<?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#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>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#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="E1:93"/><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="E2:93"/><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="E3:93"/><i><font color="Green">E27</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="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="E4:93"/><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="E5:93"/><i><font color="Green">E42</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="E6:93"/><i><font color="Green">E43</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="E7:93"/><i><font color="Green">E44</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="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E8:93"/><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#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/>
<a NAME="E9:93"/><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#T28" target="_self">Th28</a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E10:93"/><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#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#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#T28" target="_self">Th28</a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E12:93"/><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#D1" target="_self">Def1</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#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="ref" href="sprect_2.html#T29" target="_self">Th29</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="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E14:93"/>
( <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#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> &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="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="E15:93"/><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#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="topreal5.html#T23">TOPREAL5:23</a></i>;<br/>
<a NAME="E16:93"/><b>then </b><i><font color="Green">E66</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="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#T26" target="_self">Th26</a>, <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="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E17:93"/><b>then </b><i><font color="Green">E67</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="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="sprect_2.html#T32" target="_self">Th32</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:93"/><i><font color="Green">E68</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="E19:93"/><i><font color="Green">E69</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#T27" target="_self">Th27</a>, <a class="ref" href="finseq_4.html#T30">FINSEQ_4:30</a></i>;<br/>
<a NAME="E20:93"/><b>then </b><i><font color="Green">E70</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">E71</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="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#T27" target="_self">Th27</a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E22:93"/><i><font color="Green">E72</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> 1
 
<b>by </b><i><a class="ref" href="sprect_2.html#D1" target="_self">Def1</a>, <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="sprect_2.html#D2" target="_self">Def2</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<i><font color="Green">E73</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="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#T27" target="_self">Th27</a>, <a class="ref" href="finseq_4.html#T29">FINSEQ_4:29</a></i>
;<br/>
<a NAME="E24:93"/>
( <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> &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="E25:93"/><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#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="topreal5.html#T22">TOPREAL5:22</a></i>;<br/>
<a NAME="E26:93"/><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#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#D3" target="_self">Def3</a>, <a class="txt" href="sprect_2.html#E27:93"><i><font color="Green">E103</font></i></a>, <a class="txt" href="sprect_2.html#E33:93"><i><font color="Green">E106</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E27:93"/><i><font color="Green">E103</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="xxreal_0.html#NR3" target="_self">&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>
 
<b>by </b><i><a class="ref" href="sprect_2.html#D1" target="_self">Def1</a>, <a class="ref" href="sprect_2.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E28:93"/><b>then </b>
<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> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</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>,<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="ref" href="sprect_2.html#D4" target="_self">Def4</a>, <a class="txt" href="sprect_2.html#E8:93"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E30:93"/><i><font color="Green">E104</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="E31:93"/><i><font color="Green">E105</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="txt" href="sprect_2.html#E3:93"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E32:93"/>
<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="txt" href="sprect_2.html#E30:93"><i><font color="Green">E104</font></i></a>, </i>;<br/>
<a NAME="E33:93"/><b>then </b><i><font color="Green">E106</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="E34:93"/>
<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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E35:93"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</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">h</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#D3" target="_self">Def3</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E36:93"/><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>, , </i>;<br/>
<a NAME="E37:93"/><i><font color="Green">E134</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="E38:93"/>
 <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="E39:93"/><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="E40:93"/><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="sprect_2.html#T36" target="_self">Th36</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E41:93"/><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="E42:93"/><b>then </b>
<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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="sprect_2.html#T30" target="_self">Th30</a>, <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="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"><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#T32" target="_self">Th32</a>, , <a class="txt" href="sprect_2.html#E8:93"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E44:93"/><i><font color="Green">E135</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</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/>
<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#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#T29" target="_self">Th29</a>, <a class="ref" href="sprect_2.html#T31" target="_self">Th31</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="finseq_5.html#T41">FINSEQ_5:41</a></i>
;<br/>
<a NAME="E46:93"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">M</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#T29" target="_self">Th29</a>, <a class="ref" href="sprect_2.html#T31" target="_self">Th31</a>, <a class="ref" href="sprect_2.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E47:93"/><i><font color="Green">E138</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#T29" target="_self">Th29</a>, <a class="ref" href="sprect_2.html#T31" target="_self">Th31</a>, <a class="txt" href="sprect_2.html#E3:93"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E48:93"/><i><font color="Green">E139</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="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="E49:93"/>
<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="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="E50:93"/><b>then </b><i><font color="Green">E140</font></i>: 
<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/>
<a NAME="E51:93"/>
 <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#T26" target="_self">Th26</a>, <a class="txt" href="sprect_2.html#E27:93"><i><font color="Green">E103</font></i></a>, <a class="txt" href="sprect_2.html#E31:93"><i><font color="Green">E105</font></i></a>, <a class="ref" href="sprect_2.html#T34" target="_self">Th34</a>, <a class="ref" href="sprect_2.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E52:93"/>
contradiction
 <b>by </b><i><a class="ref" href="sprect_2.html#T35" target="_self">Th35</a>, , <a class="ref" href="sprect_2.html#T37" target="_self">Th37</a>, <a class="ref" href="sprect_2.html#T42" target="_self">Th42</a>, </i>;<br/>


</div>
