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

<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>set </b><font color="Maroon">P</font> =  <a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font>;<br/>
<b>thus </b><a NAME="E1:37_2_1"/>
 <a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font> is <a href="group_6.html#V1">multiplicative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1"><b>proof </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="polynom3.html#K16">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span></span>)</span>, <font color="Maroon">y'</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="polynom3.html#K16">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a><br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font>, <font color="Maroon">y</font> = <font color="Maroon">y'</font>, <font color="Maroon">xy</font> = <font color="Maroon">x'</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y'</font> as   <a href="polynom3.html#NM1">Polynomial</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="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Pxy</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y'</font></span>)</span>, <font color="Maroon">PxPy</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</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="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font></span>)</span>, <font color="Maroon">Px</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>, <font color="Maroon">Py</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E3:37_2_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">PxPy</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">Px</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">Py</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E4:37_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">xy</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">x</font> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_1"/>
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b'</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ;<br/><b>consider </b><font color="Maroon">r</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><b> such that </b><br/><a NAME="E4:37_2_1_1_1"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">xy</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">r</font> &amp; ( 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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">k</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><a NAME="E5:37_2_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">bn</font> = <font color="Maroon">b</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">xybn</font> = <font color="Maroon">xy</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> 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/><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="E9:37_2_1_1_1"/><i><font color="Green">E50</font></i>: 
( ( 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">bn</font> ) &amp; <font color="Maroon">xybn</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">bn</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">r</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E10:37_2_1_1_1"/><i><font color="Green">E51</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">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Pxy</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">R</font><br/> for <font color="Olive">fcr</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><br/> for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">i</font></span>)</span> &amp; <font color="Olive">fcr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="polynom1.html#K29">*'</a> <font color="Olive">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">bn</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">fcr</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">fcr</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span> &amp; ( 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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">fcr</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Olive">fcr</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font></span>)</span> ) ) );<br/><div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_1_1"/>
<font color="Maroon">e'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>, <font color="Maroon">q</font> = <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">e</font></span>)</span> 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/><b>consider </b><font color="Maroon">fcr</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:37_2_1_1_1_1"/><i><font color="Green">E54</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">bn</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">fcr</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fcr</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span> &amp; ( 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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fcr</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">fcr</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/><a NAME="E6:37_2_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e</font>,<font color="Maroon">fcr</font>]
 <b>by </b><i/>;<br/><a NAME="E7:37_2_1_1_1_1"/>
<font color="Maroon">fcr</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>hence </b><a NAME="E8:37_2_1_1_1_1"/>
 ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_2.html#K3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e'</font>,<font color="Olive">u</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">s</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E13:37_2_1_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>);<br/></i><a NAME="E14:37_2_1_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E15:37_2_1_1_1"/><b>then </b>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E16:37_2_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">s</font> is <a href="finseq_1.html#V1">FinSequence-like</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E17:37_2_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_2.html#K3">*</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E19:37_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="matrlin.html#D8">MATRLIN:def 8</a></i>;<br/><a NAME="E20:37_2_1_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E21:37_2_1_1_1"/><i><font color="Green">E77</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <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">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_2"><b>now </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="E1:37_2_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:37_2_1_1_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:37_2_1_1_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">s</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">sk</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>reconsider </b><font color="Maroon">sk</font> = <font color="Maroon">sk</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>, <font color="Maroon">q</font> = <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span> 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/><b>reconsider </b><font color="Maroon">pq'</font> = <font color="Maroon">p</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">q</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/><i><font color="Green">E138</font></i>: <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">k</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">pq'</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K8">Sum</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">sk</font>
