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

<b>set </b><font color="Maroon" title="c1">f1</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c2">f2</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<a NAME="E1:16_1_1"/><i><font color="Green" title="E15">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E2:16_1_1"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">f</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/>

<a NAME="E4:16_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E18">A4</font></i>: <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27" title="FINSEQ_4:th.27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E6:16_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E20">A6</font></i>: <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27" title="FINSEQ_4:th.27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E8:16_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E9:16_1_1"/><b>then </b>
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E4:16_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="goboard5.html#E6:16_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E10:16_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="sppol_2.html#T1" title="SPPOL_2:th.1">SPPOL_2:1</a></i>;<br/>
<b>hence </b><a NAME="E11:16_1_1"/>
not <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 <b>by </b><i><a class="txt" href="goboard5.html#E4:16_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="goboard5.html#E6:16_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="goboard5.html#E8:16_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></i>;<br/>

<a NAME="E12:16_1_1"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E22">A8</font></i>: <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 3
<b>by </b><i><a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T27" title="FINSEQ_4:th.27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E14:16_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E23">A9</font></i>: <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>
;<br/>
<a NAME="E16:16_1_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E24">A10</font></i>: <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>
;<br/>
<a NAME="E18:16_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
;<br/>
<a NAME="E19:16_1_1"/><b>then </b><i><font color="Green" title="E25">A11</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E20:16_1_1"/>
2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
;<br/>
<a NAME="E21:16_1_1"/><b>then </b><i><font color="Green" title="E26">A12</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E22:16_1_1"/><i><font color="Green" title="E27">A13</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E23:16_1_1"/>
4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 
;<br/>
<a NAME="E24:16_1_1"/><b>then </b><i><font color="Green" title="E28">A14</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E25:16_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">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" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a><br/>

<b>assume </b><a NAME="E1:16_1_1_6"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 ;<br/>

<a NAME="E2:16_1_1_6"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:16_1_1_6"/><b>then </b><i><font color="Green" title="E29">A15</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>assume </b><a NAME="E4:16_1_1_6"/>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<a NAME="E5:16_1_1_6"/><b>then </b><i><font color="Green" title="E30">A16</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E3:16_1_1_6"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="nat_1.html#T30" title="NAT_1:th.30">NAT_1:30</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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1_6"><i><font color="Green" title="E30">A16</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" title="E31">A17</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
0
<b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_6_1_1"><i><font color="Green" title="E31">A17</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" title="E31">A18</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  = 
1
<b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_6_1_2"><i><font color="Green" title="E31">A18</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" title="E31">A19</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 3</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
1
<b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_6_1_3"><i><font color="Green" title="E31">A19</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" title="E31">A20</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 4</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  = 
0
<b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_6_1_4"><i><font color="Green" title="E31">A20</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" title="c3">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">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" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:16_1_1_7"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 ;<br/>

<a NAME="E2:16_1_1_7"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:16_1_1_7"/><b>then </b><i><font color="Green" title="E29">A21</font></i>: 
( 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 &amp; 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 &amp; 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>assume </b><a NAME="E4:16_1_1_7"/>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<a NAME="E5:16_1_1_7"/><b>then </b><i><font color="Green" title="E30">A22</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E3:16_1_1_7"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="nat_1.html#T30" title="NAT_1:th.30">NAT_1:30</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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1_7"><i><font color="Green" title="E30">A22</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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="goboard5.html#E21:16_1_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="goboard5.html#E22:16_1_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="topreal1.html#T24" title="TOPREAL1:th.24">TOPREAL1:24</a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="goboard5.html#E24:16_1_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="topreal1.html#T22" title="TOPREAL1:th.22">TOPREAL1:22</a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">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" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a><br/>

<b>thus </b><a NAME="E28:16_1_1"/>
<font color="Maroon" title="c3">f</font> is <a href="goboard5.html#V1" title="GOBOARD5:attr.1">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" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard5.html#D4" target="_self" title="GOBOARD5:def.4">GOBOARD5:def 4</a><br/><b>let </b><font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:16_1_1_8"/><i><font color="Green" title="E29">A23</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font>
 <b>and </b><br/><a NAME="E2:16_1_1_8"/><i><font color="Green" title="E30">A24</font></i>: 
( ( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> ) or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 ;<br/>

