31" target="_self">Th31</a>, </i>
;<br/><a NAME="E9:85_5_2"/><b>then </b><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</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">z</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="sprect_2.html#E43"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</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#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&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">M</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="E11:85_5_2"/><b>then </b>
<font color="Maroon">g</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">g</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">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><a NAME="E12:85_5_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</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">g</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">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E13:85_5_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">g</font> <a href="sprect_2.html#R2" target="_self">is_a_h.c._for</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T42" target="_self">Th42</a>, <a class="ref" href="sprect_2.html#T43" target="_self">Th43</a>, </i>;<br/><a NAME="E14:85_5_2"/><i><font color="Green">E153</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">M</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"><font color="Maroon">M</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">M</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">z</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="E15:85_5_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">M</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">M</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">z</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">h</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">M</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">M</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">z</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">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E31:85"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E16:85_5_2"/><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">M</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">M</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">z</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">h</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">M</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">M</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="sprect_2.html#T35" target="_self">Th35</a>, <a class="ref" href="pscomp_1.html#T102">PSCOMP_1:102</a></i>;<br/><a NAME="E17:85_5_2"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T34" target="_self">Th34</a>, , <a class="ref" href="sprect_2.html#T45" target="_self">Th45</a>, </i>;<br/><b>hence </b><a NAME="E18:85_5_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="sprect_2.html#E32:85"><i><font color="Green">E106</font></i></a>, <a class="ref" href="sprect_2.html#T39" target="_self">Th39</a>, <a class="ref" href="sprect_2.html#T44" target="_self">Th44</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_5_3"><b>suppose </b></a><b>that </b><a NAME="E1:85_5_3"/><i><font color="Green">E154</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">z</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">z</font></span>)</span>
 <b>and </b><br/><a NAME="E2:85_5_3"/><i><font color="Green">E155</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">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <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">z</font></span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">g</font> = <span class="p1"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i>, <a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <a class="ref" href="sprect_2.html#T31" target="_self">Th31</a>, <a class="txt" href="sprect_2.html#E43"><i><font color="Green">Lemma115</font></i></a>, <a class="txt" href="sprect_2.html#E22:85"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E4:85_5_3"/><i><font color="Green">E156</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</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="E5:85_5_3"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">M</font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <a class="txt" href="sprect_2.html#E3:85"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E6:85_5_3"/>
<span class="p1"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E7:85_5_3"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">g</font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T41" target="_self">Th41</a>, <a class="txt" href="sprect_2.html#E4:85"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E8:85_5_3"/>
<font color="Maroon">g</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">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T16">FINSEQ_5:16</a></i>;<br/><a NAME="E9:85_5_3"/><b>then </b><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</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">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E10:85_5_3"/><i><font color="Green">E160</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E11:85_5_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</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#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&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">g</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">g</font></span>)</span> = 
<font color="Maroon">M</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">M</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T44" target="_self">Th44</a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">z</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#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, </i>
<br/>.= 
 <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">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T22" target="_self">Th22</a>, <a class="ref" href="finseq_5.html#T41">FINSEQ_5:41</a></i>
;<br/><a NAME="E13:85_5_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</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">g</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">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E44"><i><font color="Green">Lemma116</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E14:85_5_3"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">g</font> <a href="sprect_2.html#R2" target="_self">is_a_h.c._for</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T42" target="_self">Th42</a>, <a class="ref" href="sprect_2.html#T43" target="_self">Th43</a>, </i>;<br/><a NAME="E15:85_5_3"/><i><font color="Green">E162</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">M</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">z</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</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="E16:85_5_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">M</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">z</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">h</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">M</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">z</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">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E31:85"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E17:85_5_3"/><b>then </b><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">M</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">z</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">h</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">M</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="ref" href="sprect_2.html#T37" target="_self">Th37</a>, <a class="ref" href="pscomp_1.html#T102">PSCOMP_1:102</a></i>;<br/><a NAME="E18:85_5_3"/>
<font color="Maroon">M</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">M</font>
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/><a NAME="E19:85_5_3"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T34" target="_self">Th34</a>, <a class="ref" href="sprect_2.html#T46" target="_self">Th46</a>, <a class="ref" href="sprect_2.html#T47" target="_self">Th47</a>, </i>;<br/><b>hence </b><a NAME="E20:85_5_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="sprect_2.html#E32:85"><i><font color="Green">E106</font></i></a>, <a class="ref" href="sprect_2.html#T39" target="_self">Th39</a>, <a class="ref" href="sprect_2.html#T45" target="_self">Th45</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_5_4"><b>suppose </b></a><b>that </b><a NAME="E1:85_5_4"/><i><font color="Green">E164</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">z</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">z</font></span>)</span>