<b>by </b><i>, , , , <a class="ref" href="matrlin.html#D8">MATRLIN:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">q</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">bn</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="txt" href="hilbasis.html#E3:37_2_1_1"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/></div><b>end;</b></div><a NAME="E23:37_2_1_1_1"/><b>then </b><i><font color="Green">E151</font></i>: 
 <a href="matrlin.html#K8">Sum</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/><b>set </b><font color="Maroon">t</font> =  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font>;<br/><a NAME="E24:37_2_1_1_1"/><i><font color="Green">E152</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Pxy</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#T34">POLYNOM1:34</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E26:37_2_1_1_1"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">PxPy</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> <font color="Maroon">u</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span> &amp; ( 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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">u</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font></span>)</span> ) ) )
 <b>by </b><i>, <a class="ref" href="polynom1.html#D26">POLYNOM1:def 26</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">i</font>, <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">b1</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1  st <font color="Olive">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n1</font> &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> &amp; <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">i</font> );<br/><div><i><font color="Green">E154</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_1_3"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">n1'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font>
 ;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/> <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>
;<br/><a NAME="E4:37_2_1_1_1_3"/><b>then </b><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">b1</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E6:37_2_1_1_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">b1n</font> = <font color="Maroon">b1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><a NAME="E8:37_2_1_1_1_3"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1"><i><font color="Green">E49</font></i></a>, </i>;<br/><a NAME="E9:37_2_1_1_1_3"/><b>then </b>
<font color="Maroon">b1n</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">bn</font>
 <b>by </b><i/>;<br/><a NAME="E10:37_2_1_1_1_3"/><b>then </b>
<font color="Maroon">b1n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:37_2_1_1_1_3"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> &amp; <font color="Maroon">b1n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="hilbasis.html#E10:37_2_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>set </b><font color="Maroon">n2</font> = <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>;<br/><a NAME="E14:37_2_1_1_1_3"/>
<font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:37_2_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/><a NAME="E15:37_2_1_1_1_3"/><b>then </b><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E16:37_2_1_1_1_3"/><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E17:37_2_1_1_1_3"/><b>then </b><i><font color="Green">E186</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>, <font color="Maroon">q</font> = <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> 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="E19:37_2_1_1_1_3"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E20:37_2_1_1_1_3"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 ;<br/><a NAME="E21:37_2_1_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>then </b><i><font color="Green">E187</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>
;<br/><a NAME="E23:37_2_1_1_1_3"/><b>then </b><i><font color="Green">E188</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:37_2_1_1_1"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="polynom1.html#T33">POLYNOM1:33</a></i>;<br/><a NAME="E24:37_2_1_1_1_3"/>
for <font color="Olive">i'</font>, <font color="Olive">n1'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">b1'</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1  st <font color="Olive">n1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; <font color="Olive">b1'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">n1'</font> &amp; <font color="Olive">i'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b1'</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> &amp; <font color="Olive">i'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive">b1'</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">i'</font> )
 <b>by </b><i>, , <a class="txt" href="hilbasis.html#E10:37_2_1_1_1"><i><font color="Green">E51</font></i></a>, , <a class="txt" href="hilbasis.html#E11:37_2_1_1_1"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E25:37_2_1_1_1_3"/>
 ex <font color="Olive">n2'</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">n2'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n1'</font>,<font color="Olive">n2'</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font>, <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span><b> such that </b><br/><a NAME="E29:37_2_1_1_1"/><i><font color="Green">E189</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="hilbasis.html#E3:37_2_1_1"><i><font color="Green">E47</font></i></a>);<br/></i><div><i><font color="Green">E190</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_1_4"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font>
