<?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/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="goboard2.html#K2">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="goboard2.html#K2">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:46"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</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>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></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">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 NAME="E2:46"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a>  <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">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a>  <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 NAME="E3:46"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K33">S-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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a>  <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 NAME="E4:46"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4 <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">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">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 NAME="E5:46"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 5
 
<b>by </b><i><a class="ref" href="sprect_1.html#T90">SPRECT_1:90</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>;<br/>
<a NAME="E6:46"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E7:46"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<i><font color="Green">E35</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&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#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
2 <a href="nat_1.html#K1">+</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
 <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/>
<a NAME="E9:46"/>
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> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
 ;<br/>

<a NAME="E2:46_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 5 )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:46"><i><font color="Green">E32</font></i></a>, <a class="txt" href="sprect_3.html#E8:46"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:46_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 5 ) )
 
<b>by </b><i><a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E3:46_2"><i><font color="Green">E36</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_1"><b>suppose </b></a><a NAME="E1:46_2_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:46_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E6:46"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_2"><b>suppose </b></a><a NAME="E1:46_2_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:46_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E6:46"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_2_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_2_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_3"><b>suppose </b></a><a NAME="E1:46_2_1_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="sprect_3.html#E6:46"><i><font color="Green">E33</font></i></a>, <a class="txt" href="sprect_3.html#E7:46"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_2_1_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_4"><b>suppose </b></a><a NAME="E1:46_2_1_4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 2</span>)</span>

<br/>.= 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i><a class="txt" href="sprect_3.html#E6:46"><i><font color="Green">E33</font></i></a>, <a class="txt" href="sprect_3.html#E7:46"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_2_1_4"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1_5"><b>suppose </b></a><a NAME="E1:46_2_1_5"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 3</span>)</span>

<br/>.= 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 3
<b>by </b><i><a class="txt" href="sprect_3.html#E6:46"><i><font color="Green">E33</font></i></a>, <a class="txt" href="sprect_3.html#E7:46"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_2_1_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:46"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E8:46"><i><font color="Green">E35</font></i></a>, <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E11:46"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/>
<a NAME="E12:46"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="enumset1.html#K1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_3_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a>  )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E1:46_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_1_1"><b>suppose </b></a><a NAME="E1:46_3_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:46_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="sprect_3.html#E2:46"><i><font color="Green">E29</font></i></a>, <a class="txt" href="sprect_3.html#E3:46"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pscomp_1.html#T105">PSCOMP_1:105</a></i>;<br/><b>hence </b><a NAME="E3:46_3_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_1_2"><b>suppose </b></a><a NAME="E1:46_3_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:46_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:46"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E4:46"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pscomp_1.html#T85">PSCOMP_1:85</a></i>;<br/><b>hence </b><a NAME="E3:46_3_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_3_1_3"><b>suppose </b></a><a NAME="E1:46_3_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> 
 ;<br/><div class="add"><a NAME="E2:46_3_1_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="sprect_3.html#E5:46"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/><b>hence </b><a NAME="E3:46_3_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:46"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<b>then </b><i><font color="Green">E40</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="sprect_3.html#E12:46"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T148">FINSEQ_2:148</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E10:46"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/>
<a NAME="E15:46"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</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"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E2:46"><i><font color="Green">E29</font></i></a>, <a class="ref" href="sprect_2.html#T55">SPRECT_2:55</a></i>;<br/>
<i><font color="Green">E42</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E13:46"><i><font color="Green">E39</font></i></a>, <a class="txt" href="sprect_3.html#E14:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="sprect_3.html#E15:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>
;<br/>
<a NAME="E17:46"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> is <a href="goboard1.html#V1">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a><br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:46_6"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E2:46_6"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E3:46_6"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E4:46_6"/><i><font color="Green">E46</font></i>: 
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E5:46_6"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E11:46"><i><font color="Green">E37</font></i></a>, <a class="txt" href="sprect_3.html#E1:46_6"><i><font color="Green">E43</font></i></a>, <a class="txt" href="sprect_3.html#E2:46_6"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:46_6"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:46"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E2:46"><i><font color="Green">E29</font></i></a>, <a class="txt" href="sprect_3.html#E3:46_6"><i><font color="Green">E45</font></i></a>, <a class="txt" href="sprect_3.html#E4:46_6"><i><font color="Green">E46</font></i></a>, <a class="ref" href="sprect_2.html#T55">SPRECT_2:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:46"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="goboard2.html#K2">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</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#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E14:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="sprect_3.html#E16:46"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard2.html#D2">GOBOARD2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<a NAME="E19:46"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E20:46"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<i><font color="Green">E46</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </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#T35">FINSEQ_1:35</a></i>
<br/>.= 
<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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
2 <a href="nat_1.html#K1">+</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
 <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/>
<a NAME="E22:46"/>
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> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_8"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
 ;<br/>

