<?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">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<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="E1:64"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<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>set </b><font color="Maroon">v</font> =  <a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font>;<br/>
<a NAME="E2:64"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:64_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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</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"><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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E42</font></i>:  <a href="matrix_1.html#K1">width</a> <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> = 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, </i>
;<br/>
<a NAME="E2:64_1_1_1"/><i><font color="Green">E51</font></i>: 
<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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span> is   <a href="matrix_1.html#NM6">Matrix</a> of <font color="Maroon">m</font>,1,<font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1_2"><b>proof </b></a><div class="add">

<i><font color="Green">E53</font></i>:  <a href="finseq_1.html#K3">len</a> <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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <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>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
 <a href="matrix_1.html#K1">width</a> <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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E3:64_1_1_1_2"/>
<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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span> is   <a href="matrix_1.html#NM6">Matrix</a> of <font color="Maroon">m</font>,1,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:64_1_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></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> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><span class="p5">(<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="p5">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#T26">MATRIX_1:26</a></i>;<br/>
<a NAME="E4:64_1_1_1"/><i><font color="Green">E56</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:64_1_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:64_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
 <a href="matrix_1.html#K1">width</a> <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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, </i>
;<br/>
<a NAME="E8:64_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E9:64_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</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"><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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E60</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span> = 
<font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, </i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <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>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E3:64_1_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>;<br/>
<a NAME="E4:64_1_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <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_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_1"><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="E1:64_1_1_3_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:64_1_1_3_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3_1_1"/><i><font color="Green">E64</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E2:64_1_1_3_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_3_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <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/>


</div><b>end;</b></div>
<a NAME="E3:64_1_1_3_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <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></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<b>hence </b><a NAME="E4:64_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <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_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:64_1_1_3"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <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"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_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="E1:64_1_1_3_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:64_1_1_3_2"/><i><font color="Green">E67</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:64_1_1_3_2"/><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<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_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="polynom8.html#E2:64_1_1_3"><i><font color="Green">E65</font></i></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"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="polynom8.html#E3:64_1_1_3"><i><font color="Green">E69</font></i></a>, </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:64_1_1_3_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <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"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:64_1_1_3"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_3"><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="E1:64_1_1_3_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:64_1_1_3_3"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3_3_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3_3_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, </i>;<br/>
<a NAME="E2:64_1_1_3_3_1_1"/><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:64_1_1_3_3_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <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="txt" href="polynom8.html#E1:64_1_1_3_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:64_1_1_3_3_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:64_1_1_3_3_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>,1
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:64_1_1_3_3"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:64_1_1_3"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><font color="Olive">a</font> <a href="group_1.html#K10">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_4"><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>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:64_1_1_3_4"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:64_1_1_3_4"/>
<font color="Maroon">a</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"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3"><i><font color="Green">E69</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_3_4"/>
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3"><i><font color="Green">E69</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:64_1_1_3"/><i><font color="Green">E75</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/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_5"><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="E1:64_1_1_3_5"/><i><font color="Green">E77</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="E2:64_1_1_3_5"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <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></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3_5_1"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <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></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<b>hence </b><a NAME="E2:64_1_1_3_5_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <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></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:64_1_1_3_5"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <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_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E4:64_1_1_3_5"/><i><font color="Green">E80</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, </i>;<br/>
<a NAME="E5:64_1_1_3_5"/>
<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/>;<br/>
<a NAME="E6:64_1_1_3_5"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E4:64_1_1_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:64_1_1_3_5"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,1
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E8:64_1_1_3_5"/><i><font color="Green">E92</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:64_1_1_3_5"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:64_1_1_3_5"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<a NAME="E11:64_1_1_3_5"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:64_1_1_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>, <font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>, <font color="Maroon">c</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E13:64_1_1_3_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_5_2"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3_5_2"/><i><font color="Green">E95</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:64_1_1_3_5_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 is   <a href="nat_1.html#NM1">Nat</a>
 
<b>by </b><i><a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_5"><i><font color="Green">E78</font></i></a>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<a NAME="E4:64_1_1_3_5_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:64_1_1_3_5_2"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 is   <a href="nat_1.html#NM1">Nat</a>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<a NAME="E6:64_1_1_3_5_2"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3_5"><i><font color="Green">E79</font></i></a>, <a class="txt" href="polynom8.html#E4:64_1_1_3_5"><i><font color="Green">E80</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:64_1_1_3_5_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> )
 
;<br/>
<b>hence </b><a NAME="E8:64_1_1_3_5_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="polynom8.html#T5" target="_self">Th5</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:64_1_1_3_5"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_5_3"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3_5_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_5_3_1"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <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>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
<b>then </b> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <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>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E3:64_1_1_3_5_3_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</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#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:64_1_1_3_5_3_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:64_1_1_3_5_3"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T73">FVSUM_1:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:64_1_1_3_5"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> =  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <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> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<font color="Maroon">d</font> = 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3_5"><i><font color="Green">E79</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3_5"><i><font color="Green">E79</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E18:64_1_1_3_5"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_5"><i><font color="Green">E78</font></i></a>, <a class="txt" href="polynom8.html#E1:64_1_1_3_5"><i><font color="Green">E77</font></i></a>, <a class="txt" href="polynom8.html#E3:64_1_1_3_5"><i><font color="Green">E79</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:64_1_1_3"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:64_1_1_3_6"/><i><font color="Green">E104</font></i>: 
(  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</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#K4">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom4.html#D2">POLYNOM4:def 2</a></i>;<br/>
<a NAME="E3:64_1_1_3_6"/><i><font color="Green">E174</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>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6_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="E1:64_1_1_3_6_1"/><i><font color="Green">E175</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>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 ;<br/>

