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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<a NAME="E1:16_1_1"/><i><font color="Green">E15</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"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3 &amp;  <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"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></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>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E2:16_1_1"/><b>then </b><i><font color="Green">E16</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"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></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"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></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"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K2">*&gt;</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E4:16_1_1"/><i><font color="Green">E17</font></i>: 
1 <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"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E18</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E6:16_1_1"/><i><font color="Green">E19</font></i>: 
2 <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"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E20</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E8:16_1_1"/><i><font color="Green">E21</font></i>: 
 <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"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E9:16_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 2 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E4:16_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="goboard5.html#E6:16_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E10:16_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 2
 
<b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/>
<b>hence </b><a NAME="E11:16_1_1"/>
not <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i><a class="txt" href="goboard5.html#E4:16_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="goboard5.html#E6:16_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goboard5.html#E8:16_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="goboard1.html#D2">GOBOARD1:def 2</a></i>;<br/>

<a NAME="E12:16_1_1"/>
3 <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"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E22</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 3
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E14:16_1_1"/>
1 <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"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E23</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
;<br/>
<a NAME="E16:16_1_1"/>
2 <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"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E24</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
;<br/>
<a NAME="E18:16_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2
 
;<br/>
<a NAME="E19:16_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,1 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E20:16_1_1"/>
2 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 3
 
;<br/>
<a NAME="E21:16_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,2 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E22:16_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,3 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E23:16_1_1"/>
4 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 5
 
;<br/>
<a NAME="E24:16_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,4 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E25:16_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="topreal1.html#V1">special</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a><br/>

<b>assume </b><a NAME="E1:16_1_1_6"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:16_1_1_6"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:16_1_1_6"/><b>then </b><i><font color="Green">E29</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>assume </b><a NAME="E4:16_1_1_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:16_1_1_6"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E3:16_1_1_6"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_6_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1_6"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_6_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K21">`1</a>  = 
0
<b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:16_1_1_6_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></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>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></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>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_6_1_1"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_6_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K22">`2</a>  = 
1
<b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:16_1_1_6_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></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>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></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>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_6_1_2"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_6_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_6_1_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K21">`1</a>  = 
1
<b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:16_1_1_6_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></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>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></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>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_6_1_3"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_6_1_4"><b>suppose </b></a><a NAME="E1:16_1_1_6_1_4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K22">`2</a>  = 
0
<b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:16_1_1_6_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></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>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></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>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_6_1_4"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E26:16_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="topreal1.html#V2">unfolded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:16_1_1_7"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:16_1_1_7"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:16_1_1_7"/><b>then </b><i><font color="Green">E29</font></i>: 
( 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>assume </b><a NAME="E4:16_1_1_7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:16_1_1_7"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E3:16_1_1_7"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_7_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1_7"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_7_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_7_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard5.html#E21:16_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="topreal1.html#T21">TOPREAL1:21</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_7_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_7_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goboard5.html#E22:16_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="topreal1.html#T24">TOPREAL1:24</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_7_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_7_1_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_7_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard5.html#E24:16_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="topreal1.html#T22">TOPREAL1:22</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E27:16_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a><br/>