<a NAME="E2:46_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 5 )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:46"><i><font color="Green">E32</font></i></a>, <a class="txt" href="sprect_3.html#E21:46"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:46_8"/><b>then </b><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 5 ) )
 
<b>by </b><i><a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_8_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E3:46_8"><i><font color="Green">E47</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_8_1_1"><b>suppose </b></a><a NAME="E1:46_8_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:46_8_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E20:46"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_8_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_8_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_8_1_2"><b>suppose </b></a><a NAME="E1:46_8_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:46_8_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E20:46"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_8_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_8_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_8_1_3"><b>suppose </b></a><a NAME="E1:46_8_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:46_8_1_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E20:46"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 3
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_8_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_8_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_8_1_4"><b>suppose </b></a><a NAME="E1:46_8_1_4"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="sprect_3.html#E19:46"><i><font color="Green">E44</font></i></a>, <a class="txt" href="sprect_3.html#E20:46"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_8_1_4"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_8_1_5"><b>suppose </b></a><a NAME="E1:46_8_1_5"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 3</span>)</span>

<br/>.= 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i><a class="txt" href="sprect_3.html#E19:46"><i><font color="Green">E44</font></i></a>, <a class="txt" href="sprect_3.html#E20:46"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T86">FINSEQ_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46_8_1_5"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E23:46"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font> <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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E21:46"><i><font color="Green">E46</font></i></a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E24:46"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T29">FINSEQ_3:29</a></i>;<br/>
<a NAME="E25:46"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5
 