<a NAME="E3:16_1_1_8"/><i><font color="Green" title="E31">A25</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E4:16_1_1_8"/><i><font color="Green" title="E32">A26</font></i>: 
( <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E2:16_1_1_8"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="nat_1.html#T30" title="NAT_1:th.30">NAT_1:30</a></i>;<br/>
<a NAME="E5:16_1_1_8"/><i><font color="Green" title="E33">A27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E6:16_1_1_8"/><b>then </b><i><font color="Green" title="E34">A28</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_8"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:16_1_1_8"/><i><font color="Green" title="E35">A29</font></i>: 
( ( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 implies <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 implies <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; ( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 implies <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 
;<br/>
<a NAME="E8:16_1_1_8"/><i><font color="Green" title="E36">A30</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1_8"><i><font color="Green" title="E33">A27</font></i></a></i>;<br/>
<div><i><font color="Green" title="E37">A31</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" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_8"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="goboard5.html#E3:16_1_1_8"><i><font color="Green" title="E31">A25</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_8"><i><font color="Green" title="E32">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_8_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_8"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_8"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1_8"><i><font color="Green" title="E35">A29</font></i></a>, <a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</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" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_8_1_1_2"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_8"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_8"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1_8"><i><font color="Green" title="E35">A29</font></i></a>, <a class="ref" href="nat_1.html#T28" title="NAT_1:th.28">NAT_1:28</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" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_8_1_1_3"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E2:16_1_1_8"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="goboard5.html#E6:16_1_1_8"><i><font color="Green" title="E34">A28</font></i></a>, <a class="txt" href="goboard5.html#E7:16_1_1_8"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="goboard5.html#E8:16_1_1_8"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E9:16_1_1_8"><i><font color="Green" title="E37">A31</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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:16_1_1_8_2_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">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" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E19:16_1_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="goboard5.html#E22:16_1_1"><i><font color="Green" title="E27">A13</font></i></a>, <a class="ref" href="topreal1.html#T26" title="TOPREAL1:th.26">TOPREAL1:26</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E21:16_1_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="goboard5.html#E24:16_1_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="topreal1.html#T25" title="TOPREAL1:th.25">TOPREAL1:25</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">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" title="c4">Xf1</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">Xf2</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Xf</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E30:16_1_1"/><i><font color="Green" title="E29">A32</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E31:16_1_1"/><b>then </b><i><font color="Green" title="E30">A33</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Xf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E32:16_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E31">A34</font></i>: <font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E34:16_1_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E32">A35</font></i>: <font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E36:16_1_1"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E33">A36</font></i>: <font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E38:16_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E34">A37</font></i>: <font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E40:16_1_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E35">A38</font></i>: <font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="goboard5.html#E30:16_1_1"><i><font color="Green" title="E29">A32</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">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" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_1_1_14"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">Xf</font>
 ;<br/><a NAME="E2:16_1_1_14"/><b>then </b><i><font color="Green" title="E36">A39</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5 )
 <b>by </b><i><a class="txt" href="goboard5.html#E31:16_1_1"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1_14"><i><font color="Green" title="E36">A39</font></i></a>, <a class="ref" href="nat_1.html#T30" title="NAT_1:th.30">NAT_1:30</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" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_1"/>
<font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="goboard5.html#E33:16_1_1"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_2"/>
<font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="goboard5.html#E35:16_1_1"><i><font color="Green" title="E32">A35</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_3"/>
<font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="goboard5.html#E37:16_1_1"><i><font color="Green" title="E33">A36</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_4"/>
<font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="goboard5.html#E39:16_1_1"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_14_1_5"/>
<font color="Maroon" title="c6">Xf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="goboard5.html#E41:16_1_1"><i><font color="Green" title="E35">A38</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="E36">A40</font></i>: 
<font color="Maroon" title="c6">Xf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E31:16_1_1"><i><font color="Green" title="E30">A33</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<i><font color="Green" title="E37">A41</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,0,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> = 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">0,0,1</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T148" title="FINSEQ_2:th.148">FINSEQ_2:148</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70" title="ENUMSET1:th.70">ENUMSET1:70</a></i>
;<br/>
<a NAME="E45:16_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a></i>;<br/>
<b>then </b><i><font color="Green" title="E38">A42</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">Xf</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="txt" href="goboard5.html#E44:16_1_1"><i><font color="Green" title="E37">A41</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>

