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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E60">A1</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>set </b><font color="Maroon" title="c5">v</font> =  <a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font>;<br/>
<a NAME="E2:64"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K6" title="POLYNOM8:func.6">DFT</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c6">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">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" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">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" title="E62">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">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" title="E63">A4</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> = 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
;<br/>
<a NAME="E2:64_1_1_1"/><i><font color="Green" title="E64">A5</font></i>: 
<span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span> is   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">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" title="E65">A6</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</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" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span> is   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c3">m</font>,1,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_1_2"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:64_1_1_1"/><i><font color="Green" title="E65">A7</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p5">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_1"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="matrix_1.html#T26" title="MATRIX_1:th.26">MATRIX_1:26</a></i>;<br/>
<a NAME="E4:64_1_1_1"/><i><font color="Green" title="E66">A8</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E5:64_1_1_1"/>
<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:64_1_1_1"/><b>then </b><i><font color="Green" title="E67">A9</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E4:64_1_1_1"><i><font color="Green" title="E66">A8</font></i></a></i>;<br/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
;<br/>
<a NAME="E8:64_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="txt" href="polynom8.html#E6:64_1_1_1"><i><font color="Green" title="E67">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E63">A10</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span> = 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E3:64_1_1"/><i><font color="Green" title="E64">A11</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#D10" title="FVSUM_1:def.10">FVSUM_1:def 10</a></i>;<br/>
<a NAME="E4:64_1_1"/>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">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" title="E65">A12</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_1_1_3_1"/><i><font color="Green" title="E66">A13</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ;<br/>

<a NAME="E2:64_1_1_3_1"/>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">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" title="E67">A14</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E2:64_1_1_3_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_1"><i><font color="Green" title="E66">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_3_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_1_1"><i><font color="Green" title="E67">A14</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">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" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:64_1_1_3"/><i><font color="Green" title="E66">A15</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">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" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_1_1_3_2"/><i><font color="Green" title="E67">A16</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ;<br/>

<a NAME="E2:64_1_1_3_2"/><i><font color="Green" title="E68">A17</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:64_1_1_3_2"/><i><font color="Green" title="E69">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_2"><i><font color="Green" title="E67">A16</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_2"><i><font color="Green" title="E67">A16</font></i></a>, <a class="txt" href="polynom8.html#E1:64_1_1_3"><i><font color="Green" title="E65">A12</font></i></a></i>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_2"><i><font color="Green" title="E68">A17</font></i></a>, <a class="txt" href="polynom8.html#E3:64_1_1_3_2"><i><font color="Green" title="E69">A18</font></i></a>, <a class="ref" href="polynom8.html#D8" target="_self" title="POLYNOM8:def.8">Def8</a></i>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">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" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:64_1_1_3"/><i><font color="Green" title="E67">A19</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_1_1_3_3"/><i><font color="Green" title="E68">A20</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font>
 ;<br/>

<a NAME="E2:64_1_1_3_3"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">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" title="E69">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>;<br/>
<a NAME="E2:64_1_1_3_3_1_1"/><i><font color="Green" title="E70">A22</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_3"><i><font color="Green" title="E68">A20</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:64_1_1_3_3_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_3_1_1"><i><font color="Green" title="E70">A22</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:64_1_1_3_3_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_3_1_1"><i><font color="Green" title="E69">A21</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,1
 <b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">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" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_3"><i><font color="Green" title="E68">A20</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:64_1_1_3"/><i><font color="Green" title="E68">A23</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Olive" title="b2">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Olive" title="b4">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" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:64_1_1_3_4"/><i><font color="Green" title="E69">A24</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> )
 ;<br/>

<a NAME="E2:64_1_1_3_4"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_4"><i><font color="Green" title="E69">A24</font></i></a>, <a class="txt" href="polynom8.html#E2:64_1_1_3"><i><font color="Green" title="E66">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_3_4"/>
<font color="Maroon" title="c8">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">k</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_4"><i><font color="Green" title="E69">A24</font></i></a>, <a class="txt" href="polynom8.html#E3:64_1_1_3"><i><font color="Green" title="E67">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:64_1_1_3"/><i><font color="Green" title="E69">A25</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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" title="c7">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:64_1_1_3_5"/><i><font color="Green" title="E70">A26</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 ;<br/>