<a NAME="E2:64_1_1_3_6_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:64_1_1_3_6_1"/><b>then </b><i><font color="Green">E176</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">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:64_1_1_3_6_1"/><i><font color="Green">E177</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> )
 
<b>by </b><i>, <a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E4:64_1_1_3"><i><font color="Green">E73</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="polynom8.html#E3:64_1_1_3"><i><font color="Green">E69</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:64_1_1_3_6_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:64_1_1_3_6"/><i><font color="Green">E178</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&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/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6_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="E1:64_1_1_3_6_2"/><i><font color="Green">E179</font></i>: 
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&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="E2:64_1_1_3_6_2"/><i><font color="Green">E180</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3_6_2_1"/><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E4:64_1_1_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E2:64_1_1_3_6_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_3_6_2_1"/>
( 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> )
 <b>by </b><i><a class="txt" href="polynom8.html#E4:64_1_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="polynom8.html#E5:64_1_1_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:64_1_1_3_6_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3_6_2_2"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E2:64_1_1_3_6_2_2"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E3:64_1_1_3_6_2_2"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E4:64_1_1_3"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:64_1_1_3_6_2_2"/>
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:64_1_1_3_6_2"/>
<font color="Maroon">k</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>then </b><span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="polynom8.html#E3:64_1_1_3"><i><font color="Green">E69</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <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="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E5:64_1_1_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
;<br/>
<b>hence </b><a NAME="E6:64_1_1_3_6_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:64_1_1_3_6"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">lengthG</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E7:64_1_1_3_6"/><i><font color="Green">E185</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">lengthG</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">lengthG</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="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6_3"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</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="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>;<br/>
<a NAME="E1:64_1_1_3_6_3"/><i><font color="Green">E186</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">lengthG</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="E2:64_1_1_3_6_3"/>
 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">lengthG</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">lengthG</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>();<br/></i>
<b>hence </b><a NAME="E3:64_1_1_3_6_3"/>
 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">lengthG</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">lengthG</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="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:64_1_1_3_6"/><i><font color="Green">E187</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">lengthG</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">lengthG</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> ) )
 <b>by </b><i><a class="txt" href="polynom8.html#E4:64_1_1_3"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E10:64_1_1_3_6"/><i><font color="Green">E188</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &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#K4">dom</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> ) )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:64_1_1_3_6"/><i><font color="Green">E189</font></i>: 
 <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6_4"><b>proof </b></a><div class="add">

<a NAME="E1:64_1_1_3_6_4"/><i><font color="Green">E190</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6_4_1"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <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>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
<b>then </b><i><font color="Green">E191</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <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>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E3:64_1_1_3_6_4_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="txt" href="polynom8.html#E5:64_1_1_3"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E4:64_1_1_3_6_4_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</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#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:64_1_1_3_6_4_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:64_1_1_3_6_4"/>
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#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6_4_2"><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="E1:64_1_1_3_6_4_2"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:64_1_1_3_6_4_2"/><i><font color="Green">E193</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="txt" href="polynom8.html#E5:64_1_1_3"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E3:64_1_1_3_6_4_2"/><i><font color="Green">E194</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</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#D7">FINSEQ_1:def 7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_6_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1_3_6_4_2_1"/>
( ( 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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> ) or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&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> ) )
 </span><b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_6_4_2_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_3_6_4_2_1_1"/><i><font color="Green">E195</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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 ;<br/><div class="add"><a NAME="E2:64_1_1_3_6_4_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="polynom8.html#T5" target="_self">Th5</a>, </i>
;<br/><b>hence </b><a NAME="E4:64_1_1_3_6_4_2_1_1"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_6_4_2_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_3_6_4_2_1_2"/><i><font color="Green">E196</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR3" target="_self">&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/><div class="add"><a NAME="E2:64_1_1_3_6_4_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="polynom8.html#E5:64_1_1_3"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E3:64_1_1_3_6_4_2_1_2"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</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#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E4:64_1_1_3_6_4_2_1_2"/><b>then </b><i><font color="Green">E197</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:64_1_1_3_6_4_2_1_2"/><i><font color="Green">E198</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="polynom8.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:64_1_1_3_6_4_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E8:64_1_1_3_6_4_2_1_2"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:64_1_1_3_6_4_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> 0
 ;<br/><a NAME="E10:64_1_1_3_6_4_2_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T27">XREAL_1:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">lengthG</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E12:64_1_1_3_6_4_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="polynom8.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:64_1_1_3_6_4_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:64_1_1_3_6_4_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lengthG</font> )
 <b>by </b><i>, <a class="ref" href="polynom8.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E15:64_1_1_3_6_4_2_1_2"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom8.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="polynom8.html#T5" target="_self">Th5</a>, </i>
;<br/><b>hence </b><a NAME="E17:64_1_1_3_6_4_2_1_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:64_1_1_3_6_4"/>
 <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <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>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K4">+</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="polynom3.html#T1">POLYNOM3:1</a></i>
<br/>.= 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T7">VECTSP_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E13:64_1_1_3_6"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:64_1_1_3"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:64_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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="binom.html#K2">|^</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:64_1"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:64"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K6" target="_self">DFT</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <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#K4" target="_self">mConv</a> <font color="Maroon">p</font>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1
 ;<br/>


</div>