;<br/>
<a NAME="E47:16_1_1"/><b>then </b><i><font color="Green" title="E39">A43</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">Xf</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a></i>;<br/>
<i><font color="Green" title="E40">A44</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> = 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">Xf</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E46:16_1_1"><i><font color="Green" title="E38">A42</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>
;<br/>
<a NAME="E49:16_1_1"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">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" title="c7">n</font>, <font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a><br/>

<a NAME="E1:16_1_1_18"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E2:16_1_1_18"/><b>then </b><i><font color="Green" title="E41">A45</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:16_1_1_18"/>
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )
 ;<br/>

<a NAME="E4:16_1_1_18"/><b>then </b><i><font color="Green" title="E42">A46</font></i>: 
( ( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1_18"><i><font color="Green" title="E41">A45</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>assume </b><a NAME="E5:16_1_1_18"/>
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font>
 ;<br/>

<a NAME="E6:16_1_1_18"/><b>then </b>
( <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E4:16_1_1_18"><i><font color="Green" title="E42">A46</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">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" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">m</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E50:16_1_1"/><b>then </b><i><font color="Green" title="E41">A47</font></i>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E43:16_1_1"><i><font color="Green" title="E36">A40</font></i></a>, <a class="txt" href="goboard5.html#E47:16_1_1"><i><font color="Green" title="E39">A43</font></i></a>, <a class="txt" href="goboard5.html#E48:16_1_1"><i><font color="Green" title="E40">A44</font></i></a>, <a class="ref" href="goboard2.html#D2" title="GOBOARD2:def.2">GOBOARD2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">Yf1</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c8">Yf2</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Yf</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E52:16_1_1"/><i><font color="Green" title="E42">A48</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E53:16_1_1"/><b>then </b><i><font color="Green" title="E43">A49</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">Yf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E54:16_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E44">A50</font></i>: <font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E56:16_1_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E45">A51</font></i>: <font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E58:16_1_1"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E46">A52</font></i>: <font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E60:16_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E47">A53</font></i>: <font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E62:16_1_1"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green" title="E48">A54</font></i>: <font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="goboard5.html#E52:16_1_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">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" title="c10">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_1_1_24"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">Yf</font>
 ;<br/><a NAME="E2:16_1_1_24"/><b>then </b><i><font color="Green" title="E49">A55</font></i>: 
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5 )
 <b>by </b><i><a class="txt" href="goboard5.html#E53:16_1_1"><i><font color="Green" title="E43">A49</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1_24"><i><font color="Green" title="E49">A55</font></i></a>, <a class="ref" href="nat_1.html#T30" title="NAT_1:th.30">NAT_1:30</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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_1"/>
<font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="goboard5.html#E55:16_1_1"><i><font color="Green" title="E44">A50</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_2"/>
<font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="goboard5.html#E57:16_1_1"><i><font color="Green" title="E45">A51</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_3"/>
<font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="goboard5.html#E59:16_1_1"><i><font color="Green" title="E46">A52</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_4"/>
<font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="goboard5.html#E61:16_1_1"><i><font color="Green" title="E47">A53</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_24_1_5"/>
<font color="Maroon" title="c9">Yf</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="goboard5.html#E63:16_1_1"><i><font color="Green" title="E48">A54</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">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" title="E49">A56</font></i>: 
<font color="Maroon" title="c9">Yf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E53:16_1_1"><i><font color="Green" title="E43">A49</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<i><font color="Green" title="E50">A57</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">0,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> = 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">0,1,1</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T148" title="FINSEQ_2:th.148">FINSEQ_2:148</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,1,0</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T100" title="ENUMSET1:th.100">ENUMSET1:100</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70" title="ENUMSET1:th.70">ENUMSET1:70</a></i>
;<br/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,0</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,0</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E51">A58</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">Yf</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="goboard5.html#E66:16_1_1"><i><font color="Green" title="E50">A57</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">0,0,1</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T42" title="ENUMSET1:th.42">ENUMSET1:42</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70" title="ENUMSET1:th.70">ENUMSET1:70</a></i>
;<br/>
<a NAME="E69:16_1_1"/><b>then </b><i><font color="Green" title="E52">A59</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">Yf</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147" title="FINSEQ_2:th.147">FINSEQ_2:147</a></i>;<br/>
<i><font color="Green" title="E53">A60</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> = 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">Yf</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E68:16_1_1"><i><font color="Green" title="E51">A58</font></i></a>, <a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>
;<br/>
<a NAME="E71:16_1_1"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">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" title="c10">n</font>, <font color="Maroon" title="c11">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a><br/>