 ;<br/><b>then consider </b><font color="Maroon">b1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1, <font color="Maroon">b2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1<b> such that </b><br/><a NAME="E3:37_2_1_1_1_4"/><i><font color="Green">E192</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">u</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">xb1n</font> = <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">yb2n</font> = <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> 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="E5:37_2_1_1_1_4"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">b1n</font> = <font color="Maroon">b1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font>, <font color="Maroon">b2n</font> = <font color="Maroon">b2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><i><font color="Green">E214</font></i>: <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">Px</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E9:37_2_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="hilbasis.html#E10:37_2_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xb1n</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1n</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">Py</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xb1n</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b1n</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">yb2n</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b2n</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E8:37_2_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">b1'</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 ;<br/><a NAME="E10:37_2_1_1_1_4"/><i><font color="Green">E215</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E11:37_2_1_1_1_4"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1'</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1'</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:37_2_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="hilbasis.html#E10:37_2_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/><a NAME="E12:37_2_1_1_1_4"/><b>then </b><i><font color="Green">E216</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1'</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">b1'</font> )
 <b>by </b><i><a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/><a NAME="E13:37_2_1_1_1_4"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:37_2_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="hilbasis.html#E11:37_2_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/><a NAME="E14:37_2_1_1_1_4"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E15:37_2_1_1_1_4"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E16:37_2_1_1_1_4"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E17:37_2_1_1_1_4"/><b>then </b>
<font color="Maroon">b1n</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">bn</font>
 <b>by </b><i/>;<br/><a NAME="E18:37_2_1_1_1_4"/><b>then </b>
<font color="Maroon">b1n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:37_2_1_1_1_4"/><i><font color="Green">E220</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> &amp; <font color="Maroon">b1n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E22:37_2_1_1_1_4"/><i><font color="Green">E221</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="hilbasis.html#E9:37_2_1_1_1"><i><font color="Green">E50</font></i></a>, , </i>;<br/><a NAME="E23:37_2_1_1_1_4"/><b>then </b><i><font color="Green">E222</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T33">POLYNOM1:33</a></i>;<br/><i><font color="Green">E223</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T55">BINARITH:55</a></i>
<br/>.= 
<font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font>

;<br/><a NAME="E25:37_2_1_1_1_4"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/><a NAME="E26:37_2_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green">E225</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="polynom1.html#D6">POLYNOM1:def 6</a></i>
;<br/><a NAME="E28:37_2_1_1_1_4"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>then reconsider </b><font color="Maroon">sb1n1</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> ;<br/><a NAME="E30:37_2_1_1_1_4"/>
( <font color="Maroon">xb1n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">yb2n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">sb1n1</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E31:37_2_1_1_1_4"/><b>then </b><i><font color="Green">E227</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">xb1n</font> <a href="polynom1.html#K29">*'</a> <font color="Maroon">yb2n</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">bn</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">sb1n1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sb1n1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span> &amp; ( 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="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">sb1n1</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">sb1n1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xb1n</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">yb2n</font> <a href="polynom1.html#K15">.</a> <font color="Olive">b2</font></span>)</span> ) ) )
 <b>by </b><i>, , <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green">E59</font></i></a></i>;<br/><b>consider </b><font color="Maroon">B1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">B2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E33:37_2_1_1_1_4"/><i><font color="Green">E230</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">B2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">sb1n1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">xb1n</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">B1</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">yb2n</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">B2</font></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green">E59</font></i></a>, , </i>;<br/><a NAME="E34:37_2_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon">B1'</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> ;<br/><a NAME="E36:37_2_1_1_1_4"/><i><font color="Green">E232</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/><a NAME="E37:37_2_1_1_1_4"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">B1</font>,<font color="Maroon">B2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">B1'</font>,<span class="p2">(<span class="default"><font color="Maroon">bn</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">B1'</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/><a NAME="E38:37_2_1_1_1_4"/><b>then </b><i><font color="Green">E233</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1'</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bn</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">B1'</font> )
