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

<b>let </b><font color="Maroon">L</font> be  non <a href="vectsp_1.html#V10">degenerated</a> <a href="vectsp_2.html#NM1">comRing</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:68"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">p</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">pq</font> =  <a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1"><b>suppose </b></a><a NAME="E1:68_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:68_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21">VECTSP_1:def 21</a></i>;<br/><a NAME="E3:68_1_1"/>
<font color="Maroon">p</font> is <a href="uproots.html#V1" target="_self">non-zero</a>
 <b>by </b><i>, </i>;<br/><a NAME="E4:68_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/><a NAME="E5:68_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/><span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="polynom5.html#T41">POLYNOM5:41</a></i>
<br/>.= 
<span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom5.html#T39">POLYNOM5:39</a></i>
;<br/><a NAME="E7:68_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p4"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2"><b>now </b></a><div class="add"> <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1
<b>by </b><i>, , <a class="ref" href="polynom4.html#T13">POLYNOM4:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1
<b>by </b><i>, <a class="ref" href="polynom5.html#T41">POLYNOM5:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 1

<br/>.= 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
<b>by </b><i>, , <a class="txt" href="uproots.html#E5:68_1_1_2"><i><font color="Green">E50</font></i></a></i>
;<br/><b>hence </b><a NAME="E2:68_1_1_2"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span>
 ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E3:68_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0 = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="uproots.html#E5:68_1_1_2"><i><font color="Green">E50</font></i></a></i>
;<br/><i><font color="Green">E50</font></i>:  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> = 
 <a href="polynom4.html#K2">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="polynom5.html#D6">POLYNOM5:def 6</a></i>
<br/>.= 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,0</span>)</span>,<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="uproots.html#E5:68_1_1"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p4">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i>, </i>
;<br/><span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p2">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p4">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p5">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,1</span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p5">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p0">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,1</span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> 0
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E7:68_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i/>;<br/><a NAME="E8:68_1_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:68_1_1_2_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:68_1_1_2_5"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="txt" href="uproots.html#E5:68_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E3:68_1_1_2_5"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="txt" href="uproots.html#E5:68_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_2_5_1"/>
( <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> or <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_2_5_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> = 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="uproots.html#E7:68_1_1"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom4.html#T20">POLYNOM4:20</a></i>
;<br/><a NAME="E3:68_1_1_2_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>then </b><i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
 <a href="polynom4.html#K2">eval</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="uproots.html#E7:68_1_1"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom4.html#T20">POLYNOM4:20</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E6:68_1_1_2_5_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1]
 <b>by </b><i>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_2_5_1_2"/>
<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:68_1_1_2_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p3">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>then </b><i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="polynom4.html#K2">eval</a> <span class="p4">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
;<br/><span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D19">VECTSP_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41">VECTSP_1:41</a></i>
;<br/><b>hence </b><a NAME="E5:68_1_1_2_5_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1]
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E9:68_1_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><b>hence </b><a NAME="E10:68_1_1_2"/>
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:68_1_1"/>
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="algseq_1.html#T28">ALGSEQ_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2"><b>suppose </b></a><a NAME="E1:68_1_2"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:68_1_2"/><b>then </b>
(  <a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> &amp; <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font> )
 <b>by </b><i>, , <a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><b>hence </b><a NAME="E3:68_1_2"/>
<font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="polynom5.html#K4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">L</font></span>)</span></span><a href="polynom5.html#K4">%&gt;</a></span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="uproots.html#K6" target="_self">poly_quotient</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