<a NAME="E1:16_1_1_29"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E2:16_1_1_29"/><b>then </b><i><font color="Green" title="E54">A61</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:16_1_1_29"/>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )
 ;<br/>

<a NAME="E4:16_1_1_29"/><b>then </b><i><font color="Green" title="E55">A62</font></i>: 
( ( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1_29"><i><font color="Green" title="E54">A61</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>assume </b><a NAME="E5:16_1_1_29"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 ;<br/>

<a NAME="E6:16_1_1_29"/><b>then </b>
( <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E4:16_1_1_29"><i><font color="Green" title="E55">A62</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">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" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c11">m</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E72:16_1_1"/><b>then </b><i><font color="Green" title="E54">A63</font></i>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E65:16_1_1"><i><font color="Green" title="E49">A56</font></i></a>, <a class="txt" href="goboard5.html#E69:16_1_1"><i><font color="Green" title="E52">A59</font></i></a>, <a class="txt" href="goboard5.html#E70:16_1_1"><i><font color="Green" title="E53">A60</font></i></a>, <a class="ref" href="goboard2.html#D2" title="GOBOARD2:def.2">GOBOARD2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">M</font> = <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>;<br/>
<i><font color="Green" title="E55">A64</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> = 
2
<b>by </b><i><a class="ref" href="matrix_2.html#T3" title="MATRIX_2:th.3">MATRIX_2:3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/>
<i><font color="Green" title="E56">A65</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> = 
2
<b>by </b><i><a class="ref" href="matrix_2.html#T3" title="MATRIX_2:th.3">MATRIX_2:3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green" title="E54">A63</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E75:16_1_1"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</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" title="c11">n</font>, <font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1_32"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>
 ;<br/>

<a NAME="E2:16_1_1_32"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="matrix_2.html#T3" title="MATRIX_2:th.3">MATRIX_2:3</a></i>;<br/>
<a NAME="E3:16_1_1_32"/><b>then </b><i><font color="Green" title="E57">A66</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T25" title="MCART_1:th.25">MCART_1:25</a></i>;<br/>
<a NAME="E4:16_1_1_32"/><i><font color="Green" title="E58">A67</font></i>: 
( <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">0,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_32"><i><font color="Green" title="E57">A66</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:16_1_1_32_1_1"/><b>then </b>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">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" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green" title="E54">A63</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_32"><i><font color="Green" title="E58">A67</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:16_1_1_32_1_2"/><b>then </b>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">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" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green" title="E54">A63</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_32"><i><font color="Green" title="E58">A67</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:16_1_1_32_1_3"/><b>then </b>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">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" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green" title="E54">A63</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_32"><i><font color="Green" title="E58">A67</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">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" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:16_1_1_32_1_4"/><b>then </b>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">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" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="goboard5.html#E50:16_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="goboard5.html#E72:16_1_1"><i><font color="Green" title="E54">A63</font></i></a>, <a class="txt" href="goboard5.html#E4:16_1_1_32"><i><font color="Green" title="E58">A67</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">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" title="E57">A68</font></i>: <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">0,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,0</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> = 
 <a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E73:16_1_1"><i><font color="Green" title="E55">A64</font></i></a>, <a class="txt" href="goboard5.html#E74:16_1_1"><i><font color="Green" title="E56">A65</font></i></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></i>
<br/>.= 
 <a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="ref" href="goboard2.html#D3" title="GOBOARD2:def.3">GOBOARD2:def 3</a></i>
;<br/>
<a NAME="E77:16_1_1"/><b>then </b><i><font color="Green" title="E58">A69</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1
 
<b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></i>;<br/>
<a NAME="E78:16_1_1"/><i><font color="Green" title="E59">A70</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,2
 
<b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></i>;<br/>
<a NAME="E79:16_1_1"/><i><font color="Green" title="E60">A71</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,2
 
<b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></i>;<br/>
<a NAME="E80:16_1_1"/><i><font color="Green" title="E61">A72</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,1
 
