<?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#K27">W-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>set </b><font color="Maroon">i2</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-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">j</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">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="E1:96"/><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="E2:96"/><i><font color="Green">E26</font></i>: 
 <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#K28">W-max</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="E3:96"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-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#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-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/>

<a NAME="E4:96"/><i><font color="Green">E28</font></i>: 
 <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">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#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:96"/><i><font color="Green">E42</font></i>: 
 <a href="pscomp_1.html#K28">W-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="ref" href="sprect_2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E6:96"/><i><font color="Green">E43</font></i>: 
 <a href="pscomp_1.html#K27">W-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#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E7:96"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-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="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E8:96"/><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#K27">W-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#K27">W-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/>
<a NAME="E9:96"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-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#T29" target="_self">Th29</a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E10:96"/><b>then </b><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-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#K28">W-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/>
<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#K28">W-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#K28">W-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#D2" target="_self">Def2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K28">W-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#T29" target="_self">Th29</a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E12:96"/><i><font color="Green">E64</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="sprect_2.html#T26" target="_self">Th26</a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>
<i><font color="Green">E65</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#K27">W-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#K27">W-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#T31" target="_self">Th31</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-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#T30" target="_self">Th30</a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E14:96"/>
 <a href="pscomp_1.html#K28">W-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#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-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="txt" href="sprect_2.html#E12:96"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E15:96"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-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>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T27" target="_self">Th27</a>, <a class="ref" href="sprect_2.html#T33" target="_self">Th33</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E16:96"/><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="E17:96"/>
(  <a href="pscomp_1.html#K28">W-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="topreal1.html#K5">L~</a> <font color="Maroon">z</font> &amp; <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>  <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>, <a class="ref" href="sprect_1.html#T15">SPRECT_1:15</a></i>;<br/>
<a NAME="E18:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-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="xxreal_0.html#R1">&lt;=</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#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E19:96"/><b>then </b>
 <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#K27">W-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="txt" href="sprect_2.html#E11:96"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E20:96"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-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#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#T26" target="_self">Th26</a>, <a class="ref" href="sprect_2.html#T32" target="_self">Th32</a>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E21:96"/><i><font color="Green">E69</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="E22:96"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">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#T28" target="_self">Th28</a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E23:96"/><b>then </b><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">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#K32">E-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">E72</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#K32">E-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#K32">E-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="txt" href="sprect_2.html#E34:96"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_2.html#T28" target="_self">Th28</a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E25:96"/><b>then </b><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#K32">E-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#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="E26:96"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K28">W-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#K32">E-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="sprect_2.html#T26" target="_self">Th26</a>, , </i>;<br/>
<a NAME="E27:96"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="sprect_2.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E28:96"/>
 <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="topreal1.html#K5">L~</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T13">SPRECT_1:13</a></i>;<br/>
<a NAME="E29:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K30">N-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E30:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E31:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E32:96"/><b>then </b><i><font color="Green">E104</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">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="sprect_2.html#T26" target="_self">Th26</a>, <a class="txt" href="sprect_2.html#E35:96"><i><font color="Green">E106</font></i></a>, <a class="ref" href="sprect_2.html#D4" target="_self">Def4</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</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#K28">W-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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">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#D3" target="_self">Def3</a>, <a class="ref" href="sprect_2.html#T35" target="_self">Th35</a>, <a class="txt" href="sprect_2.html#E7:96"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E34:96"/><i><font color="Green">E105</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="E35:96"/><i><font color="Green">E106</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#D2" target="_self">Def2</a>, <a class="txt" href="sprect_2.html#E34:96"><i><font color="Green">E105</font></i></a>, <a class="txt" href="sprect_2.html#E3:96"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E36:96"/>
<font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K28">W-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#D2" target="_self">Def2</a>, <a class="ref" href="sprect_2.html#T33" target="_self">Th33</a>, <a class="txt" href="sprect_2.html#E34:96"><i><font color="Green">E105</font></i></a>, </i>;<br/>
<a NAME="E37:96"/><b>then </b><i><font color="Green">E133</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#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="E38:96"/><i><font color="Green">E134</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#D3" target="_self">Def3</a>, <a class="txt" href="sprect_2.html#E35:96"><i><font color="Green">E106</font></i></a>, <a class="ref" href="jordan4.html#T47">JORDAN4:47</a></i>;<br/>
<a NAME="E39:96"/>
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">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#D2" target="_self">Def2</a>, <a class="txt" href="sprect_2.html#E34:96"><i><font color="Green">E105</font></i></a>, </i>;<br/>