 <b>by </b><i><a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>;<br/><a NAME="E39:37_2_1_1_1_4"/><b>then </b><i><font color="Green">E234</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E40:37_2_1_1_1_4"/><b>then </b><i><font color="Green">E235</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2n</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E41:37_2_1_1_1_4"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">sb1n1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>hence </b><a NAME="E42:37_2_1_1_1_4"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, , <a class="ref" href="hilbasis.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E31:37_2_1_1_1"/><i><font color="Green">E236</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E32:37_2_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E33:37_2_1_1_1"/><b>then </b><i><font color="Green">E237</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:37_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/><a NAME="E35:37_2_1_1_1"/><b>then </b>
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">s</font></span>)</span> <a href="wsierp_1.html#K3">.</a> 1
 <b>by </b><i><a class="ref" href="polynom3.html#T4">POLYNOM3:4</a></i>;<br/><a NAME="E36:37_2_1_1_1"/><b>then </b><i><font color="Green">E238</font></i>: 
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:37_2_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_5"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span>, <font color="Maroon">q'</font> = <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> 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="E2:37_2_1_1_1_5"/>
<font color="Maroon">s</font> <a href="matrlin.html#K1">/.</a> 1 is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/><a NAME="E3:37_2_1_1_1_5"/><b>then </b><i><font color="Green">E239</font></i>: 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1 is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:37_2_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:37_2_1_1_1_5"/>
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 ;<br/><a NAME="E5:37_2_1_1_1_5"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="hilbasis.html#E10:37_2_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="hilbasis.html#E11:37_2_1_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E6:37_2_1_1_1_5"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/></div><b>end;</b></div><a NAME="E38:37_2_1_1_1"/><b>then </b>
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E39:37_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#T30">POLYNOM1:30</a></i>;<br/><a NAME="E40:37_2_1_1_1"/><b>then </b>
 <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E41:37_2_1_1_1"/><b>then </b><i><font color="Green">E240</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="txt" href="hilbasis.html#E9:37_2_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n1'</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">n2'</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_1_1_1_6"/><i><font color="Green">E241</font></i>: 
( <font color="Maroon">n1'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">n2'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2'</font> )
 ;<br/><a NAME="E2:37_2_1_1_1_6"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E11:37_2_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1'</font>, <font color="Maroon">n2</font> = <font color="Maroon">n2'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><i><font color="Green">E242</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>
;<br/><a NAME="E5:37_2_1_1_1_6"/><b>then </b><i><font color="Green">E243</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span>
 <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">b1</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E7:37_2_1_1_1_6"/><i><font color="Green">E244</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span>
 <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">b2</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E9:37_2_1_1_1_6"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">b1n</font> = <font color="Maroon">b1</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font>, <font color="Maroon">b2n</font> = <font color="Maroon">b2</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><a NAME="E11:37_2_1_1_1_6"/>
<font color="Maroon">b1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:37_2_1_1_1_6"/><b>then </b>
<font color="Maroon">b1n</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">bn</font>
 <b>by </b><i/>;<br/><a NAME="E13:37_2_1_1_1_6"/><b>then </b>
<font color="Maroon">b1n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:37_2_1_1_1_6"/><i><font color="Green">E245</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> &amp; <font color="Maroon">b1n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E17:37_2_1_1_1_6"/>
<font color="Maroon">b2</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green">E57</font></i></a>, </i>;<br/><a NAME="E18:37_2_1_1_1_6"/><b>then </b>
<font color="Maroon">b2n</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">bn</font>
 <b>by </b><i/>;<br/><a NAME="E19:37_2_1_1_1_6"/><b>then </b>
<font color="Maroon">b2n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:37_2_1_1_1_6"/><i><font color="Green">E246</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> &amp; <font color="Maroon">b2n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">i2</font> = <font color="Maroon">i2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E23:37_2_1_1_1_6"/><i><font color="Green">E247</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1"><i><font color="Green">E48</font></i></a>, , </i>;<br/><a NAME="E24:37_2_1_1_1_6"/><i><font color="Green">E248</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E25:37_2_1_1_1_6"/>
( <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">s</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom3.html#T16">POLYNOM3:16</a></i>;<br/><a NAME="E26:37_2_1_1_1_6"/><b>then </b><i><font color="Green">E249</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, , <a class="ref" href="polynom3.html#T22">POLYNOM3:22</a></i>;<br/><a NAME="E27:37_2_1_1_1_6"/><i><font color="Green">E250</font></i>: 
( <font color="Maroon">b1</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">b2</font> is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>then </b><font color="Maroon">b1</font> = 
<font color="Maroon">b1n</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, , </i>
;<br/><b>hence </b><a NAME="E29:37_2_1_1_1_6"/>
<font color="Maroon">n1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2'</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><a NAME="E43:37_2_1_1_1"/><b>then </b><i><font color="Green">E251</font></i>: 
<font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E44:37_2_1_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n1'</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:37_2_1_1_1_7"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">n1'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:37_2_1_1_1_7"/><i><font color="Green">E253</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font> )
