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<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">distributive</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">left_unital</a> non <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</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="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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>for <font color="Olive" title="b1">p</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  st  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span>;<br/>
<a NAME="E1:33"/><i><font color="Green" title="E27">A1</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 ( for <font color="Olive" title="b2">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="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:33_1"/><i><font color="Green" title="E28">A2</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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> holds <br/>for <font color="Olive" title="b2">p</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  st  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E2:33_1"/><i><font color="Green" title="E29">A3</font></i>: 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1"/>
(  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">r</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:33_1_1_1"/><i><font color="Green" title="E30">A4</font></i>: 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font>
 <b>and </b><br/><a NAME="E4:33_1_1_1"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">r</font> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>and </b><br/><a NAME="E5:33_1_1_1"/>
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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/><font color="Maroon" title="c7">r</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">n</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T19" target="_self" title="POLYNOM4:th.19">Th19</a></i>;<br/><b>set </b><font color="Maroon" title="c8">LMp</font> =  <a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c6">p</font>;<br/><b>thus </b> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">x</font> = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom3.html#K9" title="POLYNOM3:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="polynom4.html#E4:33_1_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="polynom3.html#T33" title="POLYNOM3:th.33">POLYNOM3:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T22" target="_self" title="POLYNOM4:th.22">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom4.html#E1:33_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="polynom4.html#E2:33_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="polynom4.html#E3:33_1_1_1"><i><font color="Green" title="E30">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c6">p</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom4.html#T26" target="_self" title="POLYNOM4:th.26">Th26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p3">(<span class="default"><a href="polynom4.html#K1" title="POLYNOM4:func.1">Leading-Monomial</a> <font color="Maroon" title="c6">p</font></span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom4.html#E4:33_1_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="polynom4.html#T22" target="_self" title="POLYNOM4:th.22">Th22</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_2"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:33_1_1_2"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T8" target="_self" title="POLYNOM4:th.8">Th8</a></i>;<br/><b>hence </b> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">x</font> = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12" title="POLYNOM3:func.12">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="polynom4.html#T5" target="_self" title="POLYNOM4:th.5">Th5</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="polynom4.html#T20" target="_self" title="POLYNOM4:th.20">Th20</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"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom4.html#E2:33_1_1_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="polynom4.html#T20" target="_self" title="POLYNOM4:th.20">Th20</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:33"/><i><font color="Green" title="E28">A7</font></i>: 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</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>
 
;<br/>
<a NAME="E3:33"/>
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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="polynom4.html#E1:33"><i><font color="Green" title="E27">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:33"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K15" title="POLYNOM3:func.15">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom4.html#E2:33"><i><font color="Green" title="E28">A7</font></i></a></i>;<br/>


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