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

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1"><b>suppose </b></a><a NAME="E1:65_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:65_1_1"/>
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="matrix15.html#T51" target="_self" title="MATRIX15:th.51">Th51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2"><b>suppose </b></a><a NAME="E1:65_1_2"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c3">E</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> , <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c2">K</font> <b>by </b><i><a class="ref" href="matrix_1.html#T13" title="MATRIX_1:th.13">MATRIX_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c4">A</font> =  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ;<br/><b>set </b><font color="Maroon" title="c5">B</font> =  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ;<br/><a NAME="E3:65_1_2"/><i><font color="Green" title="E55">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="matrix15.html#E1:65_1_2"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E4:65_1_2"/><b>then </b><i><font color="Green" title="E56">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>;<br/><a NAME="E5:65_1_2"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">E</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 <b>by </b><i><a class="txt" href="matrix15.html#E3:65_1_2"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/><a NAME="E6:65_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c3">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="txt" href="matrix15.html#E3:65_1_2"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="matrix15.html#E4:65_1_2"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></i>;<br/><a NAME="E7:65_1_2"/><b>then </b><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c3">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>
 <b>by </b><i><a class="txt" href="matrix15.html#E3:65_1_2"><i><font color="Green" title="E55">A2</font></i></a></i>;<br/><a NAME="E8:65_1_2"/>
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:65_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">x</font> as   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> , <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<font color="Maroon" title="c2">K</font> <b>by </b><i><a class="txt" href="matrix15.html#E1:65_1_2"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="matrix15.html#T53" target="_self" title="MATRIX15:th.53">Th53</a></i>;<br/>
<a NAME="E3:65_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E4:65_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<b>hence </b><a NAME="E5:65_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:65_1_2"/>
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix13.html#K3" title="MATRIX13:func.3">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="matrix13.html#K3" title="MATRIX13:func.3">}</a></span>
 <b>by </b><i><a class="txt" href="matrix15.html#E7:65_1_2"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
