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

<b>let </b><font color="Maroon">L</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:62"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E2:62"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="polynom8.html#R1" target="_self">is_primitive_root_of_degree</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E3:62"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:62_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:62_1"/><i><font color="Green">E51</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E3:62_1"/><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font> st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> holds <br/><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E1:62_1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> holds <br/> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>]
 
;<br/>
<a NAME="E2:62_1_1"/>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font> st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">nn</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">nn</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">nn</font>,<font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">nn</font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="polynom8.html#E8:62_1"><i><font color="Green">E64</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:62_1_1"/>
 ex <font color="Olive">G</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font> st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> holds <br/><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> ) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:62_1"/><i><font color="Green">E56</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E6:62_1"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E8:62_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:62_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:62_1_2"/><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:62_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:62_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="txt" href="polynom8.html#E8:62_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="polynom8.html#E8:62_1"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E5:62_1_2"/>
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:62_1"/><i><font color="Green">E64</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_3"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_3_1"><b>suppose </b></a><a NAME="E1:62_1_3_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:62_1_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:62_1_3_1_1"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:62_1_3_1_1_1"/><i><font color="Green">E67</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>

<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="polynom8.html#E1:62_1_3_1_1"><i><font color="Green">E66</font></i></a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="txt" href="polynom8.html#E2:62_1_3_1_1"><i><font color="Green">E68</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E3:62_1_3_1_1_1"/>
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<i><font color="Green">E68</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font>
<b>by </b><i>, , <a class="txt" href="polynom8.html#E1:62_1_3_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="polynom8.html#E1:62_1_3_1_1"><i><font color="Green">E66</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">m</font> <a href="binom.html#K5">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:62_1_3_1"><i><font color="Green">E65</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
;<br/>
<a NAME="E3:62_1_3_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:62_1_3_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E5:62_1_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="polynom8.html#E2:62_1_3_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="matrix_1.html#D12">MATRIX_1:def 12</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:62_1_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_3_2"><b>suppose </b></a><a NAME="E1:62_1_3_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:62_1_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_3_2_1"><b>proof </b></a><div class="add">

<i><font color="Green">E70</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i>, , , <a class="txt" href="polynom8.html#E5:62_1"><i><font color="Green">E56</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/>
<a NAME="E2:62_1_3_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:62_1_3_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:62_1_3_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:62_1_3_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="matrix_1.html#D12">MATRIX_1:def 12</a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:62_1_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E4:62"/>
<span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K1" target="_self">emb</a> <font color="Maroon">m</font>,<font color="Maroon">L</font></span>)</span> <a href="matrix_3.html#K6">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12">1.</a> <font color="Maroon">L</font>,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
