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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/>

<b>set </b><font color="Maroon" title="c2">v1</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">v2</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>;<br/>
<i><font color="Green" title="E32">A1</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> = 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1h.html#T22" target="_self" title="JORDAN1H:th.22">Th22</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1h.html#T15" target="_self" title="JORDAN1H:th.15">Th15</a></i>
;<br/>
<i><font color="Green" title="E33">A2</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font> = 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1h.html#T25" target="_self" title="JORDAN1H:th.25">Th25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan1h.html#T15" target="_self" title="JORDAN1H:th.15">Th15</a></i>
;<br/>
<a NAME="E3:47"/>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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> <font color="Maroon" title="c1">G</font> holds <br/><font color="Maroon" title="c1">G</font> <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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p4">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></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="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:47_3"/><i><font color="Green" title="E34">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c1">G</font>
 ;<br/>

<a NAME="E2:47_3"/><b>then </b><i><font color="Green" title="E35">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <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="c1">G</font> )
 
<b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E3:47_3"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="ref" href="jordan1h.html#T34" target="_self" title="JORDAN1H:th.34">Th34</a></i>;<br/>
<a NAME="E4:47_3"/><i><font color="Green" title="E37">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:47_3"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">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" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c5">m</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="ref" href="jordan1h.html#T35" target="_self" title="JORDAN1H:th.35">Th35</a></i>;<br/>
<a NAME="E6:47_3"/><b>then </b><i><font color="Green" title="E38">A7</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">m</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jordan1h.html#E2:47_3"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1h.html#E4:47_3"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/>
<a NAME="E7:47_3"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">m</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1h.html#E2:47_3"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="jordan1h.html#E3:47_3"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="jordan1h.html#E4:47_3"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/>
<b>hence </b><a NAME="E8:47_3"/>
<font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">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="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p4">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p4">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p5">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">m</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E6:47_3"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:47"/>
<font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <a href="jordan1h.html#K1" title="JORDAN1H:func.1">RealOrd</a> ,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:47"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="jordan1h.html#E2:47"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></i>;<br/>


</div>