<a NAME="E2:64_1_1_3_5"/><i><font color="Green" title="E71">A27</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">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" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<b>hence </b><a NAME="E2:64_1_1_3_5_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_5"><i><font color="Green" title="E70">A26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:64_1_1_3_5"/><i><font color="Green" title="E72">A28</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_5"><i><font color="Green" title="E71">A27</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E4:64_1_1_3_5"/><i><font color="Green" title="E73">A29</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>;<br/>
<a NAME="E5:64_1_1_3_5"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_5"><i><font color="Green" title="E70">A26</font></i></a></i>;<br/>
<a NAME="E6:64_1_1_3_5"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E4:64_1_1_3_5"><i><font color="Green" title="E73">A29</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:64_1_1_3_5"/><b>then </b><i><font color="Green" title="E74">A30</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">k</font>,1
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E8:64_1_1_3_5"/><i><font color="Green" title="E75">A31</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:64_1_1_3_5"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_5"><i><font color="Green" title="E70">A26</font></i></a></i>;<br/>
<a NAME="E10:64_1_1_3_5"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<a NAME="E11:64_1_1_3_5"/><b>then </b><i><font color="Green" title="E76">A32</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E8:64_1_1_3_5"><i><font color="Green" title="E75">A31</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">a</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c9">b</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3_5"><i><font color="Green" title="E72">A28</font></i></a>, <a class="txt" href="polynom8.html#E7:64_1_1_3_5"><i><font color="Green" title="E74">A30</font></i></a>, <a class="txt" href="polynom8.html#E11:64_1_1_3_5"><i><font color="Green" title="E76">A32</font></i></a></i>;<br/>
<a NAME="E13:64_1_1_3_5"/><i><font color="Green" title="E77">A33</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">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" title="E78">A34</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_5"><i><font color="Green" title="E70">A26</font></i></a></i>;<br/>
<a NAME="E2:64_1_1_3_5_2"/><b>then </b><i><font color="Green" title="E79">A35</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 
<b>by </b><i><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_5_2"><i><font color="Green" title="E78">A34</font></i></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<a NAME="E4:64_1_1_3_5_2"/><i><font color="Green" title="E80">A36</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_5"><i><font color="Green" title="E70">A26</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E5:64_1_1_3_5_2"/>
<font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<a NAME="E6:64_1_1_3_5_2"/><b>then </b><i><font color="Green" title="E81">A37</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_5_2"><i><font color="Green" title="E79">A35</font></i></a>, <a class="txt" href="polynom8.html#E4:64_1_1_3_5_2"><i><font color="Green" title="E80">A36</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:64_1_1_3_5_2"/>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">l</font> )
 
;<br/>
<b>hence </b><a NAME="E8:64_1_1_3_5_2"/>
<font color="Maroon" title="c8">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E6:64_1_1_3_5_2"><i><font color="Green" title="E81">A37</font></i></a>, <a class="txt" href="polynom8.html#E4:64_1_1_3"><i><font color="Green" title="E68">A23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:64_1_1_3_5"/><i><font color="Green" title="E78">A38</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c9">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" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
<b>then </b> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T8" title="MATRIX_3:th.8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">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" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:64_1_1_3_5_3_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_5"><i><font color="Green" title="E70">A26</font></i></a></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" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T73" title="FVSUM_1:th.73">FVSUM_1:73</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:64_1_1_3_5"/><i><font color="Green" title="E79">A39</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_5"><i><font color="Green" title="E70">A26</font></i></a>, <a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">d</font> =  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<font color="Maroon" title="c11">d</font> = 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_5"><i><font color="Green" title="E70">A26</font></i></a>, <a class="ref" href="polynom8.html#T27" target="_self" title="POLYNOM8:th.27">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E15:64_1_1_3_5"><i><font color="Green" title="E79">A39</font></i></a>, <a class="ref" href="polynom8.html#D3" target="_self" title="POLYNOM8:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E15:64_1_1_3_5"><i><font color="Green" title="E79">A39</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">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" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E14:64_1_1_3_5"><i><font color="Green" title="E78">A38</font></i></a>, <a class="txt" href="polynom8.html#E13:64_1_1_3_5"><i><font color="Green" title="E77">A33</font></i></a>, <a class="txt" href="polynom8.html#E15:64_1_1_3_5"><i><font color="Green" title="E79">A39</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:64_1_1_3"/>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">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" title="c7">F</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E2:64_1_1_3_6"/><i><font color="Green" title="E70">A40</font></i>: 
(  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c7">F</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> holds <br/><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom4.html#D2" title="POLYNOM4:def.2">POLYNOM4:def 2</a></i>;<br/>
<a NAME="E3:64_1_1_3_6"/><i><font color="Green" title="E71">A41</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> holds <br/><font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:64_1_1_3_6_1"/><i><font color="Green" title="E72">A42</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> )
 ;<br/>