<b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></i>;<br/>
<b>thus </b><a NAME="E81:16_1_1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a>,<a class="ref" href="goboard5.html#D5" target="_self" title="GOBOARD5:def.5">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" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1_34"/><i><font color="Green" title="E62">A73</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<a NAME="E2:16_1_1_34"/><b>then </b><i><font color="Green" title="E63">A74</font></i>: 
( <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5 )
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:16_1_1_34"/><i><font color="Green" title="E64">A75</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:16_1_1_34"><i><font color="Green" title="E62">A73</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1_34"><i><font color="Green" title="E63">A74</font></i></a>, <a class="txt" href="goboard5.html#E3:16_1_1_34"><i><font color="Green" title="E64">A75</font></i></a>, <a class="ref" href="nat_1.html#T30" title="NAT_1:th.30">NAT_1:30</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" title="E65">A76</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1
 <b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_1"><i><font color="Green" title="E65">A76</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">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" title="E65">A77</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,2
 <b>by </b><i><a class="txt" href="goboard5.html#E7:16_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_2"><i><font color="Green" title="E65">A77</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">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" title="E65">A78</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_3"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,2
 <b>by </b><i><a class="txt" href="goboard5.html#E13:16_1_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_3"><i><font color="Green" title="E65">A78</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">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" title="E65">A79</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_4"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,1
 <b>by </b><i><a class="txt" href="goboard5.html#E15:16_1_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_4"><i><font color="Green" title="E65">A79</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">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" title="E65">A80</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:16_1_1_34_1_5"/>
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1
 <b>by </b><i><a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_34_1_5"><i><font color="Green" title="E65">A80</font></i></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">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" title="c11">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E82:16_1_1"/><i><font color="Green" title="E62">A81</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E83:16_1_1"/><i><font color="Green" title="E63">A82</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c13">k</font>, <font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c15">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E84:16_1_1"/><i><font color="Green" title="E64">A83</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c13">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E85:16_1_1"/><i><font color="Green" title="E65">A84</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E86:16_1_1"/><i><font color="Green" title="E66">A85</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c13">k</font>
 <b>and </b><br/><a NAME="E87:16_1_1"/><i><font color="Green" title="E67">A86</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font>
 ;<br/>