<b>thus </b><a NAME="E28:16_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="goboard5.html#V1" target="_self">s.c.c.</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard5.html#D4" target="_self">GOBOARD5:def 4</a><br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:16_1_1_8"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E2:16_1_1_8"/><i><font color="Green">E30</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> ) or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:16_1_1_8"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:16_1_1_8"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:16_1_1_8"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E2:16_1_1_8"><i><font color="Green">E30</font></i></a>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/>
<a NAME="E6:16_1_1_8"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E7:16_1_1_8"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_8"><i><font color="Green">E29</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:16_1_1_8"/><i><font color="Green">E36</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 2 implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 2 implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 2 implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 
;<br/>
<a NAME="E9:16_1_1_8"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard5.html#E6:16_1_1_8"><i><font color="Green">E34</font></i></a></i>;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_8_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_8"><i><font color="Green">E29</font></i></a>, <a class="txt" href="goboard5.html#E3:16_1_1_8"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_8"><i><font color="Green">E32</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_8"><i><font color="Green">E33</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_8_1_1_1"><b>case </b></a><a NAME="E1:16_1_1_8_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_8_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_8"><i><font color="Green">E29</font></i></a>, <a class="txt" href="goboard5.html#E6:16_1_1_8"><i><font color="Green">E34</font></i></a>, <a class="txt" href="goboard5.html#E8:16_1_1_8"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_8_1_1_2"><b>case </b></a><a NAME="E1:16_1_1_8_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_8_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_8"><i><font color="Green">E29</font></i></a>, <a class="txt" href="goboard5.html#E6:16_1_1_8"><i><font color="Green">E34</font></i></a>, <a class="txt" href="goboard5.html#E8:16_1_1_8"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cqc_the1.html#T4">CQC_THE1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_8_1_1_3"><b>case </b></a><a NAME="E1:16_1_1_8_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_8_1_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E2:16_1_1_8"><i><font color="Green">E30</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1_8"><i><font color="Green">E35</font></i></a>, <a class="txt" href="goboard5.html#E8:16_1_1_8"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard5.html#E9:16_1_1_8"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cqc_the1.html#T5">CQC_THE1:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_8_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 ) )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E10:16_1_1_8"><i><font color="Green">E38</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_8_2_1"><b>suppose </b></a><a NAME="E1:16_1_1_8_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:16_1_1_8_2_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_8_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_8_2_2"><b>suppose </b></a><a NAME="E1:16_1_1_8_2_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_8_2_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E19:16_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="goboard5.html#E22:16_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="topreal1.html#T26">TOPREAL1:26</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_8_2_3"><b>suppose </b></a><a NAME="E1:16_1_1_8_2_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 4 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_8_2_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E21:16_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="goboard5.html#E24:16_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="topreal1.html#T25">TOPREAL1:25</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3">*&gt;</a></span>;<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">1,0</span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,0,1</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">1,0</span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E30:16_1_1"/><i><font color="Green">E29</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">1,0</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>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E31:16_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E32:16_1_1"/>
1 <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">0,0,1</span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E31</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E34:16_1_1"/>
2 <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">0,0,1</span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E32</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E36:16_1_1"/>
3 <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">0,0,1</span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E33</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E38:16_1_1"/>
1 <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">1,0</span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">1,0</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="goboard5.html#E30:16_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E40:16_1_1"/>
2 <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">1,0</span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E35</font></i>: <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">1,0</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="goboard5.html#E30:16_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_14"><b>now </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="E1:16_1_1_14"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:16_1_1_14"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &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="goboard5.html#E31:16_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_14_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="goboard5.html#E2:16_1_1_14"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_14_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_14_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <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>3</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="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard5.html#E33:16_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_14_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_14_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <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>3</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="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard5.html#E35:16_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_14_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_14_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <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>3</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="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goboard5.html#E37:16_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_14_1_4"><b>suppose </b></a><a NAME="E1:16_1_1_14_1_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_4"/>
<font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <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>3</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="goboard5.html#E15:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard5.html#E39:16_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_14_1_5"><b>suppose </b></a><a NAME="E1:16_1_1_14_1_5"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_5"/>
<font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#K1">.</a> <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>3</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="goboard5.html#E17:16_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="goboard5.html#E41:16_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E43:16_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E31:16_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/>
<i><font color="Green">E37</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3">*&gt;</a></span> = 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default">0,0,1</span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T148">FINSEQ_2:148</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70">ENUMSET1:70</a></i>
;<br/>
<a NAME="E45:16_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">1,0</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">0,1</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">E38</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="txt" href="goboard5.html#E44:16_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>

;<br/>
<a NAME="E47:16_1_1"/><b>then </b><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">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<i><font color="Green">E40</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E46:16_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>
;<br/>
<a NAME="E49:16_1_1"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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="16_1_1_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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/>

<a NAME="E1:16_1_1_18"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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="E2:16_1_1_18"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:16_1_1_18"/>
( <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">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <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">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> )
 ;<br/>

