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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="sprect_1.html#V1">rectangular</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>

<a NAME="E1:44"/><i><font color="Green">E26</font></i>: 
 <a href="sprect_1.html#K1">SpStSeq</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_4.html#K2">&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">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="sprect_1.html#D1">SPRECT_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="sprect_1.html#K1">SpStSeq</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:44"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="sprect_1.html#D2">SPRECT_1:def 2</a></i>;<br/>
<a NAME="E4:44"/><i><font color="Green">E28</font></i>: 
5 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T90">SPRECT_1:90</a></i>;<br/>
<i><font color="Green">E29</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_4.html#K2">&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">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E1:44"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
3 <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="finseq_4.html#K2">&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">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p4">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
3 <a href="nat_1.html#K1">+</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E6:44"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E4:44"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:44"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:44_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:44_2"/><b>then </b><i><font color="Green">E31</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</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> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green">E32</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
 <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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T91">SPRECT_1:91</a></i>
<br/>.= 
 <a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sprect_3.html#E3:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sprect_1.html#T83">SPRECT_1:83</a></i>
<br/>.= 
 <a href="pscomp_1.html#K20">NW-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E3:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sprect_1.html#T70">SPRECT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="sprect_1.html#T37">SPRECT_1:37</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E4:44"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E2:44_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_1"><b>suppose </b></a><a NAME="E1:44_2_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_2_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E3:44_2"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_2"><b>suppose </b></a><a NAME="E1:44_2_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T92">SPRECT_1:92</a></i>
<br/>.= 
 <a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sprect_3.html#E3:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sprect_1.html#T87">SPRECT_1:87</a></i>
<br/>.= 
 <a href="pscomp_1.html#K21">NE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E3:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sprect_1.html#T71">SPRECT_1:71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="sprect_3.html#E1:44_2_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="sprect_1.html#T38">SPRECT_1:38</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_3"><b>suppose </b></a><a NAME="E1:44_2_2_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="pscomp_1.html#K32">E-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T93">SPRECT_1:93</a></i>
<br/>.= 
 <a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sprect_3.html#E3:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sprect_1.html#T86">SPRECT_1:86</a></i>
<br/>.= 
 <a href="pscomp_1.html#K22">SE-corner</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E3:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sprect_1.html#T73">SPRECT_1:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="sprect_3.html#E1:44_2_2_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="sprect_1.html#T39">SPRECT_1:39</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_4"><b>suppose </b></a><a NAME="E1:44_2_2_4"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T94">SPRECT_1:94</a></i>
<br/>.= 
 <a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="sprect_3.html#E3:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sprect_1.html#T82">SPRECT_1:82</a></i>
<br/>.= 
 <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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E3:44"><i><font color="Green">E27</font></i></a>, <a class="ref" href="sprect_1.html#T72">SPRECT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="sprect_3.html#E1:44_2_2_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="sprect_1.html#T40">SPRECT_1:40</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_2_2_5"><b>suppose </b></a><a NAME="E1:44_2_2_5"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="sprect_3.html#E4:44"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="sprect_3.html#E5:44"><i><font color="Green">E29</font></i></a>, <a class="txt" href="sprect_3.html#E3:44_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="sprect_3.html#E1:44_2_2_5"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:44"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="sprect_1.html#K1">SpStSeq</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E6:44"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_5.html#T13">FINSEQ_5:13</a></i>;<br/>


</div>