<a NAME="E2:64_1_1_3_6_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6"><i><font color="Green" title="E70">A40</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:64_1_1_3_6_1"/><b>then </b><i><font color="Green" title="E73">A43</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_6_1"><i><font color="Green" title="E72">A42</font></i></a></i>;<br/>
<a NAME="E4:64_1_1_3_6_1"/><i><font color="Green" title="E74">A44</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_6_1"><i><font color="Green" title="E72">A42</font></i></a>, <a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3_6_1"><i><font color="Green" title="E73">A43</font></i></a>, <a class="txt" href="polynom8.html#E2:64_1_1_3_6"><i><font color="Green" title="E70">A40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
<b>by </b><i><a class="txt" href="polynom8.html#E4:64_1_1_3_6_1"><i><font color="Green" title="E74">A44</font></i></a>, <a class="txt" href="polynom8.html#E5:64_1_1_3"><i><font color="Green" title="E69">A25</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:64_1_1_3_6_1"/>
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:64_1_1_3_6"/><i><font color="Green" title="E72">A45</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">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" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:64_1_1_3_6_2"/><i><font color="Green" title="E73">A46</font></i>: 
(  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 ;<br/>

<a NAME="E2:64_1_1_3_6_2"/><i><font color="Green" title="E74">A47</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">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" title="E75">A48</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_6_2"><i><font color="Green" title="E73">A46</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E2:64_1_1_3_6_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>hence </b><a NAME="E3:64_1_1_3_6_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_6_2"><i><font color="Green" title="E73">A46</font></i></a>, <a class="txt" href="polynom8.html#E1:64_1_1_3_6_2_1"><i><font color="Green" title="E75">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:64_1_1_3_6_2"/><i><font color="Green" title="E75">A49</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">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" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6_2"><i><font color="Green" title="E74">A47</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E2:64_1_1_3_6_2_2"/><b>then </b><i><font color="Green" title="E76">A50</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E3:64_1_1_3_6_2_2"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_6_2"><i><font color="Green" title="E73">A46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:64_1_1_3_6_2_2"/>
<font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6_2_2"><i><font color="Green" title="E76">A50</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:64_1_1_3_6_2"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6_2"><i><font color="Green" title="E74">A47</font></i></a>, <a class="txt" href="polynom8.html#E5:64_1_1_3"><i><font color="Green" title="E69">A25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3_6_2"><i><font color="Green" title="E75">A49</font></i></a>, <a class="ref" href="algseq_1.html#T22" title="ALGSEQ_1:th.22">ALGSEQ_1:22</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">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" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">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" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">lengthG</font> = <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E7:64_1_1_3_6"/><i><font color="Green" title="E73">A51</font></i>: 
 ex <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font> st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">lengthG</font> &amp; ( for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">lengthG</font> holds <br/><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">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" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E1:64_1_1_3_6_3"/><i><font color="Green" title="E74">A52</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">lengthG</font> holds <br/> ex <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>]
 
;<br/>
<a NAME="E2:64_1_1_3_6_3"/>
 ex <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font> st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">lengthG</font> &amp; ( for <font color="Olive" title="b2">nn</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">nn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">lengthG</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">nn</font>,<font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">nn</font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="polynom8.html#E1:64_1_1_3_6_3"><i><font color="Green" title="E74">A52</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:64_1_1_3_6_3"/>
 ex <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font> st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">lengthG</font> &amp; ( for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">lengthG</font> holds <br/><font color="Olive" title="b1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E9:64_1_1_3_6"/><i><font color="Green" title="E74">A53</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">lengthG</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c8">lengthG</font> holds <br/><font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> ) )
 <b>by </b><i><a class="txt" href="polynom8.html#E7:64_1_1_3_6"><i><font color="Green" title="E73">A51</font></i></a></i>;<br/>
<a NAME="E10:64_1_1_3_6"/><i><font color="Green" title="E75">A54</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">G</font> holds <br/><font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font> ) )
 
