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<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">M</font> is  not    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font>, <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>,<font color="Maroon">D</font>
 ;<br/>

<b>set </b><font color="Maroon">m</font> =  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> or  ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font> &amp; not  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> ) )
 </span><b>by </b><i>, <a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>suppose </b></a><a NAME="E1:15_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2"><b>suppose </b></a><a NAME="E1:15_1_2"/>
 ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font> &amp; not  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E3:15_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:15_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 ;<br/><b>consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:15_1_2"/><i><font color="Green">E61</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="relat_1.html#K2">rng</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T9">MATRIX_1:9</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">p</font> as    <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E8:15_1_2"/><i><font color="Green">E62</font></i>: 
 ex <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_2_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:15_1_2_1_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E3:15_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:15_1_2_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:15_1_2_1_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D4">MATRIX_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">s</font> as    <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E6:15_1_2_1_1_2"/>
 ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:15_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:15_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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