 <b>by </b><i>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>, <font color="Maroon">q'</font> = <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> 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/>
<b>reconsider </b><font color="Maroon">bj</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E7:37_2_1_1_1_7"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E8:37_2_1_1_1_7"/><b>then </b>
( <font color="Maroon">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E9:37_2_1_1_1_7"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>then </b><i><font color="Green">E255</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">bn</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>
;<br/>
<a NAME="E11:37_2_1_1_1_7"/><b>then </b><i><font color="Green">E256</font></i>: 
<font color="Maroon">bj</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E12:37_2_1_1_1_7"/><b>then </b>
<font color="Maroon">bj</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:37_2_1_1_1_7"/><b>then </b><i><font color="Green">E257</font></i>: 
<font color="Maroon">bj</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">bn</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">bij</font> = <font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_7_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_7_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1_1_7_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or  not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_7_2_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_7_2_1_1"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_7_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_7_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1_1_7_2_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> or  not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_7_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_7_2_1_1_1_1_1"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E261</font></i>: <span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">bj</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
<b>by </b><i/>
;<br/><a NAME="E3:37_2_1_1_1_7_2_1_1_1_1_1"/>
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bn</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E4:37_2_1_1_1_7_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, <a class="ref" href="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_7_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_7_2_1_1_1_1_2"/><i><font color="Green">E262</font></i>: 
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 = 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>
;<br/><a NAME="E3:37_2_1_1_1_7_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">n</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:37_2_1_1_1_7_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E263</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:37_2_1_1_1_7_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E264</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i/>;<br/><a NAME="E6:37_2_1_1_1_7_2_1_1_1_1_2"/><i><font color="Green">E265</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:37_2_1_1_1_7_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E8:37_2_1_1_1_7_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, , , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:37_2_1_1_1_7_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_7_2_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_7_2_1_2"/><i><font color="Green">E266</font></i>: 
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:37_2_1_1_1_7_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:37_2_1_1_1_7_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><b>hence </b><a NAME="E4:37_2_1_1_1_7_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E15:37_2_1_1_1_7"/><b>then </b>
<font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D13">POLYNOM1:def 13</a></i>;<br/>
<a NAME="E16:37_2_1_1_1_7"/><b>then </b>
<font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:37_2_1_1_1_7"/><i><font color="Green">E267</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E19:37_2_1_1_1_7"/><i><font color="Green">E268</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E11:37_2_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E269</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>
;<br/>
<a NAME="E21:37_2_1_1_1_7"/>
( <font color="Maroon">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bn</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7">|</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E22:37_2_1_1_1_7"/><b>then </b><i><font color="Green">E270</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, <a class="ref" href="hilbasis.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E23:37_2_1_1_1_7"/><i><font color="Green">E271</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E24:37_2_1_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E25:37_2_1_1_1_7"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">bj</font> <a href="hilbasis.html#K1" target="_self">bag_extend</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<b>hence </b><a NAME="E26:37_2_1_1_1_7"/>
<font color="Maroon">n1'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, , <a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E45:37_2_1_1_1"/><b>then </b><i><font color="Green">E272</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E46:37_2_1_1_1"/><i><font color="Green">E273</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E11:37_2_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green">E274</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">u</font> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="polynom1.html#T7">POLYNOM1:7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
;<br/><a NAME="E48:37_2_1_1_1"/><b>then </b><i><font color="Green">E275</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E49:37_2_1_1_1"/><b>then </b>
<font color="Maroon">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T83">FUNCT_2:83</a></i>;<br/><b>hence </b><a NAME="E50:37_2_1_1_1"/>
<font color="Maroon">Pxy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PxPy</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font>
 <b>by </b><i>, , <a class="txt" href="hilbasis.html#E4:37_2_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="rlvect_2.html#T8">RLVECT_2:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:37_2_1_1"/>
<span class="p1">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</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="hilbasis.html#K6" target="_self">upm</a> <font color="Maroon">n</font>,<font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


</div><b>end;</b></div>


</div>