<b>by </b><i><a class="txt" href="polynom8.html#E9:64_1_1_3_6"><i><font color="Green" title="E74">A53</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:64_1_1_3_6"/><i><font color="Green" title="E76">A55</font></i>: 
 <a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">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" title="E77">A56</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="polynom8.html#T28" target="_self" title="POLYNOM8:th.28">Th28</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E78">A57</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T8" title="MATRIX_3:th.8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6"><i><font color="Green" title="E70">A40</font></i></a>, <a class="txt" href="polynom8.html#E10:64_1_1_3_6"><i><font color="Green" title="E75">A54</font></i></a></i>;<br/>
<a NAME="E4:64_1_1_3_6_4_1"/><b>then </b>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">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" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6_4_1"><i><font color="Green" title="E78">A57</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c10">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:64_1_1_3_6_4_2"/><i><font color="Green" title="E78">A58</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:64_1_1_3_6_4_2"/><i><font color="Green" title="E79">A59</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6"><i><font color="Green" title="E70">A40</font></i></a>, <a class="txt" href="polynom8.html#E10:64_1_1_3_6"><i><font color="Green" title="E75">A54</font></i></a></i>;<br/>
<a NAME="E3:64_1_1_3_6_4_2"/><i><font color="Green" title="E80">A60</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6_4_2"><i><font color="Green" title="E79">A59</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> ) )
 </span><b>by </b><i><a class="txt" href="polynom8.html#E3:64_1_1_3_6_4_2"><i><font color="Green" title="E80">A60</font></i></a>, <a class="txt" href="polynom8.html#E1:64_1_1_3_6_4"><i><font color="Green" title="E77">A56</font></i></a>, <a class="txt" href="polynom8.html#E1:64_1_1_3_6_4_2"><i><font color="Green" title="E78">A58</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">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" title="E81">A61</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> )
 ;<br/><div class="add"><a NAME="E2:64_1_1_3_6_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6"><i><font color="Green" title="E70">A40</font></i></a>, <a class="txt" href="polynom8.html#E1:64_1_1_3_6_4_2_1_1"><i><font color="Green" title="E81">A61</font></i></a>, <a class="txt" href="polynom8.html#E3:64_1_1_3_6"><i><font color="Green" title="E71">A41</font></i></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" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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" title="E81">A62</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6"><i><font color="Green" title="E70">A40</font></i></a>, <a class="txt" href="polynom8.html#E10:64_1_1_3_6"><i><font color="Green" title="E75">A54</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" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">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" title="E82">A63</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:64_1_1_3_6_4_2_1_2"/><i><font color="Green" title="E83">A64</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_6_4_2_1_2"><i><font color="Green" title="E81">A62</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E5:64_1_1_3_6_4_2_1_2"><i><font color="Green" title="E83">A64</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">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" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">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" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 0
 ;<br/><a NAME="E10:64_1_1_3_6_4_2_1_2"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T27" title="XREAL_1:th.27">XREAL_1:27</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">lengthG</font> = <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_6_4_2_1_2"><i><font color="Green" title="E81">A62</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">l</font> &amp; <font color="Maroon" title="c11">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">lengthG</font> )
 <b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6"><i><font color="Green" title="E70">A40</font></i></a>, <a class="txt" href="polynom8.html#E1:64_1_1_3_6_4_2_1_2"><i><font color="Green" title="E81">A62</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">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" title="E84">A65</font></i>: 
<font color="Maroon" title="c11">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">G</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E9:64_1_1_3_6"><i><font color="Green" title="E74">A53</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_6_4_2_1_2"><i><font color="Green" title="E81">A62</font></i></a>, <a class="txt" href="polynom8.html#E4:64_1_1_3_6_4_2_1_2"><i><font color="Green" title="E82">A63</font></i></a>, <a class="ref" href="finseq_1.html#T37" title="FINSEQ_1:th.37">FINSEQ_1:37</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="polynom8.html#E15:64_1_1_3_6_4_2_1_2"><i><font color="Green" title="E84">A65</font></i></a>, <a class="txt" href="polynom8.html#E9:64_1_1_3_6"><i><font color="Green" title="E74">A53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6"><i><font color="Green" title="E70">A40</font></i></a>, <a class="txt" href="polynom8.html#E1:64_1_1_3_6_4_2_1_2"><i><font color="Green" title="E81">A62</font></i></a>, <a class="txt" href="polynom8.html#E4:64_1_1_3_6"><i><font color="Green" title="E72">A45</font></i></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" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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" title="FVSUM_1:func.11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">G</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1_3_6_4"><i><font color="Green" title="E77">A56</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c9">G</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E11:64_1_1_3_6"><i><font color="Green" title="E76">A55</font></i></a>, <a class="ref" href="rlvect_1.html#T58" title="RLVECT_1:th.58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E9:64_1_1_3_6"><i><font color="Green" title="E74">A53</font></i></a>, <a class="ref" href="polynom3.html#T1" title="POLYNOM3:th.1">POLYNOM3:1</a></i>
<br/>.= 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E2:64_1_1_3_6"><i><font color="Green" title="E70">A40</font></i></a>, <a class="ref" href="vectsp_1.html#T7" title="VECTSP_1:th.7">VECTSP_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E13:64_1_1_3_6"/>
 <a href="rlvect_1.html#K9" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">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" title="RLVECT_1:func.9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11" title="FVSUM_1:func.11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">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" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="polynom8.html#E2:64_1_1"><i><font color="Green" title="E63">A10</font></i></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="polynom8.html#E3:64_1_1"><i><font color="Green" title="E64">A11</font></i></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" title="POLYNOM8:func.6">DFT</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="polynom8.html#E1:64_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="polynom8.html#D7" target="_self" title="POLYNOM8:def.7">Def7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:64"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><span class="p1">(<span class="default"><a href="polynom8.html#K6" title="POLYNOM8:func.6">DFT</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" title="POLYNOM8:NK.8">VM</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K4" title="POLYNOM8:func.4">mConv</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1
 ;<br/>


</div>