 <b>and </b><br/><a NAME="E2:85_5_4"/><i><font color="Green">E165</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">z</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <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">z</font></span>)</span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">K</font> = <span class="p1"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font>;<br/><b>reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as   <a href="sppol_2.html#NM1">S-Sequence_in_R2</a> <b>by </b><i>, <a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T27" target="_self">Th27</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <a class="ref" href="sprect_2.html#T31" target="_self">Th31</a>, <a class="txt" href="sprect_2.html#E43"><i><font color="Green">Lemma115</font></i></a>, <a class="txt" href="sprect_2.html#E44"><i><font color="Green">Lemma116</font></i></a>, <a class="txt" href="sprect_2.html#E24:85"><i><font color="Green">E73</font></i></a></i>;<br/><a NAME="E4:85_5_4"/><i><font color="Green">E166</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</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="E5:85_5_4"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">M</font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <a class="txt" href="sprect_2.html#E3:85"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E6:85_5_4"/>
<span class="p1"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E7:85_5_4"/><b>then </b><i><font color="Green">E168</font></i>: 
<span class="p1"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T41" target="_self">Th41</a>, <a class="txt" href="sprect_2.html#E4:85"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E8:85_5_4"/>
<span class="p1"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:85_5_4"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">g</font> <a href="sprect_2.html#R1" target="_self">is_in_the_area_of</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T42" target="_self">Th42</a>, <a class="txt" href="sprect_2.html#E4:85"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E10:85_5_4"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><b>then </b><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
 <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">z</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T16">FINSEQ_5:16</a></i>
;<br/><a NAME="E12:85_5_4"/><b>then </b><i><font color="Green">E170</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</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">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></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#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&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> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></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="E14:85_5_4"/><b>then </b>
<font color="Maroon">g</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">g</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">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><a NAME="E15:85_5_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</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">g</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">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E16:85_5_4"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">g</font> <a href="sprect_2.html#R2" target="_self">is_a_h.c._for</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T43" target="_self">Th43</a>, <a class="ref" href="sprect_2.html#T44" target="_self">Th44</a>, </i>;<br/><a NAME="E17:85_5_4"/><i><font color="Green">E172</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">M</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">z</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</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="E18:85_5_4"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">M</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">z</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">h</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">M</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">z</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">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E31:85"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E19:85_5_4"/><b>then </b><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">M</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">z</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">h</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">M</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="ref" href="sprect_2.html#T37" target="_self">Th37</a>, <a class="ref" href="pscomp_1.html#T102">PSCOMP_1:102</a></i>;<br/><a NAME="E20:85_5_4"/>
<font color="Maroon">M</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">M</font>
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/><a NAME="E21:85_5_4"/><b>then </b><i><font color="Green">E174</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T34" target="_self">Th34</a>, <a class="ref" href="sprect_2.html#T46" target="_self">Th46</a>, <a class="ref" href="sprect_2.html#T47" target="_self">Th47</a>, </i>;<br/><a NAME="E22:85_5_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</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#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E23:85_5_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E24:85_5_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E25:85_5_4"/><b>then </b><i><font color="Green">E175</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/><a NAME="E26:85_5_4"/><i><font color="Green">E176</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p0">(<span class="default"><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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></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">z</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="E27:85_5_4"/><i><font color="Green">E177</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E28:85_5_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</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#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&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><i><font color="Green">E178</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p5">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></span>)</span> = 
<font color="Maroon">M</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">M</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T51" target="_self">Th51</a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">z</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#K30">N-max</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="finseq_4.html#K5">..</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T24" target="_self">Th24</a>, <a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, </i>
<br/>.= 
 <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">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T22" target="_self">Th22</a>, <a class="ref" href="finseq_5.html#T41">FINSEQ_5:41</a></i>
;<br/><a NAME="E30:85_5_4"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p0">(<span class="default"><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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></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">z</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">h</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"><span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p0">(<span class="default"><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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></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">z</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">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E31:85"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E31:85_5_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p0">(<span class="default"><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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></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">z</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">h</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"><span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p0">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p0">(<span class="default"><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">z</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">M</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="sprect_2.html#T52" target="_self">Th52</a>, <a class="ref" href="pscomp_1.html#T102">PSCOMP_1:102</a></i>;<br/><a NAME="E32:85_5_4"/><b>then </b>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="sprect_2.html#T48" target="_self">Th48</a>, <a class="ref" href="sprect_2.html#T49" target="_self">Th49</a>, <a class="ref" href="sprect_2.html#T50" target="_self">Th50</a>, </i>;<br/><b>hence </b><a NAME="E33:85_5_4"/>
contradiction
 <b>by </b><i>, <a class="txt" href="sprect_2.html#E32:85"><i><font color="Green">E106</font></i></a>, <a class="ref" href="sprect_2.html#T39" target="_self">Th39</a>, <a class="ref" href="sprect_2.html#T45" target="_self">Th45</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