<a NAME="E40:96"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">h</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#T34" target="_self">Th34</a>, <a class="ref" href="sprect_2.html#T36" target="_self">Th36</a>, <a class="ref" href="sprect_2.html#T37" target="_self">Th37</a>, </i>;<br/>
<a NAME="E41:96"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">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> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-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="sprect_2.html#T26" target="_self">Th26</a>, <a class="ref" href="sprect_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E42:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-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> 1
 
<b>by </b><i><a class="txt" href="sprect_2.html#E35:96"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</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"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-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>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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="txt" href="sprect_2.html#E27:96"><i><font color="Green">E103</font></i></a>, <a class="txt" href="sprect_2.html#E8:96"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E44:96"/><i><font color="Green">E136</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#T32" target="_self">Th32</a>, , <a class="ref" href="sprect_2.html#T35" target="_self">Th35</a>, , <a class="ref" href="sprect_2.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E45:96"/><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#K27">W-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="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="sprect_2.html#T32" target="_self">Th32</a>, <a class="ref" href="jordan4.html#T20">JORDAN4:20</a></i>;<br/>
<a NAME="E46:96"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-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> <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="txt" href="sprect_2.html#E27:96"><i><font color="Green">E103</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E47:96"/><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#K27">W-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> <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="E48:96"/><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#K27">W-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> <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="E49:96"/><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#K27">W-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="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="E50:96"/><i><font color="Green">E139</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-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#T31" target="_self">Th31</a>, <a class="txt" href="sprect_2.html#E32:96"><i><font color="Green">E104</font></i></a>, </i>;<br/>
<a NAME="E51:96"/><i><font color="Green">E140</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#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#T31" target="_self">Th31</a>, <a class="txt" href="sprect_2.html#E32:96"><i><font color="Green">E104</font></i></a>, </i>;<br/>
<a NAME="E52:96"/><i><font color="Green">E141</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#T31" target="_self">Th31</a>, <a class="txt" href="sprect_2.html#E32:96"><i><font color="Green">E104</font></i></a>, <a class="txt" href="sprect_2.html#E3:96"><i><font color="Green">E27</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_4"/>
(  <a href="pscomp_1.html#K19">SW-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#K27">W-min</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#K19">SW-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#K27">W-min</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="96_4_1"><b>suppose </b></a><a NAME="E1:96_4_1"/>
 <a href="pscomp_1.html#K19">SW-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#K27">W-min</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="E2:96_4_1"/><b>then </b><i><font color="Green">E142</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#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#T43" target="_self">Th43</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E3:96_4_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#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="sprect_2.html#T44" target="_self">Th44</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E4:96_4_1"/><b>then </b>
<font color="Maroon">M</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#T45" target="_self">Th45</a>, <a class="ref" href="sprect_2.html#T46" target="_self">Th46</a>, </i>;<br/><b>hence </b><a NAME="E5:96_4_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="sprect_2.html#E44"><i><font color="Green">Lemma116</font></i></a>, <a class="ref" href="sprect_2.html#T39" target="_self">Th39</a>, <a class="ref" href="sprect_2.html#T41" target="_self">Th41</a>, <a class="ref" href="sprect_2.html#T42" target="_self">Th42</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_4_2"><b>suppose </b></a><a NAME="E1:96_4_2"/>
 <a href="pscomp_1.html#K19">SW-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#K27">W-min</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>then 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#K19">SW-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#T31" target="_self">Th31</a>, , <a class="txt" href="sprect_2.html#E32:96"><i><font color="Green">E104</font></i></a>, <a class="txt" href="sprect_2.html#E24:96"><i><font color="Green">E72</font></i></a></i>;<br/><a NAME="E3:96_4_2"/><i><font color="Green">E143</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="E4:96_4_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#K19">SW-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/>;<br/><a NAME="E5:96_4_2"/><b>then </b><i><font color="Green">E144</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#T45" target="_self">Th45</a>, <a class="txt" href="sprect_2.html#E4:96"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E6:96_4_2"/>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K19">SW-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="E7:96_4_2"/><b>then </b><i><font color="Green">E145</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#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="E8:96_4_2"/><i><font color="Green">E146</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="E9:96_4_2"/>
 <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#K19">SW-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="E10:96_4_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> <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#T49" target="_self">Th49</a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>;<br/><a NAME="E11:96_4_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#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="sprect_2.html#T44" target="_self">Th44</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E12:96_4_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">g</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#T47" target="_self">Th47</a>, <a class="ref" href="sprect_2.html#T48" target="_self">Th48</a>, </i>;<br/><a NAME="E13:96_4_2"/><i><font color="Green">E148</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#K19">SW-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="E14:96_4_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> 1</span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-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#K19">SW-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#E43"><i><font color="Green">Lemma115</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E15:96_4_2"/><b>then </b><i><font color="Green">E149</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#K19">SW-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#T43" target="_self">Th43</a>, <a class="ref" href="pscomp_1.html#T92">PSCOMP_1:92</a></i>;<br/><a NAME="E16:96_4_2"/>
<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="sprect_2.html#T41" target="_self">Th41</a>, <a class="ref" href="sprect_2.html#T42" target="_self">Th42</a>, <a class="ref" href="jordan3.html#T34">JORDAN3:34</a></i>;<br/><a NAME="E17:96_4_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#T39" target="_self">Th39</a>, <a class="ref" href="sprect_2.html#T51" target="_self">Th51</a>, <a class="ref" href="sprect_2.html#T52" target="_self">Th52</a>, </i>;<br/><b>hence </b><a NAME="E18:96_4_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="sprect_2.html#E44"><i><font color="Green">Lemma116</font></i></a>, <a class="ref" href="sprect_2.html#T46" target="_self">Th46</a>, <a class="ref" href="sprect_2.html#T50" target="_self">Th50</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
