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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="sprect_2.html#V1">clockwise_oriented</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:38"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E4:38"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E3:38"><i><font color="Green">E25</font></i></a>, <a class="ref" href="sprect_2.html#T34">SPRECT_2:34</a></i>;<br/>
<a NAME="E5:38"/><i><font color="Green">E27</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd11.html#T34">GOBRD11:34</a></i>;<br/>
<a NAME="E6:38"/><i><font color="Green">E28</font></i>: 
4 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a></i>;<br/>
<a NAME="E7:38"/><b>then </b><i><font color="Green">E29</font></i>: 
1 <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">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:38"/><b>then </b><i><font color="Green">E30</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:38"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:38"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="goboard2.html#T25">GOBOARD2:25</a></i>;<br/>
<a NAME="E12:38"/><i><font color="Green">E33</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E6:38"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:38"/><b>then </b><i><font color="Green">E34</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:38"/><i><font color="Green">E35</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:38"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:38"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:38"><i><font color="Green">E27</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/>
<a NAME="E16:38"/><i><font color="Green">E37</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E10:38"><i><font color="Green">E31</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E17:38"/><i><font color="Green">E38</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E10:38"><i><font color="Green">E31</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="sprect_3.html#E17:38"><i><font color="Green">E38</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="jordan5d.html#T42">JORDAN5D:42</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E4:38"><i><font color="Green">E26</font></i></a>, <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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E4:38_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E11:38"><i><font color="Green">E32</font></i></a>, <a class="txt" href="sprect_3.html#E16:38"><i><font color="Green">E37</font></i></a>, <a class="txt" href="sprect_3.html#E17:38"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sprect_3.html#E1:38_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="sprect_3.html#E2:38_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:38"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E16:38"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="sprect_3.html#E2:38"><i><font color="Green">E24</font></i></a>, <a class="txt" href="sprect_3.html#E8:38"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sprect_3.html#E10:38"><i><font color="Green">E31</font></i></a>, <a class="txt" href="sprect_3.html#E11:38"><i><font color="Green">E32</font></i></a>, <a class="txt" href="sprect_3.html#E13:38"><i><font color="Green">E34</font></i></a>, <a class="txt" href="sprect_3.html#E15:38"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sprect_3.html#E19:38"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard5.html#T13">GOBOARD5:13</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:38_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 2</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#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="sprect_3.html#E2:38"><i><font color="Green">E24</font></i></a>, <a class="txt" href="sprect_3.html#E3:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="sprect_3.html#E11:38"><i><font color="Green">E32</font></i></a>, <a class="txt" href="sprect_3.html#E14:38"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E16:38"><i><font color="Green">E37</font></i></a>, <a class="txt" href="sprect_3.html#E17:38"><i><font color="Green">E38</font></i></a>, <a class="txt" href="sprect_3.html#E19:38"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>hence </b><a NAME="E3:38_3"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E4:38"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pscomp_1.html#T98">PSCOMP_1:98</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E22:38"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E11:38"><i><font color="Green">E32</font></i></a>, <a class="txt" href="sprect_3.html#E19:38"><i><font color="Green">E39</font></i></a>, <a class="txt" href="sprect_3.html#E20:38"><i><font color="Green">E40</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/>

<a NAME="E23:38"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>
<a NAME="E24:38"/><i><font color="Green">E43</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">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E12:38"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E25:38"/><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E7:38"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<a NAME="E26:38"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E28:38"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E29:38"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="goboard2.html#T25">GOBOARD2:25</a></i>;<br/>
<a NAME="E30:38"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E28:38"><i><font color="Green">E46</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E31:38"/><i><font color="Green">E49</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E28:38"><i><font color="Green">E46</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E32:38"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E12:38"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E33:38"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:38"><i><font color="Green">E24</font></i></a>, <a class="txt" href="sprect_3.html#E15:38"><i><font color="Green">E36</font></i></a>, <a class="txt" href="sprect_3.html#E23:38"><i><font color="Green">E42</font></i></a>, <a class="txt" href="sprect_3.html#E24:38"><i><font color="Green">E43</font></i></a>, <a class="txt" href="sprect_3.html#E26:38"><i><font color="Green">E45</font></i></a>, <a class="txt" href="sprect_3.html#E28:38"><i><font color="Green">E46</font></i></a>, <a class="txt" href="sprect_3.html#E29:38"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard5.html#T13">GOBOARD5:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_4"/>
( (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ) or (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E33:38"><i><font color="Green">E50</font></i></a>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1"><b>suppose </b></a><b>that </b><a NAME="E1:38_4_1"/><i><font color="Green">E51</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:38_4_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><i><font color="Green">E53</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="sprect_3.html#E29:38"><i><font color="Green">E47</font></i></a>, <a class="txt" href="sprect_3.html#E30:38"><i><font color="Green">E48</font></i></a>, <a class="txt" href="sprect_3.html#E2:38_4_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:38"><i><font color="Green">E24</font></i></a>, <a class="txt" href="sprect_3.html#E3:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="sprect_3.html#E5:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="sprect_3.html#E14:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</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">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><a NAME="E4:38_4_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E7:38"><i><font color="Green">E29</font></i></a>, <a class="txt" href="sprect_3.html#E26:38"><i><font color="Green">E45</font></i></a>, <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a></i>;<br/><a NAME="E5:38_4_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E3:38_4_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="sprect_2.html#T14">SPRECT_2:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_4_1_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E1:38_4_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_2_1_1"><b>suppose </b></a><a NAME="E1:38_4_1_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:38_4_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</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#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="sprect_3.html#E2:38"><i><font color="Green">E24</font></i></a>, <a class="txt" href="sprect_3.html#E3:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="sprect_3.html#E5:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="sprect_3.html#E14:38"><i><font color="Green">E35</font></i></a>, <a class="txt" href="sprect_3.html#E29:38"><i><font color="Green">E47</font></i></a>, <a class="txt" href="sprect_3.html#E30:38"><i><font color="Green">E48</font></i></a>, <a class="txt" href="sprect_3.html#E2:38_4_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>hence </b><a NAME="E3:38_4_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E5:38_4_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="pscomp_1.html#T98">PSCOMP_1:98</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_1_2_1_2"><b>suppose </b></a><a NAME="E1:38_4_1_2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:38_4_1_2_1_2"/><b>then </b>
2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="sprect_3.html#E22:38"><i><font color="Green">E41</font></i></a>, <a class="txt" href="sprect_3.html#E25:38"><i><font color="Green">E44</font></i></a>, <a class="txt" href="sprect_3.html#E29:38"><i><font color="Green">E47</font></i></a>, <a class="txt" href="sprect_3.html#E2:38_4_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="goboard7.html#T39">GOBOARD7:39</a></i>;<br/><a NAME="E3:38_4_1_2_1_2"/><b>then </b>
2 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="sprect_3.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E4:38_4_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="sprect_3.html#E6:38"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:38_4_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_4_2"><b>suppose </b></a><b>that </b><a NAME="E1:38_4_2"/><i><font color="Green">E51</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:38_4_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E3:38_4_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="sprect_3.html#E31:38"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sprect_3.html#E1:38_4_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><b>hence </b><a NAME="E4:38_4_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E29:38"><i><font color="Green">E47</font></i></a>, <a class="txt" href="sprect_3.html#E2:38_4_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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