<b>by </b><i><a class="txt" href="sprect_3.html#E5:46"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>
<a NAME="E26:46"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_9"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="enumset1.html#K1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_9_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a>  )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E1:46_9"><i><font color="Green">E51</font></i></a>, <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_9_1_1"><b>suppose </b></a><a NAME="E1:46_9_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_9_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E25:46"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_9_1_2"><b>suppose </b></a><a NAME="E1:46_9_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:46_9_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:46"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E2:46"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pscomp_1.html#T95">PSCOMP_1:95</a></i>;<br/><b>hence </b><a NAME="E3:46_9_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="sprect_3.html#E25:46"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_9_1_3"><b>suppose </b></a><a NAME="E1:46_9_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><div class="add"><a NAME="E2:46_9_1_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="sprect_3.html#E3:46"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sprect_3.html#E4:46"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pscomp_1.html#T115">PSCOMP_1:115</a></i>;<br/><b>hence </b><a NAME="E3:46_9_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E27:46"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<b>then </b><i><font color="Green">E52</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="txt" href="sprect_3.html#E26:46"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T148">FINSEQ_2:148</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>
<br/>.= 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E23:46"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/>
<a NAME="E29:46"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E4:46"><i><font color="Green">E31</font></i></a>, <a class="txt" href="sprect_3.html#E25:46"><i><font color="Green">E49</font></i></a>, <a class="ref" href="sprect_2.html#T61">SPRECT_2:61</a></i>;<br/>
<i><font color="Green">E54</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E27:46"><i><font color="Green">E51</font></i></a>, <a class="txt" href="sprect_3.html#E28:46"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sprect_3.html#E29:46"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>
;<br/>
<a NAME="E31:46"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> is <a href="goboard1.html#V1">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a><br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:46_12"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E2:46_12"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E3:46_12"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E4:46_12"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E5:46_12"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="sprect_3.html#E24:46"><i><font color="Green">E48</font></i></a>, <a class="txt" href="sprect_3.html#E1:46_12"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sprect_3.html#E2:46_12"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:46_12"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 5</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="sprect_3.html#E1:46"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E4:46"><i><font color="Green">E31</font></i></a>, <a class="txt" href="sprect_3.html#E25:46"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sprect_3.html#E3:46_12"><i><font color="Green">E57</font></i></a>, <a class="txt" href="sprect_3.html#E4:46_12"><i><font color="Green">E58</font></i></a>, <a class="ref" href="sprect_2.html#T61">SPRECT_2:61</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:46"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="goboard2.html#K2">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</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#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E25:46"><i><font color="Green">E49</font></i></a>, <a class="txt" href="sprect_3.html#E28:46"><i><font color="Green">E52</font></i></a>, <a class="txt" href="sprect_3.html#E30:46"><i><font color="Green">E54</font></i></a>, <a class="ref" href="goboard2.html#D2">GOBOARD2:def 2</a></i>;<br/>
<i><font color="Green">E56</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> = 
2
<b>by </b><i><a class="ref" href="matrix_2.html#T3">MATRIX_2:3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E18:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<i><font color="Green">E57</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> = 
2
<b>by </b><i><a class="ref" href="matrix_2.html#T3">MATRIX_2:3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="sprect_3.html#E32:46"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E35:46"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E18:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E36:46"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E18:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E37:46"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E32:46"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E38:46"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E32:46"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E39:46"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E4:46"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pscomp_1.html#T85">PSCOMP_1:85</a></i>;<br/>
<a NAME="E40:46"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E2:46"><i><font color="Green">E29</font></i></a>, <a class="txt" href="sprect_3.html#E3:46"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pscomp_1.html#T105">PSCOMP_1:105</a></i>;<br/>
<a NAME="E41:46"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E3:46"><i><font color="Green">E30</font></i></a>, <a class="txt" href="sprect_3.html#E4:46"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pscomp_1.html#T115">PSCOMP_1:115</a></i>;<br/>
<a NAME="E42:46"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="txt" href="sprect_3.html#E1:46"><i><font color="Green">E28</font></i></a>, <a class="txt" href="sprect_3.html#E2:46"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pscomp_1.html#T95">PSCOMP_1:95</a></i>;<br/>
<a NAME="E43:46"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:46_15"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:46_15"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,2</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="sprect_3.html#T12" target="_self">Th12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_15_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="txt" href="sprect_3.html#E2:46_15"><i><font color="Green">E66</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_15_1_1"><b>suppose </b></a><a NAME="E1:46_15_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:46_15_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="sprect_3.html#E35:46"><i><font color="Green">E58</font></i></a>, <a class="txt" href="sprect_3.html#E37:46"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sprect_3.html#E39:46"><i><font color="Green">E62</font></i></a>, <a class="txt" href="sprect_3.html#E2:46_15_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_15_1_2"><b>suppose </b></a><a NAME="E1:46_15_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:46_15_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="sprect_3.html#E35:46"><i><font color="Green">E58</font></i></a>, <a class="txt" href="sprect_3.html#E38:46"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sprect_3.html#E2:46_15_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_15_1_3"><b>suppose </b></a><a NAME="E1:46_15_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:46_15_1_3"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="sprect_3.html#E36:46"><i><font color="Green">E59</font></i></a>, <a class="txt" href="sprect_3.html#E37:46"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sprect_3.html#E40:46"><i><font color="Green">E63</font></i></a>, <a class="txt" href="sprect_3.html#E41:46"><i><font color="Green">E64</font></i></a>, <a class="txt" href="sprect_3.html#E2:46_15_1_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_15_1_4"><b>suppose </b></a><a NAME="E1:46_15_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:46_15_1_4"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="txt" href="sprect_3.html#E36:46"><i><font color="Green">E59</font></i></a>, <a class="txt" href="sprect_3.html#E38:46"><i><font color="Green">E61</font></i></a>, <a class="txt" href="sprect_3.html#E42:46"><i><font color="Green">E65</font></i></a>, <a class="txt" href="sprect_3.html#E2:46_15_1_4"><i><font color="Green">E67</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E44:46"/><b>then </b>
 <a href="goboard2.html#K1">GoB</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_3.html#E33:46"><i><font color="Green">E56</font></i></a>, <a class="txt" href="sprect_3.html#E34:46"><i><font color="Green">E57</font></i></a>, <a class="ref" href="goboard2.html#D1">GOBOARD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E45:46"/>
 <a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="matrix_2.html#K6">][</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="goboard2.html#D3">GOBOARD2:def 3</a></i>;<br/>


</div>
