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

<b>let </b><font color="Maroon">G</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/>

<b>set </b><font color="Maroon">v1</font> =  <a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">v2</font> =  <a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span>;<br/>
<i><font color="Green">E59</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E60</font></i>:  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E3:47"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">n</font>,<font color="Olive">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p4">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p4">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:47_3"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:47_3"/><b>then </b><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E3:47_3"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,1</span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:47_3"/><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E5:47_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">m</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:47_3"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/>
<a NAME="E7:47_3"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , , <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/>
<b>hence </b><a NAME="E8:47_3"/>
<font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p4">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p4">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="goboard2.html#K1">GoB</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <a href="jordan1h.html#K1" target="_self">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" target="_self">Values</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="goboard2.html#D1">GOBOARD2:def 1</a></i>;<br/>


</div>
