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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E55</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:56"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:56"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:56"/><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> ;<br/>
<a NAME="E7:56"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E4:56"><i><font color="Green">E58</font></i></a>, <a class="txt" href="sprect_2.html#E5:56"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:56"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E3:56"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:56"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="sprect_2.html#E1:56"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sprect_2.html#E2:56"><i><font color="Green">E56</font></i></a>, <a class="ref" href="sprect_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E10:56"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E1:56"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sprect_2.html#E2:56"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:56"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E3:56"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:56"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="sprect_2.html#E4:56"><i><font color="Green">E58</font></i></a>, <a class="txt" href="sprect_2.html#E5:56"><i><font color="Green">E59</font></i></a>, <a class="ref" href="sprect_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E13:56"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:56_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:56_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:56_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E5:56_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>
<b>assume </b><a NAME="E6:56_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:56_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E9:56_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E10:56_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<a NAME="E11:56_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E5:56_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:56_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E3:56"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:56_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E9:56_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:56_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E5:56"><i><font color="Green">E59</font></i></a>, <a class="txt" href="sprect_2.html#E10:56_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:56_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="sprect_2.html#E10:56_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:56_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="sprect_2.html#E6:56"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sprect_2.html#E4:56_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E17:56_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="sprect_2.html#E3:56_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="sprect_2.html#E8:56_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="sprect_2.html#E13:56_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="sprect_2.html#E14:56_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:56"/>
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sprect_2.html#E9:56"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sprect_2.html#E12:56"><i><font color="Green">E62</font></i></a>, <a class="ref" href="sprect_2.html#T4" target="_self">Th4</a></i>;<br/>


</div>
