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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</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#V2">unital</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:35"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">PNR</font> =  <a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">CPNR</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">CR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>,   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>] <b>means </b>for <font color="Olive">p</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font>  st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>;<br/>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E2:35_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>]
 ;<br/><b>hence </b><a NAME="E3:35_1"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:35"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E2:35_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p'</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p'</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:35_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
for <font color="Olive">p</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font> holds  <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">F'</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span><br/> for <font color="Olive">p1'</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">F'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">p1'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">F'</font> holds <br/><font color="Olive">p1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Olive">F'</font></span>)</span>;<br/>
<a NAME="E7:35"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F'</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>;<br/><b>let </b><font color="Maroon">p1'</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/>



<b>assume </b><a NAME="E1:35_3"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">p1'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F'</font> )
 ;<br/>

<a NAME="E2:35_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E3:35_3"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">F'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then </b><i><font color="Green">E50</font></i>: <font color="Maroon">p1'</font> = 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
<br/>.= 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/>
<a NAME="E5:35_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>
 
;<br/>
<a NAME="E6:35_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F'</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:35_3"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F'</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E8:35_3"/><b>then </b>
<font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<a NAME="E9:35_3"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E10:35_3"/>
<font color="Maroon">p1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F'</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:35"/><i><font color="Green">E51</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>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4"><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="E1:35_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">F'</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>;<br/><b>let </b><font color="Maroon">p1'</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:35_4_1"/><i><font color="Green">E54</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">p1'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F'</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_4_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_4_1_1_1"><b>suppose </b></a><a NAME="E1:35_4_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:35_4_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:35_4_1_1_1"/>
<font color="Maroon">p1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_4_1_1_2"><b>suppose </b></a><a NAME="E1:35_4_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:35_4_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E4:35_4_1_1_2"/><i><font color="Green">E56</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T25">FINSEQ_2:25</a></i>;<br/><a NAME="E5:35_4_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F'</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/><a NAME="E6:35_4_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">p'</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">p</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><a NAME="E9:35_4_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:35_4_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F'</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/><a NAME="E11:35_4_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E13:35_4_1_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:35_4_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> ;<br/><a NAME="E16:35_4_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">q'</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E17:35_4_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">q'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/><b>reconsider </b><font color="Maroon">q''</font> = <font color="Maroon">q'</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">q</font> = 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q'</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T39">FINSEQ_2:39</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i/>
;<br/><a NAME="E20:35_4_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q''</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/><a NAME="E21:35_4_1_1_2"/><i><font color="Green">E77</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>;<br/><a NAME="E22:35_4_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseqop.html#T10">FINSEQOP:10</a></i>;<br/><a NAME="E23:35_4_1_1_2"/>
<font color="Maroon">p1'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p'</font> <a href="polynom1.html#K22">+</a> <font color="Maroon">q''</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, , <a class="txt" href="hilbasis.html#E1:35"><i><font color="Green">E44</font></i></a>, <a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><font color="Maroon">p1'</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F'</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E2:35"><i><font color="Green">E45</font></i></a>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:35_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:35"/>
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>2</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>
<a NAME="E10:35"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>]
 
;<br/>
<b>hence </b><a NAME="E11:35"/>
<font color="Maroon">p1</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
