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

<a NAME="E1:41_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E2:41_1_1"/>
not <font color="Maroon">G</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>assume </b><a NAME="E3:41_1_1"/>
not <font color="Maroon">G</font> is <a href="relat_1.html#V2">non-empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:41_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E6:41_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a></i>;<br/>
<a NAME="E7:41_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">s0</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:41_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E10:41_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s0</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">n0</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:41_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> holds <br/> ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n0</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D1">MATRIX_1:def 1</a></i>;<br/>
<a NAME="E13:41_1_1"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">s1</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E15:41_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E16:41_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n0</font>
 <b>by </b><i/>;<br/>
<a NAME="E17:41_1_1"/>
 ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n0</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E18:41_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>


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