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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="algseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="polynom4.html#K2" target="_self">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E1:32"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="algseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="polynom4.html#K2" target="_self">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:32_1"/><i><font color="Green">E28</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:32_1_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:32_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:32_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="polynom4.html#T19" target="_self">Th19</a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b> <a href="polynom4.html#K2" target="_self">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="polynom4.html#K2" target="_self">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="polynom4.html#E4:32_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="polynom3.html#T32">POLYNOM3:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom4.html#E1:32_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="polynom4.html#E2:32_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="polynom4.html#E3:32_1_1_1"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></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="polynom4.html#K2" target="_self">eval</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom4.html#E32"><i><font color="Green">Lemma24</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18">VECTSP_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom4.html#E4:32_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="polynom4.html#T22" target="_self">Th22</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_2"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:32_1_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="polynom4.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b> <a href="polynom4.html#K2" target="_self">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="polynom4.html#K2" target="_self">eval</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="polynom4.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="polynom4.html#E2:32_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polynom4.html#T20" target="_self">Th20</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:32"/><i><font color="Green">E27</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E3:32"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>(<a class="txt" href="polynom4.html#E1:32"><i><font color="Green">E26</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:32"/>
 <a href="polynom4.html#K2" target="_self">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <span class="p2">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" target="_self">eval</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom4.html#E2:32"><i><font color="Green">E27</font></i></a></i>;<br/>


</div>
