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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E7</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E8</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E3:6"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan15.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:6"/><b>then </b><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan15.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan15.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan15.html#E4:6"><i><font color="Green">E9</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/>
<a NAME="E6:6"/><b>then </b><i><font color="Green">E10</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="sppol_1.html#V1">horizontal</a>
 
<b>by </b><i><a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a></i>;<br/>
<a NAME="E7:6"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan15.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:6"/><b>then </b><i><font color="Green">E11</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan15.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan15.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/>
<a NAME="E10:6"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="sppol_1.html#V1">horizontal</a>
 
<b>by </b><i><a class="ref" href="sppol_1.html#T36">SPPOL_1:36</a></i>;<br/>
<i><font color="Green">E13</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan15.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan15.html#E4:6"><i><font color="Green">E9</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan15.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan15.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/>
<a NAME="E12:6"/><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan15.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan15.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan15.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/>
<a NAME="E13:6"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan15.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan15.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan15.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/>
<a NAME="E14:6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan15.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan15.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan15.html#E8:6"><i><font color="Green">E11</font></i></a>, <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/>
<b>hence </b><a NAME="E15:6"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan15.html#E6:6"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan15.html#E10:6"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan15.html#E11:6"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan15.html#E12:6"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan15.html#E13:6"><i><font color="Green">E15</font></i></a>, <a class="ref" href="jordan15.html#T6" target="_self">Th6</a></i>;<br/>


</div>
