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

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:17_1_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1">MATRIX_1:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="relat_1.html#V3">empty-yielding</a>
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E3:17_1_1_1"/><i><font color="Green">E14</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E4:17_1_1_1"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E5:17_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard1.html#E3:17_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E6:17_1_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/><a NAME="E7:17_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard1.html#E2:17_1_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="finseq_1.html#NM1">FinSequence</a> ;<br/><a NAME="E9:17_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="goboard1.html#E1:17_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="goboard1.html#E6:17_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_1.html#T184">RELAT_1:184</a></i>;<br/><a NAME="E10:17_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/><b>hence </b><a NAME="E11:17_1_1_1"/>
0 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard1.html#E4:17_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="goboard1.html#E6:17_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:17_1_1"/><i><font color="Green">E13</font></i>: 
( 0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> or 0 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_1_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:17_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/><a NAME="E3:17_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E4:17_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E6:17_1_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:17_1_1_2"/><i><font color="Green">E16</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="goboard1.html#E4:17_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/><a NAME="E8:17_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="goboard1.html#E2:17_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="goboard1.html#E6:17_1_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E9:17_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="goboard1.html#E2:17_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="goboard1.html#E1:17_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="goboard1.html#E7:17_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E10:17_1_1_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E11:17_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:17_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="relat_1.html#V3">empty-yielding</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T184">RELAT_1:184</a></i>;<br/>


</div>