<a NAME="E4:16_1_1_18"/><b>then </b><i><font color="Green">E42</font></i>: 
( ( <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="goboard5.html#E2:16_1_1_18"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>assume </b><a NAME="E5:16_1_1_18"/>
<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="E6:16_1_1_18"/><b>then </b>
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E4:16_1_1_18"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_1_18"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E50:16_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <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>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E43:16_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="goboard5.html#E47:16_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="goboard5.html#E48:16_1_1"><i><font color="Green">E40</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#K3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,1,1</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">0,0</span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E52:16_1_1"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,0</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>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E53:16_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E54:16_1_1"/>
1 <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">0,1,1</span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E44</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E56:16_1_1"/>
2 <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">0,1,1</span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E45</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E58:16_1_1"/>
3 <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">0,1,1</span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E46</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E60:16_1_1"/>
1 <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">0,0</span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E47</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,0</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="goboard5.html#E52:16_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E62:16_1_1"/>
2 <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">0,0</span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E48</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,0</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="goboard5.html#E52:16_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_24"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:16_1_1_24"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:16_1_1_24"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 5 )
 <b>by </b><i><a class="txt" href="goboard5.html#E53:16_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_24_1"/>
( <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 or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1_24"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_24_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_24_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard5.html#E55:16_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_24_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_24_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard5.html#E57:16_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_24_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_24_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goboard5.html#E59:16_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_24_1_4"><b>suppose </b></a><a NAME="E1:16_1_1_24_1_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard5.html#E61:16_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_24_1_5"><b>suppose </b></a><a NAME="E1:16_1_1_24_1_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="goboard5.html#E63:16_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E65:16_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E53:16_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/>
<i><font color="Green">E50</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3">*&gt;</a></span> = 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default">0,1,1</span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T148">FINSEQ_2:148</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default">1,1,0</span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T100">ENUMSET1:100</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70">ENUMSET1:70</a></i>
;<br/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,0</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/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>
;<br/>
<b>then </b><i><font color="Green">E51</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="goboard5.html#E66:16_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default">0,0,1</span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T42">ENUMSET1:42</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70">ENUMSET1:70</a></i>
;<br/>
<a NAME="E69:16_1_1"/><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">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<i><font color="Green">E53</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E68:16_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>
;<br/>
<a NAME="E71:16_1_1"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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="16_1_1_29"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</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/>

<a NAME="E1:16_1_1_29"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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="E2:16_1_1_29"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</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_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:16_1_1_29"/>
( <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">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">c<sub>11</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">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> )
 ;<br/>

<a NAME="E4:16_1_1_29"/><b>then </b><i><font color="Green">E55</font></i>: 
( ( <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 ) &amp; ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1_29"><i><font color="Green">E54</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>assume </b><a NAME="E5:16_1_1_29"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E6:16_1_1_29"/><b>then </b>
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E4:16_1_1_29"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E7:16_1_1_29"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>10</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">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E72:16_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <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>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E65:16_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="goboard5.html#E69:16_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="goboard5.html#E70:16_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="goboard2.html#D2">GOBOARD2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span>;<br/>
<i><font color="Green">E55</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></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>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<i><font color="Green">E56</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></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>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E75:16_1_1"/>
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"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></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>3</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>3</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="16_1_1_32"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_1_1_32"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/>

<a NAME="E2:16_1_1_32"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_2.html#T3">MATRIX_2:3</a></i>;<br/>
<a NAME="E3:16_1_1_32"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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="mcart_1.html#T25">MCART_1:25</a></i>;<br/>
<a NAME="E4:16_1_1_32"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_32_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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>11</sub></font>,<font color="Maroon">c<sub>12</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>11</sub></font>,<font color="Maroon">c<sub>12</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>11</sub></font>,<font color="Maroon">c<sub>12</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="goboard5.html#E3:16_1_1_32"><i><font color="Green">E57</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="16_1_1_32_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_32_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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="E2:16_1_1_32_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>12</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><a NAME="E3:16_1_1_32_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>11</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>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_32_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_32_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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="E2:16_1_1_32_1_2"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>12</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><a NAME="E3:16_1_1_32_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>11</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>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_32_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_32_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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="E2:16_1_1_32_1_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>12</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><a NAME="E3:16_1_1_32_1_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>11</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>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_32_1_4"><b>suppose </b></a><a NAME="E1:16_1_1_32_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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="E2:16_1_1_32_1_4"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>12</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><a NAME="E3:16_1_1_32_1_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</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>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>11</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>3</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then </b><i><font color="Green">E57</font></i>: <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">0,1</span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,0</span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default">1,1</span><a href="euclid.html#K23">]|</a></span> = 
 <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>3</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>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E73:16_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="goboard5.html#E74:16_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="goboard2.html#D1">GOBOARD2:def 1</a></i>
<br/>.= 
 <a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="goboard2.html#D3">GOBOARD2:def 3</a></i>
;<br/>
<a NAME="E77:16_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1
 