<a NAME="E88:16_1_1"/><i><font color="Green" title="E68">A87</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="goboard5.html#E82:16_1_1"><i><font color="Green" title="E62">A81</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E89:16_1_1"/>
<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="goboard5.html#E2:16_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="goboard5.html#E83:16_1_1"><i><font color="Green" title="E63">A82</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E90:16_1_1"/><b>then </b><i><font color="Green" title="E69">A88</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">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" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E88:16_1_1"><i><font color="Green" title="E68">A87</font></i></a>, <a class="txt" href="goboard5.html#E90:16_1_1"><i><font color="Green" title="E69">A88</font></i></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</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" title="E70">A89</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_1_35_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><a NAME="E3:16_1_1_35_1"/><b>then </b><i><font color="Green" title="E71">A90</font></i>: 
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E77:16_1_1"><i><font color="Green" title="E58">A69</font></i></a>, <a class="txt" href="goboard5.html#E84:16_1_1"><i><font color="Green" title="E64">A83</font></i></a>, <a class="txt" href="goboard5.html#E86:16_1_1"><i><font color="Green" title="E66">A85</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E4:16_1_1_35_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><a NAME="E5:16_1_1_35_1"/><b>then </b><i><font color="Green" title="E72">A91</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E78:16_1_1"><i><font color="Green" title="E59">A70</font></i></a>, <a class="txt" href="goboard5.html#E85:16_1_1"><i><font color="Green" title="E65">A84</font></i></a>, <a class="txt" href="goboard5.html#E87:16_1_1"><i><font color="Green" title="E67">A86</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_1"><i><font color="Green" title="E70">A89</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E6:16_1_1_35_1"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_1"><i><font color="Green" title="E71">A90</font></i></a>, <a class="ref" href="goboard1.html#T1" title="GOBOARD1:th.1">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" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_1"><i><font color="Green" title="E71">A90</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_35_1"><i><font color="Green" title="E72">A91</font></i></a>, <a class="ref" href="goboard1.html#T2" title="GOBOARD1:th.2">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" title="E70">A92</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:16_1_1_35_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><a NAME="E3:16_1_1_35_2"/><b>then </b><i><font color="Green" title="E71">A93</font></i>: 
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="goboard5.html#E78:16_1_1"><i><font color="Green" title="E59">A70</font></i></a>, <a class="txt" href="goboard5.html#E84:16_1_1"><i><font color="Green" title="E64">A83</font></i></a>, <a class="txt" href="goboard5.html#E86:16_1_1"><i><font color="Green" title="E66">A85</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_2"><i><font color="Green" title="E70">A92</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E4:16_1_1_35_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><a NAME="E5:16_1_1_35_2"/><b>then </b><i><font color="Green" title="E72">A94</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="goboard5.html#E79:16_1_1"><i><font color="Green" title="E60">A71</font></i></a>, <a class="txt" href="goboard5.html#E85:16_1_1"><i><font color="Green" title="E65">A84</font></i></a>, <a class="txt" href="goboard5.html#E87:16_1_1"><i><font color="Green" title="E67">A86</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_2"><i><font color="Green" title="E70">A92</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E6:16_1_1_35_2"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_2"><i><font color="Green" title="E71">A93</font></i></a>, <a class="ref" href="goboard1.html#T1" title="GOBOARD1:th.1">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" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_2"><i><font color="Green" title="E71">A93</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_35_2"><i><font color="Green" title="E72">A94</font></i></a>, <a class="ref" href="goboard1.html#T2" title="GOBOARD1:th.2">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" title="E70">A95</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:16_1_1_35_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><a NAME="E3:16_1_1_35_3"/><b>then </b><i><font color="Green" title="E71">A96</font></i>: 
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E79:16_1_1"><i><font color="Green" title="E60">A71</font></i></a>, <a class="txt" href="goboard5.html#E84:16_1_1"><i><font color="Green" title="E64">A83</font></i></a>, <a class="txt" href="goboard5.html#E86:16_1_1"><i><font color="Green" title="E66">A85</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_3"><i><font color="Green" title="E70">A95</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E4:16_1_1_35_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><a NAME="E5:16_1_1_35_3"/><b>then </b><i><font color="Green" title="E72">A97</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E80:16_1_1"><i><font color="Green" title="E61">A72</font></i></a>, <a class="txt" href="goboard5.html#E85:16_1_1"><i><font color="Green" title="E65">A84</font></i></a>, <a class="txt" href="goboard5.html#E87:16_1_1"><i><font color="Green" title="E67">A86</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_3"><i><font color="Green" title="E70">A95</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E6:16_1_1_35_3"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_3"><i><font color="Green" title="E71">A96</font></i></a>, <a class="ref" href="goboard1.html#T1" title="GOBOARD1:th.1">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" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_3"><i><font color="Green" title="E71">A96</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_35_3"><i><font color="Green" title="E72">A97</font></i></a>, <a class="ref" href="goboard1.html#T2" title="GOBOARD1:th.2">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" title="E70">A98</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:16_1_1_35_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><a NAME="E3:16_1_1_35_4"/><b>then </b><i><font color="Green" title="E71">A99</font></i>: 
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E80:16_1_1"><i><font color="Green" title="E61">A72</font></i></a>, <a class="txt" href="goboard5.html#E84:16_1_1"><i><font color="Green" title="E64">A83</font></i></a>, <a class="txt" href="goboard5.html#E86:16_1_1"><i><font color="Green" title="E66">A85</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_4"><i><font color="Green" title="E70">A98</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E4:16_1_1_35_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E76:16_1_1"><i><font color="Green" title="E57">A68</font></i></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></i>;<br/><a NAME="E5:16_1_1_35_4"/><b>then </b><i><font color="Green" title="E72">A100</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E5:16_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="goboard5.html#E17:16_1_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="goboard5.html#E77:16_1_1"><i><font color="Green" title="E58">A69</font></i></a>, <a class="txt" href="goboard5.html#E85:16_1_1"><i><font color="Green" title="E65">A84</font></i></a>, <a class="txt" href="goboard5.html#E87:16_1_1"><i><font color="Green" title="E67">A86</font></i></a>, <a class="txt" href="goboard5.html#E1:16_1_1_35_4"><i><font color="Green" title="E70">A98</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E6:16_1_1_35_4"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_4"><i><font color="Green" title="E71">A99</font></i></a>, <a class="ref" href="goboard1.html#T1" title="GOBOARD1:th.1">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" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard5.html#E3:16_1_1_35_4"><i><font color="Green" title="E71">A99</font></i></a>, <a class="txt" href="goboard5.html#E5:16_1_1_35_4"><i><font color="Green" title="E72">A100</font></i></a>, <a class="ref" href="goboard1.html#T2" title="GOBOARD1:th.2">GOBOARD1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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