<b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/>
<a NAME="E78:16_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,2
 
<b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/>
<a NAME="E79:16_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 2,2
 
<b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/>
<a NAME="E80:16_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 2,1
 
<b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/>
<b>thus </b><a NAME="E81:16_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a>,<a class="ref" href="goboard5.html#D5" target="_self">GOBOARD5:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_34"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:16_1_1_34"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:16_1_1_34"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 5 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:16_1_1_34"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_34"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_34_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1_34"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard5.html#E3:16_1_1_34"><i><font color="Green">E64</font></i></a>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_34_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_34_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:16_1_1_34_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1
 <b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_34_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_34_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>take </b>
2
;<br/><b>thus </b><a NAME="E2:16_1_1_34_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,2
 <b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_34_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_34_1_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>take </b>
2
;<br/><b>take </b>
2
;<br/><b>thus </b><a NAME="E2:16_1_1_34_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 2,2
 <b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_34_1_4"><b>suppose </b></a><a NAME="E1:16_1_1_34_1_4"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>take </b>
2
;<br/><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:16_1_1_34_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 2,1
 <b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_4"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_34_1_5"><b>suppose </b></a><a NAME="E1:16_1_1_34_1_5"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:16_1_1_34_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1
 <b>by </b><i><a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_5"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E82:16_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E83:16_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E84:16_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E85:16_1_1"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E86:16_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E87:16_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E88:16_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="goboard5.html#E82:16_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E89:16_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="goboard5.html#E83:16_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E90:16_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_35"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_35"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E88:16_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="goboard5.html#E90:16_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="cqc_the1.html#T5">CQC_THE1:5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_35_1"><b>suppose </b></a><a NAME="E1:16_1_1_35_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_1_35_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E3:16_1_1_35_1"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E77:16_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard5.html#E84:16_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard5.html#E86:16_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E4:16_1_1_35_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E5:16_1_1_35_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E78:16_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="goboard5.html#E85:16_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="goboard5.html#E87:16_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E6:16_1_1_35_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><b>hence </b><a NAME="E7:16_1_1_35_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_35_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_35_2"><b>suppose </b></a><a NAME="E1:16_1_1_35_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:16_1_1_35_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E3:16_1_1_35_2"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="txt" href="goboard5.html#E78:16_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="goboard5.html#E84:16_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard5.html#E86:16_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E4:16_1_1_35_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E5:16_1_1_35_2"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="txt" href="goboard5.html#E79:16_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard5.html#E85:16_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="goboard5.html#E87:16_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E6:16_1_1_35_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><b>hence </b><a NAME="E7:16_1_1_35_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_35_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_35_3"><b>suppose </b></a><a NAME="E1:16_1_1_35_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:16_1_1_35_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E3:16_1_1_35_3"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E79:16_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="goboard5.html#E84:16_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard5.html#E86:16_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E4:16_1_1_35_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E5:16_1_1_35_3"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E80:16_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard5.html#E85:16_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="goboard5.html#E87:16_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E6:16_1_1_35_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><b>hence </b><a NAME="E7:16_1_1_35_3"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_35_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_35_4"><b>suppose </b></a><a NAME="E1:16_1_1_35_4"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:16_1_1_35_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E3:16_1_1_35_4"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E80:16_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard5.html#E84:16_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="goboard5.html#E86:16_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_4"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E4:16_1_1_35_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E5:16_1_1_35_4"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="goboard5.html#E77:16_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard5.html#E85:16_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="goboard5.html#E87:16_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_4"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E6:16_1_1_35_4"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_4"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><b>hence </b><a NAME="E7:16_1_1_35_4"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_4"><i><font color="Green">E71</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_35_4"><i><font color="Green">E72</font></i></a>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
